5}_{14})^{\widetilde{\omega}_{(1)}}&=\Sigma_{5}(W_{1})\bigoplus\Sigma_{5}(W_{11})\bigoplus\Sigma_{5}(W_{41})\bigoplus\Sigma_{5}(W_{51})\\\
&\quad\bigoplus\Sigma_{5}(W_{71},W_{81},W_{106},W_{111})\bigoplus\mathbb{V},\end{array}$

where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\Sigma_{5}(W_{1})=\langle\\{[W_{j}]:\,1\leq j\leq 10\\}\rangle,\ \
\Sigma_{5}(W_{11})=\langle\\{[W_{j}]:\,11\leq j\leq 40\\}\rangle,\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt&\Sigma_{5}(W_{41})=\langle\\{[W_{j}]:\,41\leq
j\leq 50\\}\rangle,\ \ \Sigma_{5}(W_{51})=\langle\\{[W_{j}]:\,51\leq j\leq
70\\}\rangle,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\Sigma_{5}(W_{71},W_{81},W_{106},W_{111})=\langle\\{[W_{j}]:\,71\leq
j\leq 115\\}\rangle,\\\ &\mathbb{V}=\langle\\{[W_{j}]:\,116\leq j\leq
130\\}\rangle.\end{array}$

Then, a direct computation shows that

  * (i)

$[\Sigma_{5}(W_{1})]^{\Sigma_{5}}=\langle[\widehat{p_{1}}]\rangle$ with
$\widehat{p_{1}}:=\sum_{1\leq j\leq 10}W_{j},$

  * (ii)

$[\Sigma_{5}(W_{11})]^{\Sigma_{5}}=\langle[\widehat{p_{2}}]\rangle$ with
$\widehat{p_{2}}:=\sum_{11\leq j\leq 40}W_{j},$

  * (iii)

$[\Sigma_{5}(W_{41})]^{\Sigma_{5}}=\langle[\widehat{p_{3}}]\rangle$ with
$\widehat{p_{3}}:=\sum_{41\leq j\leq 50}W_{j},$

  * (iv)

$[\Sigma_{5}(W_{51})]^{\Sigma_{5}}=\langle[\widehat{p_{4}}]\rangle$ with
$\widehat{p_{4}}:=\sum_{51\leq j\leq 70}W_{j},$

  * (v)

$[\Sigma_{5}(W_{71},W_{81},W_{106},W_{111})]^{\Sigma_{5}}=\langle[\widehat{p_{5}}]\rangle$
with $\widehat{p_{5}}:=W_{71}+W_{73}+W_{75}+W_{76}+W_{77}+\sum_{111\leq j\leq
115}W_{j},$

  * (vi)

$[\mathbb{V}]^{\Sigma_{5}}=\langle[\widehat{p_{6}}]\rangle$ with
$\widehat{p_{6}}:=\sum_{116\leq j\leq
119}W_{j}+W_{121}+W_{122}+W_{124}+W_{125}+\sum_{127\leq j\leq 130}W_{j}.$

Indeed, for simplicity, we prove the result (vi) in detail. The others can be
proved by the similar computations. An admissible monomial basis for
$\mathbb{V}$ is the set $\\{[W_{j}]:\,116\leq j\leq 130\\}.$ Suppose that
$[F]\in[\mathbb{V}]^{\Sigma_{5}},$ then

$F\equiv\sum_{116\leq j\leq 130}\gamma_{j}W_{j}$

with $\gamma_{j}\in\mathbb{Z}_{2}.$ Acting the homomorphisms
$\sigma_{d}:\mathbb{P}^{\otimes 5}_{14}\to\mathbb{P}^{\otimes 5}_{14}$ on both
sides of this equality and using the relations $\sigma_{d}(F)+F\equiv 0,$ for
$1\leq d\leq 4,$ we get

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\sigma_{1}(F)+F&\equiv(\gamma_{119}+\gamma_{121})W_{116}+\gamma_{120}W_{117}+\gamma_{123}W_{118}+(\gamma_{122}+\gamma_{130})(W_{122}+W_{130})\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{125}+\gamma_{127})(W_{125}+W_{127})+(\gamma_{126}+\gamma_{128}+\gamma_{130})W_{126}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{122}+\gamma_{126}+\gamma_{128})W_{128}\equiv 0,\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt\sigma_{2}(F)+F&\equiv(\gamma_{116}+\gamma_{119})(W_{116}+W_{119})+(\gamma_{117}+\gamma_{120}+\gamma_{129})(W_{117}+W_{120})\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{118}+\gamma_{122})(W_{118}+W_{122})+(\gamma_{123}+\gamma_{126})(W_{123}+W_{126})\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{124}+\gamma_{125}+\gamma_{126})W_{124}+(\gamma_{123}+\gamma_{124}+\gamma_{125})W_{125}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{128}+\gamma_{130})(W_{128}+W_{130})\equiv 0,\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt\sigma_{3}(F)+F&\equiv(\gamma_{117}+\gamma_{118}+\gamma_{123})W_{117}+(\gamma_{117}+\gamma_{118}+\gamma_{120})W_{118}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{119}+\gamma_{121})(W_{119}+W_{121})+(\gamma_{120}+\gamma_{123})(W_{120}+W_{123})\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{124}+\gamma_{129})(W_{124}+W_{129})+(\gamma_{122}+\gamma_{125}+\gamma_{126})(W_{125}+W_{126})\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{127}+\gamma_{128})(W_{127}+W_{128})\equiv 0,\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt\sigma_{4}(F)+F&\equiv(\gamma_{116}+\gamma_{117})(W_{116}+W_{117})+(\gamma_{119}+\gamma_{120}+\gamma_{129})W_{119}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{119}+\gamma_{120}+\gamma_{121})W_{120}+(\gamma_{121}+\gamma_{129})(W_{121}+W_{129})\\\
&\quad+\gamma_{123}W_{124}+\gamma_{126}W_{125}+(\gamma_{128}+\gamma_{130})W_{127}\equiv
0.\end{array}$

These equalities show that $\gamma_{120}=\gamma_{123}=\gamma_{126}=0$ and
$\gamma_{j}=\gamma_{121}$ for $j\neq 120,\,123,\,126.$ This confirms the above
statement. Now, let $G\in(\mathbb{P}^{\otimes
5}_{14})^{\widetilde{\omega}_{(1)}}$ such that $[G]\in[(Q^{\otimes
5}_{14})^{\widetilde{\omega}_{(1)}}]^{GL_{5}}.$ Because $\Sigma_{5}\subset
GL_{5},$ from the above calculations, we have

$G\equiv\beta_{1}\widehat{p_{1}}+\beta_{2}\widehat{p_{2}}+\beta_{3}\widehat{p_{3}}+\beta_{4}\widehat{p_{4}}+\beta_{5}\widehat{p_{5}}+\beta_{6}\widehat{p_{6}}$

in which $\beta_{j}\in\mathbb{Z}_{2}.$ Using the relation
$\sigma_{5}(G)+G\equiv 0,$ we get

$(\beta_{1}+\beta_{2})W_{4}+(\beta_{2}+\beta_{5})W_{14}+\beta_{2}W_{20}+(\beta_{2}+\beta_{3})W_{32}+(\beta_{4}+\beta_{6})W_{111}+\
\mbox{other terms}\equiv 0.$

This equality implies that $\beta_{4}=\beta_{6}$ and $\beta_{j}=0$ for $j\neq
4,\,6.$ This means that

$[(Q^{\otimes
5}_{14})^{\widetilde{\omega}_{(1)}}]^{GL_{5}}=\langle[\zeta:=\widehat{p_{4}}+\widehat{p_{6}}]\rangle.$

Now we prove the proposition for the case $j=2.$ According to Remark 2.5.2, an
admissible basis of $(Q^{\otimes 5}_{14})^{\widetilde{\omega}_{(2)}}$ is the
set $\\{[W_{j}]_{\widetilde{\omega}_{(2)}}:\,131\leq j\leq 145\\}.$ From this,
a direct computation shows that

$(Q^{\otimes 5}_{14})^{\widetilde{\omega}_{(2)}}=(Q^{\otimes
5}_{14})^{(\widetilde{\omega}_{(2)})^{>0}}\cong\Sigma_{5}(W_{131})\bigoplus\Sigma_{5}(W_{136}),$

where
$\Sigma_{5}(W_{131})=\langle\\{[W_{j}]_{\widetilde{\omega}_{(2)}}:\,131\leq
j\leq 135\\}\rangle$ and
$\Sigma_{5}(W_{136})=\langle\\{[W_{j}]_{\widetilde{\omega}_{(2)}}:\,136\leq
j\leq 145\\}\rangle$ and that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt[\Sigma_{5}(W_{131})]^{\Sigma_{5}}&=\langle[\sum_{131\leq j\leq
135}W_{j}]_{\widetilde{\omega}_{(2)}}\rangle\\\
\mbox{[}\Sigma_{5}(W_{136})\mbox{]}^{\Sigma_{5}}&=\langle[\sum_{136\leq j\leq
145}W_{j}]_{\widetilde{\omega}_{(2)}}\rangle.\end{array}$

Then if $H\in(\mathbb{P}^{\otimes 5}_{14})^{\widetilde{\omega}_{(2)}}$ such
that $[H]\in[(Q^{\otimes 5}_{14})^{\widetilde{\omega}_{(2)}}]^{GL_{5}},$ then

$H\equiv_{\widetilde{\omega}_{(2)}}\xi_{1}\sum_{131\leq j\leq
135}W_{j}+\xi_{2}\sum_{136\leq j\leq 145}W_{j}$

with $\xi_{1}$ and $\xi_{2}$ belong to $\mathbb{Z}_{2}.$ Straightforward
calculations indicate that

$\sigma_{5}(H)+H\equiv_{\widetilde{\omega}_{(2)}}(\beta_{1}+\beta_{2})W_{134}+\beta_{2}(W_{139}+W_{140}+W_{141})\equiv_{\widetilde{\omega}_{(2)}}0$

which implies $\xi_{1}=\xi_{2}=0.$ So $[(Q^{\otimes
5}_{14})^{\widetilde{\omega}_{(2)}}]^{GL_{5}}$ is trivial.

Now because $Q^{\otimes 5}_{14}\cong\bigoplus_{1\leq j\leq 3}(Q^{\otimes
5}_{14})^{\widetilde{\omega}_{(j)}}$ (see Remark 2.5.2) and
$[P_{\mathscr{A}_{2}}((\mathbb{P}_{14}^{\otimes
5})^{*})]_{GL_{5}}\cong[Q^{\otimes 5}_{14}]^{GL_{5}},$ by the above
calculations, we conclude that

$\dim[P_{\mathscr{A}_{2}}((\mathbb{P}_{14}^{\otimes
5})^{*})]_{GL_{5}}\leq\sum_{1\leq j\leq 3}\dim[(Q^{\otimes
5}_{14})^{\widetilde{\omega}_{(j)}}]^{GL_{5}}=1.$

The proof of the proposition is completed. ∎

Now following Lin [15] and Chen [7], it implies that ${\rm
Ext}_{\mathscr{A}_{2}}^{5,5+14}(\mathbb{Z}_{2},\mathbb{Z}_{2})=\langle
h_{0}d_{0}\rangle$ with $h_{0}d_{0}\neq 0.$ Further, $h_{0}\in{\rm
Im}(Tr_{1}^{\mathscr{A}_{2}})$ (see Singer [30]), $d_{0}\in{\rm
Im}(Tr_{4}^{\mathscr{A}_{2}})$ (see Hà [9]) and the ”total” transfer
$Tr^{\mathscr{A}_{2}}$ is an algebraic homomorphism. So, $h_{0}d_{0}\in{\rm
Im}(Tr_{5}^{\mathscr{A}_{2}}),$ that is,
$\dim[P_{\mathscr{A}_{2}}((\mathbb{P}_{14}^{\otimes 5})^{*})]_{GL_{5}}\geq 1.$
This and Proposition 2.5.3 show that
$[P_{\mathscr{A}_{2}}((\mathbb{P}_{14}^{\otimes 5})^{*})]_{GL_{5}}$ is one-
dimensional, which confirms the result for the case $n=14.$ ∎

###### Proof of Case n = 31.

Let us recall that Kameko’s squaring operation
$\widetilde{Sq^{0}_{*}}:Q^{\otimes 5}_{31}\longrightarrow Q^{\otimes 5}_{13}$
is an epimorphism of $\mathbb{Z}_{2}GL_{5}$-modules. So, $Q^{\otimes
5}_{31}\cong{\rm Ker}\widetilde{Sq^{0}_{*}}\bigoplus Q^{\otimes 5}_{13}.$ By
our previous result in [26] that the invariant space $[Q^{\otimes
5}_{13}]^{GL_{5}}$ is trivial, and we therefore deduce an estimate

$\dim[P_{\mathscr{A}_{2}}((\mathbb{P}_{31}^{\otimes
5})^{*})]_{GL_{5}}=\dim[Q^{\otimes 5}_{31}]^{GL_{5}}\leq\dim[{\rm
Ker}\widetilde{Sq^{0}_{*}}]^{GL_{5}}.$

On the other hand, the works by Lin [15] and Chen [7] show that ${\rm
Ext}_{\mathscr{A}_{2}}^{5,5+31}(\mathbb{Z}_{2},\mathbb{Z}_{2})=\langle
n_{0},h_{0}^{4}h_{5}\rangle.$ Moreover, the non-zero elements $h_{0}^{4}h_{5}$
and $n_{0}$ are detected by the rank 5 transfer (see Singer [30], Chơn-Hà
[8]). Hence, $\dim[P_{\mathscr{A}_{2}}((\mathbb{P}_{31}^{\otimes
5})^{*})]_{GL_{5}}\geq 2.$ Thus, to prove that
$[P_{\mathscr{A}_{2}}((\mathbb{P}_{31}^{\otimes 5})^{*})]_{GL_{5}}$ has
dimension $2,$ we will show that $\dim[{\rm
Ker}\widetilde{Sq^{0}_{*}}]^{GL_{5}}\leq 2.$ But this is immediate from
Theorem 1.6 and the following proposition.

######  Proposition 2.5.4.

The following holds:

$\dim[(Q^{\otimes
5}_{31})^{\omega_{(j)}}]^{GL_{5}}=\left\\{\begin{array}[]{ll}1&\mbox{if
$j=1,\,2$},\\\ 0&\mbox{if $j=3$}.\end{array}\right.$

###### Proof.

We prove the proposition for $j=1,\,3$ and leave the rest to the reader. First
of all, we consider the space $(Q^{\otimes 5}_{31})^{\omega_{(1)}}=(Q^{\otimes
5}_{31})^{\omega_{(1)}^{0}}\bigoplus(Q^{\otimes 5}_{31})^{\omega_{(1)}^{>0}},$
where as shown in the proofs of Theorem 1.6 and Proposition 2.2.1, the readers
can notice that $Q^{\otimes 5}_{31})^{\omega_{(1)}^{0}}$ is a
$\mathbb{Z}_{2}$-vector space of dimension $30$ with a basis consisting all
the classes represented by the following admissible monomials:

$L_{1}=x_{5}^{31}$, |  $L_{2}=x_{4}^{31}$, |  $L_{3}=x_{3}^{31}$, |  $L_{4}=x_{2}^{31}$, |  $L_{5}=x_{1}^{31}$,
---|---|---|---|---
$L_{6}=x_{4}x_{5}^{30}$, |  $L_{7}=x_{3}x_{5}^{30}$, |  $L_{8}=x_{3}x_{4}^{30}$, |  $L_{9}=x_{2}x_{5}^{30}$, |  $L_{10}=x_{2}x_{4}^{30}$,
$L_{11}=x_{2}x_{3}^{30}$, |  $L_{12}=x_{1}x_{5}^{30}$, |  $L_{13}=x_{1}x_{4}^{30}$, |  $L_{14}=x_{1}x_{3}^{30}$, |  $L_{15}=x_{1}x_{2}^{30}$,
$L_{16}=x_{3}x_{4}^{2}x_{5}^{28}$, |  $L_{17}=x_{2}x_{4}^{2}x_{5}^{28}$, |  $L_{18}=x_{2}x_{3}^{2}x_{5}^{28}$, |  $L_{19}=x_{2}x_{3}^{2}x_{4}^{28}$, |  $L_{20}=x_{1}x_{4}^{2}x_{5}^{28}$,
$L_{21}=x_{1}x_{3}^{2}x_{5}^{28}$, |  $L_{22}=x_{1}x_{3}^{2}x_{4}^{28}$, |  $L_{23}=x_{1}x_{2}^{2}x_{5}^{28}$, |  $L_{24}=x_{1}x_{2}^{2}x_{4}^{28}$, |  $L_{25}=x_{1}x_{2}^{2}x_{3}^{28}$,
$L_{26}=x_{2}x_{3}^{2}x_{4}^{4}x_{5}^{24}$, |  $L_{27}=x_{1}x_{3}^{2}x_{4}^{4}x_{5}^{24}$, |  $L_{28}=x_{1}x_{2}^{2}x_{4}^{4}x_{5}^{24}$, |  $L_{29}=x_{1}x_{2}^{2}x_{3}^{4}x_{5}^{24}$, | $L_{30}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{24}$

and that $Q^{\otimes 5}_{31})^{\omega_{(1)}^{>0}}$ is 1-dimensional with the
basis $\\{[L_{31}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{8}x_{5}^{16}]\\}.$ Thus, an
admissible monomial basis of $(Q^{\otimes 5}_{31})^{\omega_{(1)}}$ is the set
$\\{[L_{j}]_{\omega_{(1)}}=[L_{j}]:\,1\leq j\leq 31\\}.$ Then it is not
difficult to check that

$(Q^{\otimes
5}_{31})^{\omega_{(1)}}\cong\Sigma_{5}(L_{1})\bigoplus\Sigma_{5}(L_{6})\bigoplus\Sigma_{5}(L_{16})\bigoplus\Sigma_{5}(L_{26})\bigoplus\Sigma_{5}(L_{31}),$

where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\Sigma_{5}(L_{1})&=\langle\\{[L_{j}]:\,1\leq j\leq 5\\}\rangle,\ \
\Sigma_{5}(L_{6})=\langle\\{[L_{j}]:\,6\leq j\leq 15\\}\rangle\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\Sigma_{5}(L_{16})&=\langle\\{[L_{j}]:\,16\leq j\leq
25\\}\rangle,\ \ \Sigma_{5}(L_{26})=\langle\\{[L_{j}]:\,26\leq j\leq
30\\}\rangle,\\\ \Sigma_{5}(L_{31})&=\langle[L_{31}]\rangle.\end{array}$

Then, by using the homomorphisms $\sigma_{d}:\mathbb{P}^{\otimes
5}_{31}\to\mathbb{P}^{\otimes 5}_{31}$ for $1\leq d\leq 4$ and similar
computations as in the proof of Proposition 2.5.3, we obtain

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt[\Sigma_{5}(L_{1})]^{\Sigma_{5}}&=\langle[\sum_{1\leq j\leq
5}L_{j}]\rangle,\ \
\mbox{[}\Sigma_{5}(L_{6})\mbox{]}^{\Sigma_{5}}=\langle[\sum_{6\leq j\leq
15}L_{j}]\rangle,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\mbox{[}\Sigma_{5}(L_{16})\mbox{]}^{\Sigma_{5}}&=\langle[\sum_{16\leq
j\leq 25}L_{j}]\rangle,\ \
\mbox{[}\Sigma_{5}(L_{26})\mbox{]}^{\Sigma_{5}}=\langle[\sum_{26\leq j\leq
30}L_{j}]\rangle,\\\
\mbox{[}\Sigma_{5}(L_{31})\mbox{]}^{\Sigma_{5}}&=\langle[L_{31}]\rangle.\end{array}$

These show that if $[T]\in[(Q^{\otimes 5}_{31})^{\omega_{(1)}}]^{GL_{5}}$ then

$T\equiv\gamma_{1}\sum_{1\leq j\leq 5}L_{j}+\gamma_{2}\sum_{6\leq j\leq
15}L_{j}+\gamma_{3}\sum_{16\leq j\leq 25}L_{j}+\gamma_{4}\sum_{26\leq j\leq
30}L_{j}+\gamma_{5}L_{31}$

where $\gamma_{j}\in\mathbb{Z}_{2},\,1\leq j\leq 5.$ Acting the homomorphism
$\sigma_{5}:\mathbb{P}^{\otimes 5}_{31}\to\mathbb{P}^{\otimes 5}_{31}$ on both
sides of this equality and using the relation $\sigma_{5}(T)+T\equiv 0,$ we
get

$(\gamma_{1}+\gamma_{2})L_{4}+(\gamma_{2}+\gamma_{3})(L_{9}+L_{10}+L_{11})+(\gamma_{3}+\gamma_{4})(L_{17}+L_{18}+L_{19})+(\gamma_{4}+\gamma_{5})L_{26}\equiv
0$

and therefore $\gamma_{1}=\gamma_{2}=\gamma_{3}=\gamma_{4}=\gamma_{5},$ that
is $[(Q^{\otimes 5}_{31})^{\omega_{(1)}}]^{GL_{5}}=\langle[\sum_{1\leq j\leq
31}L_{j}]\rangle.$ Therefore $[(Q^{\otimes 5}_{31})^{\omega_{(1)}}]^{GL_{5}}$
has dimension 1.

Now, we prove the proposition for $j=3.$ Recall that the set
$\\{[Y_{j}]_{\omega_{(3)}}:\,1\leq j\leq 70\\}$ is an admissible monomial
basis of $(Q^{\otimes 5}_{31})^{\omega_{(3)}}=(Q^{\otimes
5}_{31})^{\omega_{(1)}^{>0}}.$ By a simple computation, we find that
$(Q^{\otimes
5}_{31})^{\omega_{(3)}}\cong\Sigma_{5}(Y_{1})\bigoplus\Sigma_{5}(Y_{21})\bigoplus\mathbb{U},$
where $\Sigma_{5}(Y_{1})=\langle\\{[Y_{j}]_{\omega_{(3)}}:\,1\leq j\leq
20\\}\rangle,$ $\Sigma_{5}(Y_{21})=\langle\\{[Y_{j}]_{\omega_{(3)}}:\,21\leq
j\leq 50\\}\rangle$ and $\mathbb{U}=\langle\\{[Y_{j}]_{\omega_{(3)}}:\,51\leq
j\leq 70\\}\rangle,$ and that
$[\Sigma_{5}(Y_{1})]^{\Sigma_{5}}=\langle[\sum_{1\leq j\leq
20}Y_{j}]_{\omega_{(3)}}\rangle,$
$[\Sigma_{5}(Y_{21})]^{\Sigma_{5}}=\langle[\sum_{21\leq j\leq
50}Y_{j}]_{\omega_{(3)}}\rangle$ and $[\mathbb{U}]^{\Sigma_{5}}=0.$ From these
calculations, we find that for any $W\in\mathbb{P}^{\otimes 5}_{31}$ such that
$[W]_{\omega_{(3)}}\in[(Q^{\otimes 5}_{31})^{\omega_{(3)}}]^{GL_{5}},$ then
$W\equiv_{\omega_{(3)}}\beta_{1}\sum_{1\leq j\leq
20}Y_{j}+\beta_{2}\sum_{21\leq j\leq 50}Y_{j}$ where $\beta_{1}$ and
$\beta_{2}$ belong to $\mathbb{Z}_{2}.$ Now, it is not too difficult to check
that

$\sigma_{5}(W)+W\equiv_{\omega_{(3)}}\beta_{1}(\sum_{1\leq j\leq
4}Y_{j})+\beta_{2}(Y_{8}+Y_{17}+Y_{18})+\mbox{other terms}\
\equiv_{\omega_{(3)}}0$

and so $\beta_{1}=\beta_{2}=0.$ This means that $[(Q^{\otimes
5}_{31})^{\omega_{(3)}}]^{GL_{5}}$ is trivial. The proposition follows. ∎

Thus, as argued above, we claim that the coinvariant
$[P_{\mathscr{A}_{2}}((\mathbb{P}_{31}^{\otimes 5})^{*})]_{GL_{5}}$ is
2-dimensional. This completes the proof of the case $n=31.$ ∎

###### Proof of Case n = 32.

It suffices to show that the spaces of $GL_{5}$-invariants $[(Q^{\otimes
5}_{32})^{\omega^{\prime}_{(j)}}]^{GL_{5}}$ are trivial, for any $j,$ $1\leq
j\leq 3.$ Indeed, we prove this statement for $j=1.$ The others can be
obtained by similar computations. Let us recall that from the proof of Theorem
1.12, we have $(Q^{\otimes 5}_{32})^{\omega^{\prime}_{(1)}}\cong(Q^{\otimes
5}_{32})^{(\omega^{\prime}_{(1)})^{0}}\bigoplus(Q^{\otimes
5}_{32})^{(\omega^{\prime}_{(1)})^{>0}},$ where $\dim(Q^{\otimes
5}_{32})^{(\omega^{\prime}_{(1)})^{0}}=115$ and $\dim(Q^{\otimes
5}_{32})^{(\omega^{\prime}_{(1)})^{>0}}=9.$ A simple computation shows that a
basis of $(Q^{\otimes 5}_{32})^{(\omega^{\prime}_{(1)})^{0}}$ is the set of
all classes represented by the admissible monomials $\widetilde{Z_{j}}$ for
$1\leq j\leq 115$ (see Subsect.3.6). We put
$A(\omega^{\prime}_{(1)})=\\{\widetilde{Z_{j}}:\,1\leq j\leq 115\\},$ then
from the calculations in Theorem 1.12, we see that the set
$[A(\omega^{\prime}_{(1)})\cup B(\omega^{\prime}_{(1)})\cup
C(\omega^{\prime}_{(1)})]_{\omega^{\prime}_{(1)}}=[A(\omega^{\prime}_{(1)})\cup
B(\omega^{\prime}_{(1)})\cup C(\omega^{\prime}_{(1)})]$ is an admissible basis
of $(Q^{\otimes 5}_{32})^{\omega^{\prime}_{(1)}}$ and therefore we have a
direct summand decomposition of the $\Sigma_{5}$-modules:

$(Q^{\otimes
5}_{32})^{\omega^{\prime}_{(1)}}=\Sigma_{5}(\widetilde{Z_{1}})\bigoplus\Sigma_{5}(\widetilde{Z_{21}})\bigoplus\Sigma_{5}(\widetilde{Z_{31}},\widetilde{Z_{51}})\bigoplus\Sigma_{5}(\widetilde{Z_{81}},\widetilde{Z_{101}})\bigoplus\mathbb{M},$

where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\Sigma_{5}(\widetilde{Z_{1}})=\langle\\{[\widetilde{Z_{j}}]:\,1\leq
j\leq 20\\}\rangle,\ \
\Sigma_{5}(\widetilde{Z_{21}})=\langle\\{[\widetilde{Z_{j}}]:\,21\leq j\leq
30\\}\rangle,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\Sigma_{5}(\widetilde{Z_{31}},\widetilde{Z_{51}})=\langle\\{[\widetilde{Z_{j}}]:\,31\leq
j\leq 80\\}\rangle,\ \
\Sigma_{5}(\widetilde{Z_{81}},\widetilde{Z_{101}})=\langle\\{[\widetilde{Z_{j}}]:\,81\leq
j\leq 115\\}\rangle,\\\ &\mathbb{M}=\langle\\{[Z_{j}]:\,1\leq j\leq
9\\}\rangle.\end{array}$

Then, based on the homomorphisms
$\sigma_{1},\ldots,\sigma_{4}:\mathbb{P}^{\otimes
5}_{32}\to\mathbb{P}^{\otimes 5}_{32}$ and direct calculations, it may be
concluded that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&[\Sigma_{5}(\widetilde{Z_{1}})]^{\Sigma_{5}}=\langle[\widehat{q_{1}}]\rangle,\
\mbox{with $\widehat{q_{1}}:=\sum_{1\leq j\leq 20}\widetilde{Z_{j}}$},\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\mbox{[}\Sigma_{5}(\widetilde{Z_{21}})\mbox{]}^{\Sigma_{5}}=\langle[\widehat{q_{2}}]\rangle,\
\mbox{with $\widehat{q_{2}}:=\sum_{21\leq j\leq 30}\widetilde{Z_{j}}$},\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\mbox{[}\Sigma_{5}(\widetilde{Z_{31}},\widetilde{Z_{51}})\mbox{]}^{\Sigma_{5}}=\langle[\widehat{q_{3}}]\rangle,\
\mbox{with
$\widehat{q_{3}}:=\widetilde{Z_{32}}+\widetilde{Z_{34}}+\widetilde{Z_{37}}+\widetilde{Z_{38}}+\widetilde{Z_{40}}+\widetilde{Z_{43}}+\widetilde{Z_{44}}+\sum_{48\leq
j\leq 80}\widetilde{Z_{j}}$},\\\
&\mbox{[}\Sigma_{5}(\widetilde{Z_{81}},\widetilde{Z_{101}})\mbox{]}^{\Sigma_{5}}=\langle[\widehat{q_{4}}+\widehat{q_{5}}],[\widehat{q_{5}}+\widehat{q_{6}}]\rangle,\
\mbox{where}\\\
&\qquad\widehat{q_{4}}:=\widetilde{Z_{82}}+\widetilde{Z_{83}}+\widetilde{Z_{85}}+\widetilde{Z_{86}}+\sum_{90\leq
j\leq 95}\widetilde{Z_{j}},\\\
&\qquad\widehat{q_{5}}:=\widetilde{Z_{81}}+\widetilde{Z_{84}}+\widetilde{Z_{87}}+\widetilde{Z_{88}}+\widetilde{Z_{89}},\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt&\qquad\widehat{q_{6}}:=\sum_{96\leq j\leq
115}\widetilde{Z_{j}},\\\
&\mbox{[}\mathbb{M}\mbox{]}^{\Sigma_{5}}=\langle[\widehat{q_{7}}]\rangle,\
\mbox{with $\widehat{q_{7}}:=\sum_{2\leq j\leq 9}Z_{j}$}.\end{array}$

Now let us consider $K\in\mathbb{P}^{\otimes 5}_{32}$ such that
$[K]\in[(Q^{\otimes 5}_{32})^{\omega^{\prime}_{(1)}}]^{GL_{5}}.$ Then, since
$\Sigma_{5}\subset GL_{5},$ by the above computations, we have
$K\equiv\sum_{1\leq i\leq
3}\xi_{i}\widehat{q_{i}}+\xi_{4}(\widehat{q_{4}}+\widehat{q_{5}})+\xi_{5}(\widehat{q_{5}}+\widehat{q_{6}})+\xi_{6}\widehat{q_{7}}$
where $\xi_{i}\in\mathbb{Z}_{2}.$ By direct computations using the relation
$\sigma_{5}(K)\equiv K,$ we obtain

$\xi_{1}\widetilde{Z_{7}}+(\xi_{1}+\xi_{3})\widetilde{Z_{10}}+(\xi_{1}+\xi_{2})\widetilde{Z_{16}}+\xi_{4}\widetilde{Z_{33}}+(\xi_{3}+\xi_{5})\widetilde{Z_{52}}+\xi_{6}Z_{8}+\
\mbox{other terms}\ \equiv 0,$

which leads to $\xi_{i}=0$ for all $i.$ This claims the result when $n=32,$
and therefore, the proof of Theorem 1.13 is completed. ∎

##  3 Appendix

In this section, we describe the following sets: $(\mathscr{C}^{\otimes
5}_{14})^{\widetilde{\omega}_{(j)}}$ for $1\leq j\leq 3,$
$(\mathscr{C}^{\otimes 5}_{31})^{\omega_{(j)}^{>0}}$ for $j=2,3$ and
$(\mathscr{C}^{\otimes 5}_{32})^{\omega^{\prime}_{(j)}}$ for $1\leq j\leq 3.$
At the same time, we provide an algorithm in MAGMA for verifying the
dimensions of $Q^{\otimes 5}_{31}$ and $Q^{\otimes 5}_{32}.$

###  3.1 Admissible monomials in $(\mathscr{C}^{\otimes
5}_{14})^{\widetilde{\omega}_{(1)}}$

$W_{1}=x_{4}^{7}x_{5}^{7}$, |  $W_{2}=x_{3}^{7}x_{5}^{7}$, |  $W_{3}=x_{3}^{7}x_{4}^{7}$, |  $W_{4}=x_{2}^{7}x_{5}^{7}$,
---|---|---|---
$W_{5}=x_{2}^{7}x_{4}^{7}$, |  $W_{6}=x_{2}^{7}x_{3}^{7}$, |  $W_{7}=x_{1}^{7}x_{5}^{7}$, |  $W_{8}=x_{1}^{7}x_{4}^{7}$,
$W_{9}=x_{1}^{7}x_{3}^{7}$, |  $W_{10}=x_{1}^{7}x_{2}^{7}$, |  $W_{11}=x_{3}x_{4}^{6}x_{5}^{7}$, |  $W_{12}=x_{3}x_{4}^{7}x_{5}^{6}$,
$W_{13}=x_{3}^{7}x_{4}x_{5}^{6}$, |  $W_{14}=x_{2}x_{4}^{6}x_{5}^{7}$, |  $W_{15}=x_{2}x_{4}^{7}x_{5}^{6}$, |  $W_{16}=x_{2}x_{3}^{6}x_{5}^{7}$,
$W_{17}=x_{2}x_{3}^{6}x_{4}^{7}$, |  $W_{18}=x_{2}x_{3}^{7}x_{5}^{6}$, |  $W_{19}=x_{2}x_{3}^{7}x_{4}^{6}$, |  $W_{20}=x_{2}^{7}x_{4}x_{5}^{6}$,
$W_{21}=x_{2}^{7}x_{3}x_{5}^{6}$, |  $W_{22}=x_{2}^{7}x_{3}x_{4}^{6}$, |  $W_{23}=x_{1}x_{4}^{6}x_{5}^{7}$, |  $W_{24}=x_{1}x_{4}^{7}x_{5}^{6}$,
$W_{25}=x_{1}x_{3}^{6}x_{5}^{7}$, |  $W_{26}=x_{1}x_{3}^{6}x_{4}^{7}$, |  $W_{27}=x_{1}x_{3}^{7}x_{5}^{6}$, |  $W_{28}=x_{1}x_{3}^{7}x_{4}^{6}$,
$W_{29}=x_{1}x_{2}^{6}x_{5}^{7}$, |  $W_{30}=x_{1}x_{2}^{6}x_{4}^{7}$, |  $W_{31}=x_{1}x_{2}^{6}x_{3}^{7}$, |  $W_{32}=x_{1}x_{2}^{7}x_{5}^{6}$,
$W_{33}=x_{1}x_{2}^{7}x_{4}^{6}$, |  $W_{34}=x_{1}x_{2}^{7}x_{3}^{6}$, |  $W_{35}=x_{1}^{7}x_{4}x_{5}^{6}$, |  $W_{36}=x_{1}^{7}x_{3}x_{5}^{6}$,
$W_{37}=x_{1}^{7}x_{3}x_{4}^{6}$, |  $W_{38}=x_{1}^{7}x_{2}x_{5}^{6}$, |  $W_{39}=x_{1}^{7}x_{2}x_{4}^{6}$, |  $W_{40}=x_{1}^{7}x_{2}x_{3}^{6}$,
$W_{41}=x_{3}^{3}x_{4}^{5}x_{5}^{6}$, |  $W_{42}=x_{2}^{3}x_{4}^{5}x_{5}^{6}$, |  $W_{43}=x_{2}^{3}x_{3}^{5}x_{5}^{6}$, |  $W_{44}=x_{2}^{3}x_{3}^{5}x_{4}^{6}$,
$W_{45}=x_{1}^{3}x_{4}^{5}x_{5}^{6}$, |  $W_{46}=x_{1}^{3}x_{3}^{5}x_{5}^{6}$, |  $W_{47}=x_{1}^{3}x_{3}^{5}x_{4}^{6}$, |  $W_{48}=x_{1}^{3}x_{2}^{5}x_{5}^{6}$,
$W_{49}=x_{1}^{3}x_{2}^{5}x_{4}^{6}$, |  $W_{50}=x_{1}^{3}x_{2}^{5}x_{3}^{6}$, |  $W_{51}=x_{2}x_{3}^{2}x_{4}^{4}x_{5}^{7}$, |  $W_{52}=x_{2}x_{3}^{2}x_{4}^{7}x_{5}^{4}$,
$W_{53}=x_{2}x_{3}^{7}x_{4}^{2}x_{5}^{4}$, |  $W_{54}=x_{2}^{7}x_{3}x_{4}^{2}x_{5}^{4}$, |  $W_{55}=x_{1}x_{3}^{2}x_{4}^{4}x_{5}^{7}$, |  $W_{56}=x_{1}x_{3}^{2}x_{4}^{7}x_{5}^{4}$,
$W_{57}=x_{1}x_{3}^{7}x_{4}^{2}x_{5}^{4}$, |  $W_{58}=x_{1}x_{2}^{2}x_{4}^{4}x_{5}^{7}$, |  $W_{59}=x_{1}x_{2}^{2}x_{4}^{7}x_{5}^{4}$, |  $W_{60}=x_{1}x_{2}^{2}x_{3}^{4}x_{5}^{7}$,
$W_{61}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{7}$, |  $W_{62}=x_{1}x_{2}^{2}x_{3}^{7}x_{5}^{4}$, |  $W_{63}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{4}$, |  $W_{64}=x_{1}x_{2}^{7}x_{4}^{2}x_{5}^{4}$,
$W_{65}=x_{1}x_{2}^{7}x_{3}^{2}x_{5}^{4}$, |  $W_{66}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{4}$, |  $W_{67}=x_{1}^{7}x_{3}x_{4}^{2}x_{5}^{4}$, |  $W_{68}=x_{1}^{7}x_{2}x_{4}^{2}x_{5}^{4}$,
$W_{69}=x_{1}^{7}x_{2}x_{3}^{2}x_{5}^{4}$, |  $W_{70}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{4}$, |  $W_{71}=x_{2}x_{3}x_{4}^{6}x_{5}^{6}$, |  $W_{72}=x_{2}x_{3}^{6}x_{4}x_{5}^{6}$,
$W_{73}=x_{1}x_{3}x_{4}^{6}x_{5}^{6}$, |  $W_{74}=x_{1}x_{3}^{6}x_{4}x_{5}^{6}$, |  $W_{75}=x_{1}x_{2}x_{4}^{6}x_{5}^{6}$, |  $W_{76}=x_{1}x_{2}x_{3}^{6}x_{5}^{6}$,
$W_{77}=x_{1}x_{2}x_{3}^{6}x_{4}^{6}$, |  $W_{78}=x_{1}x_{2}^{6}x_{4}x_{5}^{6}$, |  $W_{79}=x_{1}x_{2}^{6}x_{3}x_{5}^{6}$, |  $W_{80}=x_{1}x_{2}^{6}x_{3}x_{4}^{6}$,
$W_{81}=x_{2}x_{3}^{2}x_{4}^{5}x_{5}^{6}$, |  $W_{82}=x_{1}x_{3}^{2}x_{4}^{5}x_{5}^{6}$, |  $W_{83}=x_{1}x_{2}^{2}x_{4}^{5}x_{5}^{6}$, |  $W_{84}=x_{1}x_{2}^{2}x_{3}^{5}x_{5}^{6}$,
$W_{85}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{6}$, |  $W_{86}=x_{2}x_{3}^{3}x_{4}^{4}x_{5}^{6}$, |  $W_{87}=x_{2}x_{3}^{3}x_{4}^{6}x_{5}^{4}$, |  $W_{88}=x_{2}^{3}x_{3}x_{4}^{4}x_{5}^{6}$,
$W_{89}=x_{2}^{3}x_{3}x_{4}^{6}x_{5}^{4}$, |  $W_{90}=x_{1}x_{3}^{3}x_{4}^{4}x_{5}^{6}$, |  $W_{91}=x_{1}x_{3}^{3}x_{4}^{6}x_{5}^{4}$, |  $W_{92}=x_{1}x_{2}^{3}x_{4}^{4}x_{5}^{6}$,
$W_{93}=x_{1}x_{2}^{3}x_{4}^{6}x_{5}^{4}$, |  $W_{94}=x_{1}x_{2}^{3}x_{3}^{4}x_{5}^{6}$, |  $W_{95}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{6}$, |  $W_{96}=x_{1}x_{2}^{3}x_{3}^{6}x_{5}^{4}$,
$W_{97}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{4}$, |  $W_{98}=x_{1}^{3}x_{3}x_{4}^{4}x_{5}^{6}$, |  $W_{99}=x_{1}^{3}x_{3}x_{4}^{6}x_{5}^{4}$, |  $W_{100}=x_{1}^{3}x_{2}x_{4}^{4}x_{5}^{6}$,
$W_{101}=x_{1}^{3}x_{2}x_{4}^{6}x_{5}^{4}$, |  $W_{102}=x_{1}^{3}x_{2}x_{3}^{4}x_{5}^{6}$, |  $W_{103}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{6}$, |  $W_{104}=x_{1}^{3}x_{2}x_{3}^{6}x_{5}^{4}$,
$W_{105}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{4}$, |  $W_{106}=x_{2}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{4}$, |  $W_{107}=x_{1}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{4}$, |  $W_{108}=x_{1}^{3}x_{2}^{5}x_{4}^{2}x_{5}^{4}$,
$W_{109}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{5}^{4}$, |  $W_{110}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{4}$, |  $W_{111}=x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{4}$, |  $W_{112}=x_{1}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{4}$,
$W_{113}=x_{1}^{3}x_{2}^{3}x_{4}^{4}x_{5}^{4}$, |  $W_{114}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{5}^{4}$, |  $W_{115}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{4}$, |  $W_{116}=x_{1}x_{2}x_{3}^{2}x_{4}^{4}x_{5}^{6}$,
$W_{117}=x_{1}x_{2}x_{3}^{2}x_{4}^{6}x_{5}^{4}$, |  $W_{118}=x_{1}x_{2}x_{3}^{6}x_{4}^{2}x_{5}^{4}$, |  $W_{119}=x_{1}x_{2}^{2}x_{3}x_{4}^{4}x_{5}^{6}$, |  $W_{120}=x_{1}x_{2}^{2}x_{3}x_{4}^{6}x_{5}^{4}$,
$W_{121}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}x_{5}^{6}$, |  $W_{122}=x_{1}x_{2}^{6}x_{3}x_{4}^{2}x_{5}^{4}$, |  $W_{123}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{2}x_{5}^{4}$, |  $W_{124}=x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{4}x_{5}^{4}$,
$W_{125}=x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{4}x_{5}^{4}$, |  $W_{126}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{2}x_{5}^{4}$, |  $W_{127}=x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{4}x_{5}^{4}$, |  $W_{128}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{2}x_{5}^{4}$,
$W_{129}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{3}x_{5}^{4}$, |  $W_{130}=x_{1}^{3}x_{2}^{4}x_{3}x_{4}^{2}x_{5}^{4}$. |  |

###  3.2 Admissible monomials in $(\mathscr{C}^{\otimes
5}_{14})^{\widetilde{\omega}_{(2)}}$

$W_{131}=x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{7}x_{5}^{2}$, |  $W_{132}=x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{2}x_{5}^{7}$, |  $W_{133}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{2}x_{5}^{2}$, |  $W_{134}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{2}x_{5}^{2}$,
---|---|---|---
$W_{135}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{2}x_{5}^{2}$, |  $W_{136}=x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{3}x_{5}^{6}$, |  $W_{137}=x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{2}x_{5}^{6}$, |  $W_{138}=x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{6}x_{5}^{2}$,
$W_{139}=x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{2}x_{5}^{6}$, |  $W_{140}=x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{6}x_{5}^{2}$, |  $W_{141}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{2}x_{5}^{2}$, |  $W_{142}=x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{2}x_{5}^{6}$,
$W_{143}=x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{6}x_{5}^{2}$, |  $W_{144}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{2}x_{5}^{2}$, |  $W_{145}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{2}x_{5}^{2}$. |

###  3.3 Admissible monomials in $(\mathscr{C}^{\otimes
5}_{14})^{\widetilde{\omega}_{(3)}}$

$W_{146}=x_{2}x_{3}^{3}x_{4}^{3}x_{5}^{7}$, |  $W_{147}=x_{2}x_{3}^{3}x_{4}^{7}x_{5}^{3}$, |  $W_{148}=x_{2}x_{3}^{7}x_{4}^{3}x_{5}^{3}$, |  $W_{149}=x_{2}^{3}x_{3}x_{4}^{3}x_{5}^{7}$,
---|---|---|---
$W_{150}=x_{2}^{3}x_{3}x_{4}^{7}x_{5}^{3}$, |  $W_{151}=x_{2}^{3}x_{3}^{3}x_{4}x_{5}^{7}$, |  $W_{152}=x_{2}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{5}$, |  $W_{153}=x_{2}^{3}x_{3}^{3}x_{4}^{5}x_{5}^{3}$,
$W_{154}=x_{2}^{3}x_{3}^{3}x_{4}^{7}x_{5}$, |  $W_{155}=x_{2}^{3}x_{3}^{5}x_{4}^{3}x_{5}^{3}$, |  $W_{156}=x_{2}^{3}x_{3}^{7}x_{4}x_{5}^{3}$, |  $W_{157}=x_{2}^{3}x_{3}^{7}x_{4}^{3}x_{5}$,
$W_{158}=x_{2}^{7}x_{3}x_{4}^{3}x_{5}^{3}$, |  $W_{159}=x_{2}^{7}x_{3}^{3}x_{4}x_{5}^{3}$, |  $W_{160}=x_{2}^{7}x_{3}^{3}x_{4}^{3}x_{5}$, |  $W_{161}=x_{1}x_{3}^{3}x_{4}^{3}x_{5}^{7}$,
$W_{162}=x_{1}x_{3}^{3}x_{4}^{7}x_{5}^{3}$, |  $W_{163}=x_{1}x_{3}^{7}x_{4}^{3}x_{5}^{3}$, |  $W_{164}=x_{1}x_{2}^{3}x_{4}^{3}x_{5}^{7}$, |  $W_{165}=x_{1}x_{2}^{3}x_{4}^{7}x_{5}^{3}$,
$W_{166}=x_{1}x_{2}^{3}x_{3}^{3}x_{5}^{7}$, |  $W_{167}=x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{7}$, |  $W_{168}=x_{1}x_{2}^{3}x_{3}^{7}x_{5}^{3}$, |  $W_{169}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{3}$,
$W_{170}=x_{1}x_{2}^{7}x_{4}^{3}x_{5}^{3}$, |  $W_{171}=x_{1}x_{2}^{7}x_{3}^{3}x_{5}^{3}$, |  $W_{172}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{3}$, |  $W_{173}=x_{1}^{3}x_{3}x_{4}^{3}x_{5}^{7}$,
$W_{174}=x_{1}^{3}x_{3}x_{4}^{7}x_{5}^{3}$, |  $W_{175}=x_{1}^{3}x_{3}^{3}x_{4}x_{5}^{7}$, |  $W_{176}=x_{1}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{5}$, |  $W_{177}=x_{1}^{3}x_{3}^{3}x_{4}^{5}x_{5}^{3}$,
$W_{178}=x_{1}^{3}x_{3}^{3}x_{4}^{7}x_{5}$, |  $W_{179}=x_{1}^{3}x_{3}^{5}x_{4}^{3}x_{5}^{3}$, |  $W_{180}=x_{1}^{3}x_{3}^{7}x_{4}x_{5}^{3}$, |  $W_{181}=x_{1}^{3}x_{3}^{7}x_{4}^{3}x_{5}$,
$W_{182}=x_{1}^{3}x_{2}x_{4}^{3}x_{5}^{7}$, |  $W_{183}=x_{1}^{3}x_{2}x_{4}^{7}x_{5}^{3}$, |  $W_{184}=x_{1}^{3}x_{2}x_{3}^{3}x_{5}^{7}$, |  $W_{185}=x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{7}$,
$W_{186}=x_{1}^{3}x_{2}x_{3}^{7}x_{5}^{3}$, |  $W_{187}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{3}$, |  $W_{188}=x_{1}^{3}x_{2}^{3}x_{4}x_{5}^{7}$, |  $W_{189}=x_{1}^{3}x_{2}^{3}x_{4}^{3}x_{5}^{5}$,
$W_{190}=x_{1}^{3}x_{2}^{3}x_{4}^{5}x_{5}^{3}$, |  $W_{191}=x_{1}^{3}x_{2}^{3}x_{4}^{7}x_{5}$, |  $W_{192}=x_{1}^{3}x_{2}^{3}x_{3}x_{5}^{7}$, |  $W_{193}=x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{7}$,
$W_{194}=x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{5}^{5}$, |  $W_{195}=x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{5}$, |  $W_{196}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{5}^{3}$, |  $W_{197}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{3}$,
$W_{198}=x_{1}^{3}x_{2}^{3}x_{3}^{7}x_{5}$, |  $W_{199}=x_{1}^{3}x_{2}^{3}x_{3}^{7}x_{4}$, |  $W_{200}=x_{1}^{3}x_{2}^{5}x_{4}^{3}x_{5}^{3}$, |  $W_{201}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{5}^{3}$,
$W_{202}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{3}$, |  $W_{203}=x_{1}^{3}x_{2}^{7}x_{4}x_{5}^{3}$, |  $W_{204}=x_{1}^{3}x_{2}^{7}x_{4}^{3}x_{5}$, |  $W_{205}=x_{1}^{3}x_{2}^{7}x_{3}x_{5}^{3}$,
$W_{206}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{3}$, |  $W_{207}=x_{1}^{3}x_{2}^{7}x_{3}^{3}x_{5}$, |  $W_{208}=x_{1}^{3}x_{2}^{7}x_{3}^{3}x_{4}$, |  $W_{209}=x_{1}^{7}x_{3}x_{4}^{3}x_{5}^{3}$,
$W_{210}=x_{1}^{7}x_{3}^{3}x_{4}x_{5}^{3}$, |  $W_{211}=x_{1}^{7}x_{3}^{3}x_{4}^{3}x_{5}$, |  $W_{212}=x_{1}^{7}x_{2}x_{4}^{3}x_{5}^{3}$, |  $W_{213}=x_{1}^{7}x_{2}x_{3}^{3}x_{5}^{3}$,
$W_{214}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{3}$, |  $W_{215}=x_{1}^{7}x_{2}^{3}x_{4}x_{5}^{3}$, |  $W_{216}=x_{1}^{7}x_{2}^{3}x_{4}^{3}x_{5}$, |  $W_{217}=x_{1}^{7}x_{2}^{3}x_{3}x_{5}^{3}$,
$W_{218}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{3}$, |  $W_{219}=x_{1}^{7}x_{2}^{3}x_{3}^{3}x_{5}$, |  $W_{220}=x_{1}^{7}x_{2}^{3}x_{3}^{3}x_{4}$, |  $W_{221}=x_{1}x_{2}x_{3}^{2}x_{4}^{3}x_{5}^{7}$,
$W_{222}=x_{1}x_{2}x_{3}^{2}x_{4}^{7}x_{5}^{3}$, |  $W_{223}=x_{1}x_{2}x_{3}^{3}x_{4}^{2}x_{5}^{7}$, |  $W_{224}=x_{1}x_{2}x_{3}^{3}x_{4}^{3}x_{5}^{6}$, |  $W_{225}=x_{1}x_{2}x_{3}^{3}x_{4}^{6}x_{5}^{3}$,
$W_{226}=x_{1}x_{2}x_{3}^{3}x_{4}^{7}x_{5}^{2}$, |  $W_{227}=x_{1}x_{2}x_{3}^{6}x_{4}^{3}x_{5}^{3}$, |  $W_{228}=x_{1}x_{2}x_{3}^{7}x_{4}^{2}x_{5}^{3}$, |  $W_{229}=x_{1}x_{2}x_{3}^{7}x_{4}^{3}x_{5}^{2}$,
$W_{230}=x_{1}x_{2}^{2}x_{3}x_{4}^{3}x_{5}^{7}$, |  $W_{231}=x_{1}x_{2}^{2}x_{3}x_{4}^{7}x_{5}^{3}$, |  $W_{232}=x_{1}x_{2}^{2}x_{3}^{3}x_{4}x_{5}^{7}$, |  $W_{233}=x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{3}x_{5}^{5}$,
$W_{234}=x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{5}x_{5}^{3}$, |  $W_{235}=x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{7}x_{5}$, |  $W_{236}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{3}x_{5}^{3}$, |  $W_{237}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}x_{5}^{3}$,
$W_{238}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{3}x_{5}$, |  $W_{239}=x_{1}x_{2}^{3}x_{3}x_{4}^{2}x_{5}^{7}$, |  $W_{240}=x_{1}x_{2}^{3}x_{3}x_{4}^{3}x_{5}^{6}$, |  $W_{241}=x_{1}x_{2}^{3}x_{3}x_{4}^{6}x_{5}^{3}$,
$W_{242}=x_{1}x_{2}^{3}x_{3}x_{4}^{7}x_{5}^{2}$, |  $W_{243}=x_{1}x_{2}^{3}x_{3}^{2}x_{4}x_{5}^{7}$, |  $W_{244}=x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{3}x_{5}^{5}$, |  $W_{245}=x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{5}x_{5}^{3}$,
$W_{246}=x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{7}x_{5}$, |  $W_{247}=x_{1}x_{2}^{3}x_{3}^{3}x_{4}x_{5}^{6}$, |  $W_{248}=x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{2}x_{5}^{5}$, |  $W_{249}=x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{4}$,
$W_{250}=x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{3}$, |  $W_{251}=x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{5}x_{5}^{2}$, |  $W_{252}=x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{6}x_{5}$, |  $W_{253}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{3}x_{5}^{3}$,
$W_{254}=x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{3}$, |  $W_{255}=x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{3}x_{5}^{2}$, |  $W_{256}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}x_{5}^{3}$, |  $W_{257}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{3}x_{5}$,
$W_{258}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}x_{5}^{2}$, |  $W_{259}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{2}x_{5}$, |  $W_{260}=x_{1}x_{2}^{6}x_{3}x_{4}^{3}x_{5}^{3}$, |  $W_{261}=x_{1}x_{2}^{6}x_{3}^{3}x_{4}x_{5}^{3}$,
$W_{262}=x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{3}x_{5}$, |  $W_{263}=x_{1}x_{2}^{7}x_{3}x_{4}^{2}x_{5}^{3}$, |  $W_{264}=x_{1}x_{2}^{7}x_{3}x_{4}^{3}x_{5}^{2}$, |  $W_{265}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}x_{5}^{3}$,
$W_{266}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{3}x_{5}$, |  $W_{267}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}x_{5}^{2}$, |  $W_{268}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{2}x_{5}$, |  $W_{269}=x_{1}^{3}x_{2}x_{3}x_{4}^{2}x_{5}^{7}$,
$W_{270}=x_{1}^{3}x_{2}x_{3}x_{4}^{3}x_{5}^{6}$, |  $W_{271}=x_{1}^{3}x_{2}x_{3}x_{4}^{6}x_{5}^{3}$, |  $W_{272}=x_{1}^{3}x_{2}x_{3}x_{4}^{7}x_{5}^{2}$, |  $W_{273}=x_{1}^{3}x_{2}x_{3}^{2}x_{4}x_{5}^{7}$,
$W_{274}=x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{3}x_{5}^{5}$, |  $W_{275}=x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{5}x_{5}^{3}$, |  $W_{276}=x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{7}x_{5}$, |  $W_{277}=x_{1}^{3}x_{2}x_{3}^{3}x_{4}x_{5}^{6}$,
$W_{278}=x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{2}x_{5}^{5}$, |  $W_{279}=x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{3}x_{5}^{4}$, |  $W_{280}=x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{4}x_{5}^{3}$, |  $W_{281}=x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{5}x_{5}^{2}$,
$W_{282}=x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{6}x_{5}$, |  $W_{283}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{3}x_{5}^{3}$, |  $W_{284}=x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{2}x_{5}^{3}$, |  $W_{285}=x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{3}x_{5}^{2}$,
$W_{286}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}x_{5}^{3}$, |  $W_{287}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{3}x_{5}$, |  $W_{288}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}x_{5}^{2}$, |  $W_{289}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{2}x_{5}$,
$W_{290}=x_{1}^{3}x_{2}^{3}x_{3}x_{4}x_{5}^{6}$, |  $W_{291}=x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{2}x_{5}^{5}$, |  $W_{292}=x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{3}x_{5}^{4}$, |  $W_{293}=x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{4}x_{5}^{3}$,
$W_{294}=x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{5}x_{5}^{2}$, |  $W_{295}=x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{6}x_{5}$, |  $W_{296}=x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}x_{5}^{4}$, |  $W_{297}=x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}$,
$W_{298}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}x_{5}^{3}$, |  $W_{299}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{3}x_{5}$, |  $W_{300}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}x_{5}^{2}$, |  $W_{301}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{2}x_{5}$,
$W_{302}=x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{2}x_{5}^{3}$, |  $W_{303}=x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{3}x_{5}^{2}$, |  $W_{304}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}x_{5}^{3}$, |  $W_{305}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{3}x_{5}$,

$W_{306}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}x_{5}^{2}$, |  $W_{307}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{2}x_{5}$, |  $W_{308}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}x_{5}^{2}$, |  $W_{309}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{2}x_{5}$,
---|---|---|---
$W_{310}=x_{1}^{7}x_{2}x_{3}x_{4}^{2}x_{5}^{3}$, |  $W_{311}=x_{1}^{7}x_{2}x_{3}x_{4}^{3}x_{5}^{2}$, |  $W_{312}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}x_{5}^{3}$, |  $W_{313}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{3}x_{5}$,
$W_{314}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}x_{5}^{2}$, |  $W_{315}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{2}x_{5}$, |  $W_{316}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}x_{5}^{2}$, |  $W_{317}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{2}x_{5}$,
$W_{318}=x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}x_{5}^{3}$, |  $W_{319}=x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}^{3}x_{5}$, |  $W_{320}=x_{1}^{3}x_{2}^{4}x_{3}x_{4}^{3}x_{5}^{3}$. |

###  3.4 Admissible monomials in $(\mathscr{C}^{\otimes
5}_{31})^{\omega_{(2)}^{>0}}$

$X_{1}=x_{1}x_{2}^{2}x_{3}x_{4}^{12}x_{5}^{15}$, |  $X_{2}=x_{1}x_{2}^{2}x_{3}x_{4}^{15}x_{5}^{12}$, |  $X_{3}=x_{1}x_{2}^{2}x_{3}^{15}x_{4}x_{5}^{12}$, |  $X_{4}=x_{1}x_{2}^{2}x_{3}x_{4}^{13}x_{5}^{14}$,
---|---|---|---
$X_{5}=x_{1}x_{2}^{2}x_{3}^{13}x_{4}x_{5}^{14}$, |  $X_{6}=x_{1}x_{2}^{2}x_{3}^{13}x_{4}^{14}x_{5}$, |  $X_{7}=x_{1}x_{2}^{14}x_{3}x_{4}x_{5}^{14}$, |  $X_{8}=x_{1}x_{2}^{14}x_{3}x_{4}^{14}x_{5}$,
$X_{9}=x_{1}x_{2}x_{3}^{14}x_{4}x_{5}^{14}$, |  $X_{10}=x_{1}x_{2}x_{3}^{14}x_{4}^{14}x_{5}$, |  $X_{11}=x_{1}x_{2}x_{3}^{2}x_{4}^{12}x_{5}^{15}$, |  $X_{12}=x_{1}x_{2}x_{3}^{2}x_{4}^{15}x_{5}^{12}$,
$X_{13}=x_{1}x_{2}x_{3}^{2}x_{4}^{13}x_{5}^{14}$, |  $X_{14}=x_{1}x_{2}^{3}x_{3}^{14}x_{4}x_{5}^{12}$, |  $X_{15}=x_{1}x_{2}^{3}x_{3}^{12}x_{4}x_{5}^{14}$, |  $X_{16}=x_{1}x_{2}^{3}x_{3}^{12}x_{4}^{14}x_{5}$,
$X_{17}=x_{1}x_{2}^{15}x_{3}^{2}x_{4}x_{5}^{12}$, |  $X_{18}=x_{1}x_{2}x_{3}x_{4}^{14}x_{5}^{14}$, |  $X_{19}=x_{1}x_{2}x_{3}^{3}x_{4}^{14}x_{5}^{12}$, |  $X_{20}=x_{1}x_{2}x_{3}^{15}x_{4}^{2}x_{5}^{12}$,
$X_{21}=x_{1}x_{2}x_{3}^{3}x_{4}^{12}x_{5}^{14}$, |  $X_{22}=x_{1}x_{2}^{3}x_{3}x_{4}^{14}x_{5}^{12}$, |  $X_{23}=x_{1}x_{2}^{3}x_{3}x_{4}^{12}x_{5}^{14}$, |  $X_{24}=x_{1}x_{2}^{3}x_{3}^{13}x_{4}^{2}x_{5}^{12}$,
$X_{25}=x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{10}x_{5}^{12}$, |  $X_{26}=x_{1}x_{2}^{15}x_{3}x_{4}^{2}x_{5}^{12}$, |  $X_{27}=x_{1}^{3}x_{2}x_{3}^{14}x_{4}x_{5}^{12}$, |  $X_{28}=x_{1}^{3}x_{2}x_{3}^{12}x_{4}x_{5}^{14}$,
$X_{29}=x_{1}^{3}x_{2}x_{3}^{12}x_{4}^{14}x_{5}$, |  $X_{30}=x_{1}^{3}x_{2}x_{3}x_{4}^{14}x_{5}^{12}$, |  $X_{31}=x_{1}^{3}x_{2}x_{3}x_{4}^{12}x_{5}^{14}$, |  $X_{32}=x_{1}^{3}x_{2}x_{3}^{13}x_{4}^{2}x_{5}^{12}$,
$X_{33}=x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{10}x_{5}^{12}$, |  $X_{34}=x_{1}^{3}x_{2}^{13}x_{3}x_{4}^{2}x_{5}^{12}$, |  $X_{35}=x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{10}x_{5}^{12}$, |  $X_{36}=x_{1}^{3}x_{2}^{13}x_{3}^{2}x_{4}x_{5}^{12}$,
$X_{37}=x_{1}^{3}x_{2}^{5}x_{3}^{10}x_{4}x_{5}^{12}$, |  $X_{38}=x_{1}^{15}x_{2}x_{3}^{2}x_{4}x_{5}^{12}$, |  $X_{39}=x_{1}^{15}x_{2}x_{3}x_{4}^{2}x_{5}^{12}$, |  $X_{40}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{9}x_{5}^{15}$,
$X_{41}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{15}x_{5}^{9}$, |  $X_{42}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{8}x_{5}^{15}$, |  $X_{43}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{15}x_{5}^{8}$, |  $X_{44}=x_{1}x_{2}^{2}x_{3}^{12}x_{4}x_{5}^{15}$,
$X_{45}=x_{1}x_{2}^{2}x_{3}^{12}x_{4}^{15}x_{5}$, |  $X_{46}=x_{1}x_{2}^{2}x_{3}^{15}x_{4}^{4}x_{5}^{9}$, |  $X_{47}=x_{1}x_{2}^{2}x_{3}^{15}x_{4}^{5}x_{5}^{8}$, |  $X_{48}=x_{1}x_{2}^{2}x_{3}^{15}x_{4}^{12}x_{5}$,
$X_{49}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{8}x_{5}^{15}$, |  $X_{50}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{15}x_{5}^{8}$, |  $X_{51}=x_{1}x_{2}^{3}x_{3}^{14}x_{4}^{12}x_{5}$, |  $X_{52}=x_{1}x_{2}^{3}x_{3}^{15}x_{4}^{4}x_{5}^{8}$,
$X_{53}=x_{1}x_{2}^{15}x_{3}^{2}x_{4}^{4}x_{5}^{9}$, |  $X_{54}=x_{1}x_{2}^{15}x_{3}^{2}x_{4}^{5}x_{5}^{8}$, |  $X_{55}=x_{1}x_{2}^{15}x_{3}^{2}x_{4}^{12}x_{5}$, |  $X_{56}=x_{1}x_{2}^{15}x_{3}^{3}x_{4}^{4}x_{5}^{8}$,
$X_{57}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{8}x_{5}^{15}$, |  $X_{58}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{15}x_{5}^{8}$, |  $X_{59}=x_{1}^{3}x_{2}x_{3}^{14}x_{4}^{12}x_{5}$, |  $X_{60}=x_{1}^{3}x_{2}x_{3}^{15}x_{4}^{4}x_{5}^{8}$,
$X_{61}=x_{1}^{3}x_{2}^{5}x_{3}^{10}x_{4}^{12}x_{5}$, |  $X_{62}=x_{1}^{3}x_{2}^{13}x_{3}^{2}x_{4}^{12}x_{5}$, |  $X_{63}=x_{1}^{3}x_{2}^{15}x_{3}x_{4}^{4}x_{5}^{8}$, |  $X_{64}=x_{1}^{15}x_{2}x_{3}^{2}x_{4}^{4}x_{5}^{9}$,
$X_{65}=x_{1}^{15}x_{2}x_{3}^{2}x_{4}^{5}x_{5}^{8}$, |  $X_{66}=x_{1}^{15}x_{2}x_{3}^{2}x_{4}^{12}x_{5}$, |  $X_{67}=x_{1}^{15}x_{2}x_{3}^{3}x_{4}^{4}x_{5}^{8}$, |  $X_{68}=x_{1}^{15}x_{2}^{3}x_{3}x_{4}^{4}x_{5}^{8}$,
$X_{69}=x_{1}x_{2}^{2}x_{3}x_{4}^{14}x_{5}^{13}$, |  $X_{70}=x_{1}x_{2}^{2}x_{3}^{13}x_{4}^{2}x_{5}^{13}$, |  $X_{71}=x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{12}x_{5}^{13}$, |  $X_{72}=x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{13}x_{5}^{12}$,
$X_{73}=x_{1}x_{2}^{2}x_{3}^{13}x_{4}^{3}x_{5}^{12}$, |  $X_{74}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{11}x_{5}^{13}$, |  $X_{75}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{10}x_{5}^{13}$, |  $X_{76}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{11}x_{5}^{12}$,
$X_{77}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{9}x_{5}^{12}$, |  $X_{78}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{8}x_{5}^{13}$, |  $X_{79}=x_{1}x_{2}^{6}x_{3}x_{4}^{10}x_{5}^{13}$, |  $X_{80}=x_{1}x_{2}^{6}x_{3}x_{4}^{11}x_{5}^{12}$,
$X_{81}=x_{1}x_{2}^{6}x_{3}^{11}x_{4}x_{5}^{12}$, |  $X_{82}=x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{9}x_{5}^{12}$, |  $X_{83}=x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{8}x_{5}^{13}$, |  $X_{84}=x_{1}x_{2}^{6}x_{3}^{7}x_{4}^{8}x_{5}^{9}$,
$X_{85}=x_{1}x_{2}^{6}x_{3}^{7}x_{4}^{9}x_{5}^{8}$, |  $X_{86}=x_{1}x_{2}^{14}x_{3}x_{4}^{2}x_{5}^{13}$, |  $X_{87}=x_{1}x_{2}^{14}x_{3}x_{4}^{3}x_{5}^{12}$, |  $X_{88}=x_{1}x_{2}^{14}x_{3}^{3}x_{4}x_{5}^{12}$,
$X_{89}=x_{1}x_{2}x_{3}^{14}x_{4}^{2}x_{5}^{13}$, |  $X_{90}=x_{1}x_{2}x_{3}^{14}x_{4}^{3}x_{5}^{12}$, |  $X_{91}=x_{1}x_{2}x_{3}^{2}x_{4}^{14}x_{5}^{13}$, |  $X_{92}=x_{1}x_{2}x_{3}^{6}x_{4}^{10}x_{5}^{13}$,
$X_{93}=x_{1}x_{2}x_{3}^{6}x_{4}^{11}x_{5}^{12}$, |  $X_{94}=x_{1}x_{2}^{3}x_{3}^{12}x_{4}^{2}x_{5}^{13}$, |  $X_{95}=x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{12}x_{5}^{13}$, |  $X_{96}=x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{13}x_{5}^{12}$,
$X_{97}=x_{1}x_{2}^{3}x_{3}^{12}x_{4}^{3}x_{5}^{12}$, |  $X_{98}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{10}x_{5}^{13}$, |  $X_{99}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{11}x_{5}^{12}$, |  $X_{100}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{9}x_{5}^{12}$,
$X_{101}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{8}x_{5}^{13}$, |  $X_{102}=x_{1}x_{2}^{7}x_{3}^{10}x_{4}x_{5}^{12}$, |  $X_{103}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{9}x_{5}^{12}$, |  $X_{104}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{8}x_{5}^{13}$,
$X_{105}=x_{1}x_{2}^{7}x_{3}^{6}x_{4}^{8}x_{5}^{9}$, |  $X_{106}=x_{1}x_{2}^{7}x_{3}^{6}x_{4}^{9}x_{5}^{8}$, |  $X_{107}=x_{1}x_{2}x_{3}^{7}x_{4}^{10}x_{5}^{12}$, |  $X_{108}=x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{12}x_{5}^{12}$,
$X_{109}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{8}x_{5}^{12}$, |  $X_{110}=x_{1}x_{2}^{7}x_{3}x_{4}^{10}x_{5}^{12}$, |  $X_{111}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{8}x_{5}^{12}$, |  $X_{112}=x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{8}x_{5}^{8}$,
$X_{113}=x_{1}^{3}x_{2}x_{3}^{12}x_{4}^{2}x_{5}^{13}$, |  $X_{114}=x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{12}x_{5}^{13}$, |  $X_{115}=x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{13}x_{5}^{12}$, |  $X_{116}=x_{1}^{3}x_{2}x_{3}^{12}x_{4}^{3}x_{5}^{12}$,
$X_{117}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{10}x_{5}^{13}$, |  $X_{118}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{11}x_{5}^{12}$, |  $X_{119}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{9}x_{5}^{12}$, |  $X_{120}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{8}x_{5}^{13}$,
$X_{121}=x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{12}x_{5}^{12}$, |  $X_{122}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{8}x_{5}^{12}$, |  $X_{123}=x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{12}x_{5}^{12}$, |  $X_{124}=x_{1}^{3}x_{2}^{4}x_{3}x_{4}^{10}x_{5}^{13}$,
$X_{125}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{8}x_{5}^{12}$, |  $X_{126}=x_{1}^{3}x_{2}^{4}x_{3}x_{4}^{11}x_{5}^{12}$, |  $X_{127}=x_{1}^{3}x_{2}^{3}x_{3}^{12}x_{4}x_{5}^{12}$, |  $X_{128}=x_{1}^{3}x_{2}^{4}x_{3}^{11}x_{4}x_{5}^{12}$,
$X_{129}=x_{1}^{3}x_{2}^{7}x_{3}^{8}x_{4}x_{5}^{12}$, |  $X_{130}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{9}x_{5}^{12}$, |  $X_{131}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{8}x_{5}^{13}$, |  $X_{132}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{8}x_{5}^{9}$,
$X_{133}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{9}x_{5}^{8}$, |  $X_{134}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{8}x_{5}^{12}$, |  $X_{135}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{8}x_{5}^{8}$, |  $X_{136}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{8}x_{5}^{12}$,
$X_{137}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{8}x_{5}^{8}$, |  $X_{138}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{9}x_{5}^{12}$, |  $X_{139}=x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}^{9}x_{5}^{12}$, |  $X_{140}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{8}x_{5}^{13}$,
$X_{141}=x_{1}^{7}x_{2}x_{3}^{10}x_{4}x_{5}^{12}$, |  $X_{142}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{9}x_{5}^{12}$, |  $X_{143}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{8}x_{5}^{13}$, |  $X_{144}=x_{1}^{7}x_{2}x_{3}^{6}x_{4}^{8}x_{5}^{9}$,
$X_{145}=x_{1}^{7}x_{2}x_{3}^{6}x_{4}^{9}x_{5}^{8}$, |  $X_{146}=x_{1}^{7}x_{2}x_{3}x_{4}^{10}x_{5}^{12}$, |  $X_{147}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{8}x_{5}^{12}$, |  $X_{148}=x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{8}x_{5}^{8}$,
$X_{149}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{8}x_{5}^{12}$, |  $X_{150}=x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{8}x_{5}^{8}$, |  $X_{151}=x_{1}^{7}x_{2}^{3}x_{3}^{8}x_{4}x_{5}^{12}$, |  $X_{152}=x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}x_{5}^{8}$,
$X_{153}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{8}x_{5}^{8}$, |  $X_{154}=x_{1}x_{2}^{7}x_{3}^{10}x_{4}^{5}x_{5}^{8}$, |  $X_{155}=x_{1}x_{2}^{2}x_{3}^{12}x_{4}^{3}x_{5}^{13}$, |  $X_{156}=x_{1}x_{2}^{7}x_{3}^{10}x_{4}^{4}x_{5}^{9}$,
$X_{157}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{9}x_{5}^{14}$, |  $X_{158}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{14}x_{5}^{9}$, |  $X_{159}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{13}x_{5}^{8}$, |  $X_{160}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{13}x_{5}^{8}$,
$X_{161}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{13}x_{5}^{8}$, |  $X_{162}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{12}x_{5}^{9}$, |  $X_{163}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{12}x_{5}^{9}$, |  $X_{164}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{12}x_{5}^{9}$,

$X_{165}=x_{1}x_{2}^{7}x_{3}^{11}x_{4}^{4}x_{5}^{8}$, |  $X_{166}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{9}x_{5}^{14}$, |  $X_{167}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{14}x_{5}^{9}$, |  $X_{168}=x_{1}x_{2}^{3}x_{3}^{14}x_{4}^{4}x_{5}^{9}$,
---|---|---|---
$X_{169}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{12}x_{5}^{8}$, |  $X_{170}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{12}x_{5}^{8}$, |  $X_{171}=x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{8}x_{5}^{14}$, |  $X_{172}=x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{14}x_{5}^{8}$,
$X_{173}=x_{1}x_{2}^{3}x_{3}^{14}x_{4}^{5}x_{5}^{8}$, |  $X_{174}=x_{1}^{3}x_{2}^{5}x_{3}^{10}x_{4}^{4}x_{5}^{9}$, |  $X_{175}=x_{1}^{3}x_{2}^{5}x_{3}^{10}x_{4}^{5}x_{5}^{8}$, |  $X_{176}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{12}x_{5}^{9}$,
$X_{177}=x_{1}^{3}x_{2}^{13}x_{3}^{2}x_{4}^{4}x_{5}^{9}$, |  $X_{178}=x_{1}^{3}x_{2}^{5}x_{3}^{11}x_{4}^{4}x_{5}^{8}$, |  $X_{179}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{13}x_{5}^{8}$, |  $X_{180}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{12}x_{5}^{9}$,
$X_{181}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{13}x_{5}^{8}$, |  $X_{182}=x_{1}^{3}x_{2}^{13}x_{3}^{2}x_{4}^{5}x_{5}^{8}$, |  $X_{183}=x_{1}^{3}x_{2}^{3}x_{3}^{12}x_{4}^{4}x_{5}^{9}$, |  $X_{184}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{12}x_{5}^{9}$,
$X_{185}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{12}x_{5}^{8}$, |  $X_{186}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{12}x_{5}^{8}$, |  $X_{187}=x_{1}^{3}x_{2}^{3}x_{3}^{12}x_{4}^{5}x_{5}^{8}$, |  $X_{188}=x_{1}^{3}x_{2}^{13}x_{3}^{3}x_{4}^{4}x_{5}^{8}$,
$X_{189}=x_{1}^{3}x_{2}^{3}x_{3}^{13}x_{4}^{4}x_{5}^{8}$, |  $X_{190}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{13}x_{5}^{8}$, |  $X_{191}=x_{1}^{3}x_{2}^{7}x_{3}^{9}x_{4}^{4}x_{5}^{8}$, |  $X_{192}=x_{1}^{3}x_{2}^{7}x_{3}^{4}x_{4}^{9}x_{5}^{8}$,
$X_{193}=x_{1}^{3}x_{2}^{7}x_{3}^{4}x_{4}^{8}x_{5}^{9}$, |  $X_{194}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{9}x_{5}^{14}$, |  $X_{195}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{14}x_{5}^{9}$, |  $X_{196}=x_{1}^{3}x_{2}x_{3}^{14}x_{4}^{4}x_{5}^{9}$,
$X_{197}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{12}x_{5}^{8}$, |  $X_{198}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{12}x_{5}^{8}$, |  $X_{199}=x_{1}^{3}x_{2}x_{3}^{14}x_{4}^{5}x_{5}^{8}$, |  $X_{200}=x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{8}x_{5}^{14}$,
$X_{201}=x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{14}x_{5}^{8}$, |  $X_{202}=x_{1}^{7}x_{2}x_{3}^{10}x_{4}^{5}x_{5}^{8}$, |  $X_{203}=x_{1}^{7}x_{2}^{9}x_{3}^{2}x_{4}^{5}x_{5}^{8}$, |  $X_{204}=x_{1}^{7}x_{2}^{9}x_{3}^{2}x_{4}^{4}x_{5}^{9}$,
$X_{205}=x_{1}^{7}x_{2}x_{3}^{10}x_{4}^{4}x_{5}^{9}$, |  $X_{206}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{13}x_{5}^{8}$, |  $X_{207}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{12}x_{5}^{9}$, |  $X_{208}=x_{1}^{7}x_{2}x_{3}^{11}x_{4}^{4}x_{5}^{8}$,
$X_{209}=x_{1}^{7}x_{2}^{11}x_{3}x_{4}^{4}x_{5}^{8}$, |  $X_{210}=x_{1}^{7}x_{2}^{3}x_{3}^{9}x_{4}^{4}x_{5}^{8}$, |  $X_{211}=x_{1}^{7}x_{2}^{9}x_{3}^{3}x_{4}^{4}x_{5}^{8}$, |  $X_{212}=x_{1}^{7}x_{2}^{3}x_{3}^{4}x_{4}^{9}x_{5}^{8}$,
$X_{213}=x_{1}^{7}x_{2}^{3}x_{3}^{4}x_{4}^{8}x_{5}^{9}$, |  $X_{214}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{12}x_{5}^{8}$, |  $X_{215}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{12}x_{5}^{8}$. |

###  3.5 Admissible monomials in $(\mathscr{C}^{\otimes
5}_{31})^{\omega_{(3)}^{>0}}$

$Y_{1}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{6}x_{5}^{15}$, |  $Y_{2}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{15}x_{5}^{6}$, |  $Y_{3}=x_{1}x_{2}^{3}x_{3}^{15}x_{4}^{6}x_{5}^{6}$, |  $Y_{4}=x_{1}x_{2}^{15}x_{3}^{3}x_{4}^{6}x_{5}^{6}$,
---|---|---|---
$Y_{5}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{6}x_{5}^{15}$, |  $Y_{6}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{15}x_{5}^{6}$, |  $Y_{7}=x_{1}^{3}x_{2}x_{3}^{15}x_{4}^{6}x_{5}^{6}$, |  $Y_{8}=x_{1}^{3}x_{2}^{15}x_{3}x_{4}^{6}x_{5}^{6}$,
$Y_{9}=x_{1}^{15}x_{2}x_{3}^{3}x_{4}^{6}x_{5}^{6}$, |  $Y_{10}=x_{1}^{15}x_{2}^{3}x_{3}x_{4}^{6}x_{5}^{6}$, |  $Y_{11}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{6}x_{5}^{15}$, |  $Y_{12}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{15}x_{5}^{6}$,
$Y_{13}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{2}x_{5}^{15}$, |  $Y_{14}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{15}x_{5}^{2}$, |  $Y_{15}=x_{1}^{3}x_{2}^{5}x_{3}^{15}x_{4}^{2}x_{5}^{6}$, |  $Y_{16}=x_{1}^{3}x_{2}^{5}x_{3}^{15}x_{4}^{6}x_{5}^{2}$,
$Y_{17}=x_{1}^{3}x_{2}^{15}x_{3}^{5}x_{4}^{2}x_{5}^{6}$, |  $Y_{18}=x_{1}^{3}x_{2}^{15}x_{3}^{5}x_{4}^{6}x_{5}^{2}$, |  $Y_{19}=x_{1}^{15}x_{2}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{6}$, |  $Y_{20}=x_{1}^{15}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{2}$,
$Y_{21}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{7}x_{5}^{14}$, |  $Y_{22}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{6}x_{5}^{14}$, |  $Y_{23}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{14}x_{5}^{6}$, |  $Y_{24}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{6}x_{5}^{14}$,
$Y_{25}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{14}x_{5}^{6}$, |  $Y_{26}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{7}x_{5}^{14}$, |  $Y_{27}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{6}x_{5}^{14}$, |  $Y_{28}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{14}x_{5}^{6}$,
$Y_{29}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{6}x_{5}^{14}$, |  $Y_{30}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{14}x_{5}^{6}$, |  $Y_{31}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{6}x_{5}^{14}$, |  $Y_{32}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{14}x_{5}^{6}$,
$Y_{33}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{6}x_{5}^{14}$, |  $Y_{34}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{14}x_{5}^{6}$, |  $Y_{35}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{7}x_{5}^{14}$, |  $Y_{36}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{2}x_{5}^{14}$,
$Y_{37}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{14}x_{5}^{2}$, |  $Y_{38}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{2}x_{5}^{14}$, |  $Y_{39}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{14}x_{5}^{2}$, |  $Y_{40}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{14}$,
$Y_{41}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{14}x_{5}^{2}$, |  $Y_{42}=x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{4}^{2}x_{5}^{6}$, |  $Y_{43}=x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{4}^{6}x_{5}^{2}$, |  $Y_{44}=x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{4}^{2}x_{5}^{6}$,
$Y_{45}=x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{4}^{6}x_{5}^{2}$, |  $Y_{46}=x_{1}x_{2}^{7}x_{3}^{11}x_{4}^{6}x_{5}^{6}$, |  $Y_{47}=x_{1}^{7}x_{2}x_{3}^{11}x_{4}^{6}x_{5}^{6}$, |  $Y_{48}=x_{1}^{7}x_{2}^{11}x_{3}x_{4}^{6}x_{5}^{6}$,
$Y_{49}=x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{4}^{2}x_{5}^{6}$, |  $Y_{50}=x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{4}^{6}x_{5}^{2}$, |  $Y_{51}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{14}$, |  $Y_{52}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{14}x_{5}^{6}$,
$Y_{53}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{6}x_{5}^{14}$, |  $Y_{54}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{14}x_{5}^{6}$, |  $Y_{55}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{3}x_{5}^{14}$, |  $Y_{56}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{14}x_{5}^{3}$,
$Y_{57}=x_{1}^{3}x_{2}^{3}x_{3}^{13}x_{4}^{6}x_{5}^{6}$, |  $Y_{58}=x_{1}^{3}x_{2}^{13}x_{3}^{3}x_{4}^{6}x_{5}^{6}$, |  $Y_{59}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{6}x_{5}^{11}$, |  $Y_{60}=x_{1}^{3}x_{2}^{5}x_{3}^{11}x_{4}^{6}x_{5}^{6}$,
$Y_{61}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{7}x_{5}^{10}$, |  $Y_{62}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{6}x_{5}^{10}$, |  $Y_{63}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{10}x_{5}^{6}$, |  $Y_{64}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{6}x_{5}^{10}$,
$Y_{65}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{10}x_{5}^{6}$, |  $Y_{66}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{10}$, |  $Y_{67}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{10}x_{5}^{6}$, |  $Y_{68}=x_{1}^{3}x_{2}^{7}x_{3}^{9}x_{4}^{6}x_{5}^{6}$,
$Y_{69}=x_{1}^{7}x_{2}^{3}x_{3}^{9}x_{4}^{6}x_{5}^{6}$, |  $Y_{70}=x_{1}^{7}x_{2}^{9}x_{3}^{3}x_{4}^{6}x_{5}^{6}$. |  |

###  3.6 Admissible monomials in $(\mathscr{C}^{\otimes
5}_{32})^{\omega^{\prime}_{(1)}}$

We have $(\mathscr{C}^{\otimes
5}_{32})^{\omega^{\prime}_{(1)}}=(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(1)})^{0}}\cup(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(1)})^{>0}},$ where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt&(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(1)})^{0}}=A(\omega^{\prime}_{(1)})=\\{\widetilde{Z_{j}}:\,1\leq
j\leq 115\\},\\\ &(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(1)})^{>0}}=B(\omega^{\prime}_{(1)})\cup
C(\omega^{\prime}_{(1)})=\\{Z_{j}:\,1\leq j\leq 9\\}.\end{array}$

The admissible monomials $\widetilde{Z_{j}}$ and $Z_{j}$ are determined as
follows:

$\widetilde{Z_{1}}=x_{4}x_{5}^{31}$, |  $\widetilde{Z_{2}}=x_{4}^{31}x_{5}$, |  $\widetilde{Z_{3}}=x_{3}x_{5}^{31}$, |  $\widetilde{Z_{4}}=x_{3}x_{4}^{31}$,
---|---|---|---
$\widetilde{Z_{5}}=x_{3}^{31}x_{5}$, |  $\widetilde{Z_{6}}=x_{3}^{31}x_{4}$, |  $\widetilde{Z_{7}}=x_{2}x_{5}^{31}$, |  $\widetilde{Z_{8}}=x_{2}x_{4}^{31}$,
$\widetilde{Z_{9}}=x_{2}x_{3}^{31}$, |  $\widetilde{Z_{10}}=x_{2}^{31}x_{5}$, |  $\widetilde{Z_{11}}=x_{2}^{31}x_{4}$, |  $\widetilde{Z_{12}}=x_{2}^{31}x_{3}$,
$\widetilde{Z_{13}}=x_{1}x_{5}^{31}$, |  $\widetilde{Z_{14}}=x_{1}x_{4}^{31}$, |  $\widetilde{Z_{15}}=x_{1}x_{3}^{31}$, |  $\widetilde{Z_{16}}=x_{1}x_{2}^{31}$,

$\widetilde{Z_{17}}=x_{1}^{31}x_{5}$, |  $\widetilde{Z_{18}}=x_{1}^{31}x_{4}$, |  $\widetilde{Z_{19}}=x_{1}^{31}x_{3}$, |  $\widetilde{Z_{20}}=x_{1}^{31}x_{2}$,
---|---|---|---
$\widetilde{Z_{21}}=x_{4}^{3}x_{5}^{29}$, |  $\widetilde{Z_{22}}=x_{3}^{3}x_{5}^{29}$, |  $\widetilde{Z_{23}}=x_{3}^{3}x_{4}^{29}$, |  $\widetilde{Z_{24}}=x_{2}^{3}x_{5}^{29}$,
$\widetilde{Z_{25}}=x_{2}^{3}x_{4}^{29}$, |  $\widetilde{Z_{26}}=x_{2}^{3}x_{3}^{29}$, |  $\widetilde{Z_{27}}=x_{1}^{3}x_{5}^{29}$, |  $\widetilde{Z_{28}}=x_{1}^{3}x_{4}^{29}$,
$\widetilde{Z_{29}}=x_{1}^{3}x_{3}^{29}$, |  $\widetilde{Z_{30}}=x_{1}^{3}x_{2}^{29}$, |  $\widetilde{Z_{31}}=x_{3}x_{4}x_{5}^{30}$, |  $\widetilde{Z_{32}}=x_{3}x_{4}^{30}x_{5}$,
$\widetilde{Z_{33}}=x_{2}x_{4}x_{5}^{30}$, |  $\widetilde{Z_{34}}=x_{2}x_{4}^{30}x_{5}$, |  $\widetilde{Z_{35}}=x_{2}x_{3}x_{5}^{30}$, |  $\widetilde{Z_{36}}=x_{2}x_{3}x_{4}^{30}$,
$\widetilde{Z_{37}}=x_{2}x_{3}^{30}x_{5}$, |  $\widetilde{Z_{38}}=x_{2}x_{3}^{30}x_{4}$, |  $\widetilde{Z_{39}}=x_{1}x_{4}x_{5}^{30}$, |  $\widetilde{Z_{40}}=x_{1}x_{4}^{30}x_{5}$,
$\widetilde{Z_{41}}=x_{1}x_{3}x_{5}^{30}$, |  $\widetilde{Z_{42}}=x_{1}x_{3}x_{4}^{30}$, |  $\widetilde{Z_{43}}=x_{1}x_{3}^{30}x_{5}$, |  $\widetilde{Z_{44}}=x_{1}x_{3}^{30}x_{4}$,
$\widetilde{Z_{45}}=x_{1}x_{2}x_{5}^{30}$, |  $\widetilde{Z_{46}}=x_{1}x_{2}x_{4}^{30}$, |  $\widetilde{Z_{47}}=x_{1}x_{2}x_{3}^{30}$, |  $\widetilde{Z_{48}}=x_{1}x_{2}^{30}x_{5}$,
$\widetilde{Z_{49}}=x_{1}x_{2}^{30}x_{4}$, |  $\widetilde{Z_{50}}=x_{1}x_{2}^{30}x_{3}$, |  $\widetilde{Z_{51}}=x_{3}x_{4}^{2}x_{5}^{29}$, |  $\widetilde{Z_{52}}=x_{2}x_{4}^{2}x_{5}^{29}$,
$\widetilde{Z_{53}}=x_{2}x_{3}^{2}x_{5}^{29}$, |  $\widetilde{Z_{54}}=x_{2}x_{3}^{2}x_{4}^{29}$, |  $\widetilde{Z_{55}}=x_{1}x_{4}^{2}x_{5}^{29}$, |  $\widetilde{Z_{56}}=x_{1}x_{3}^{2}x_{5}^{29}$,
$\widetilde{Z_{57}}=x_{1}x_{3}^{2}x_{4}^{29}$, |  $\widetilde{Z_{58}}=x_{1}x_{2}^{2}x_{5}^{29}$, |  $\widetilde{Z_{59}}=x_{1}x_{2}^{2}x_{4}^{29}$, |  $\widetilde{Z_{60}}=x_{1}x_{2}^{2}x_{3}^{29}$,
$\widetilde{Z_{61}}=x_{3}x_{4}^{3}x_{5}^{28}$, |  $\widetilde{Z_{62}}=x_{3}^{3}x_{4}x_{5}^{28}$, |  $\widetilde{Z_{63}}=x_{2}x_{4}^{3}x_{5}^{28}$, |  $\widetilde{Z_{64}}=x_{2}x_{3}^{3}x_{5}^{28}$,
$\widetilde{Z_{65}}=x_{2}x_{3}^{3}x_{4}^{28}$, |  $\widetilde{Z_{66}}=x_{2}^{3}x_{4}x_{5}^{28}$, |  $\widetilde{Z_{67}}=x_{2}^{3}x_{3}x_{5}^{28}$, |  $\widetilde{Z_{68}}=x_{2}^{3}x_{3}x_{4}^{28}$,
$\widetilde{Z_{69}}=x_{1}x_{4}^{3}x_{5}^{28}$, |  $\widetilde{Z_{70}}=x_{1}x_{3}^{3}x_{5}^{28}$, |  $\widetilde{Z_{71}}=x_{1}x_{3}^{3}x_{4}^{28}$, |  $\widetilde{Z_{72}}=x_{1}x_{2}^{3}x_{5}^{28}$,
$\widetilde{Z_{73}}=x_{1}x_{2}^{3}x_{4}^{28}$, |  $\widetilde{Z_{74}}=x_{1}x_{2}^{3}x_{3}^{28}$, |  $\widetilde{Z_{75}}=x_{1}^{3}x_{4}x_{5}^{28}$, |  $\widetilde{Z_{76}}=x_{1}^{3}x_{3}x_{5}^{28}$,
$\widetilde{Z_{77}}=x_{1}^{3}x_{3}x_{4}^{28}$, |  $\widetilde{Z_{78}}=x_{1}^{3}x_{2}x_{5}^{28}$, |  $\widetilde{Z_{79}}=x_{1}^{3}x_{2}x_{4}^{28}$, |  $\widetilde{Z_{80}}=x_{1}^{3}x_{2}x_{3}^{28}$,
$\widetilde{Z_{81}}=x_{2}x_{3}x_{4}^{2}x_{5}^{28}$, |  $\widetilde{Z_{82}}=x_{2}x_{3}^{2}x_{4}x_{5}^{28}$, |  $\widetilde{Z_{83}}=x_{2}x_{3}^{2}x_{4}^{28}x_{5}$, |  $\widetilde{Z_{84}}=x_{1}x_{3}x_{4}^{2}x_{5}^{28}$,
$\widetilde{Z_{85}}=x_{1}x_{3}^{2}x_{4}x_{5}^{28}$, |  $\widetilde{Z_{86}}=x_{1}x_{3}^{2}x_{4}^{28}x_{5}$, |  $\widetilde{Z_{87}}=x_{1}x_{2}x_{4}^{2}x_{5}^{28}$, |  $\widetilde{Z_{88}}=x_{1}x_{2}x_{3}^{2}x_{5}^{28}$,
$\widetilde{Z_{89}}=x_{1}x_{2}x_{3}^{2}x_{4}^{28}$, |  $\widetilde{Z_{90}}=x_{1}x_{2}^{2}x_{4}x_{5}^{28}$, |  $\widetilde{Z_{91}}=x_{1}x_{2}^{2}x_{4}^{28}x_{5}$, |  $\widetilde{Z_{92}}=x_{1}x_{2}^{2}x_{3}x_{5}^{28}$,
$\widetilde{Z_{93}}=x_{1}x_{2}^{2}x_{3}x_{4}^{28}$, |  $\widetilde{Z_{94}}=x_{1}x_{2}^{2}x_{3}^{28}x_{5}$, |  $\widetilde{Z_{95}}=x_{1}x_{2}^{2}x_{3}^{28}x_{4}$, |  $\widetilde{Z_{96}}=x_{2}x_{3}^{2}x_{4}^{4}x_{5}^{25}$,
$\widetilde{Z_{97}}=x_{1}x_{3}^{2}x_{4}^{4}x_{5}^{25}$, |  $\widetilde{Z_{98}}=x_{1}x_{2}^{2}x_{4}^{4}x_{5}^{25}$, |  $\widetilde{Z_{99}}=x_{1}x_{2}^{2}x_{3}^{4}x_{5}^{25}$, |  $\widetilde{Z_{100}}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{25}$,
$\widetilde{Z_{101}}=x_{2}x_{3}^{2}x_{4}^{5}x_{5}^{24}$, |  $\widetilde{Z_{102}}=x_{1}x_{3}^{2}x_{4}^{5}x_{5}^{24}$, |  $\widetilde{Z_{103}}=x_{1}x_{2}^{2}x_{4}^{5}x_{5}^{24}$, |  $\widetilde{Z_{104}}=x_{1}x_{2}^{2}x_{3}^{5}x_{5}^{24}$,
$\widetilde{Z_{105}}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{24}$, |  $\widetilde{Z_{106}}=x_{2}x_{3}^{3}x_{4}^{4}x_{5}^{24}$, |  $\widetilde{Z_{107}}=x_{2}^{3}x_{3}x_{4}^{4}x_{5}^{24}$, |  $\widetilde{Z_{108}}=x_{1}x_{3}^{3}x_{4}^{4}x_{5}^{24}$,
$\widetilde{Z_{109}}=x_{1}x_{2}^{3}x_{4}^{4}x_{5}^{24}$, |  $\widetilde{Z_{110}}=x_{1}x_{2}^{3}x_{3}^{4}x_{5}^{24}$, |  $\widetilde{Z_{111}}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{24}$, |  $\widetilde{Z_{112}}=x_{1}^{3}x_{3}x_{4}^{4}x_{5}^{24}$,
$\widetilde{Z_{113}}=x_{1}^{3}x_{2}x_{4}^{4}x_{5}^{24}$, |  $\widetilde{Z_{114}}=x_{1}^{3}x_{2}x_{3}^{4}x_{5}^{24}$, |  $\widetilde{Z_{115}}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{24}$, |  $Z_{1}=x_{1}x_{2}x_{3}^{2}x_{4}^{4}x_{5}^{24}$,
$Z_{2}=x_{1}x_{2}^{2}x_{3}x_{4}^{4}x_{5}^{24}$, |  $Z_{3}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}x_{5}^{24}$, |  $Z_{4}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{24}x_{5}$, |  $Z_{5}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{8}x_{5}^{17}$,
$Z_{6}=x_{1}x_{2}^{2}x_{3}^{4}x_{4}^{9}x_{5}^{16}$, |  $Z_{7}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{8}x_{5}^{16}$, |  $Z_{8}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{8}x_{5}^{16}$, |  $Z_{9}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{8}x_{5}^{16}$.

###  3.7 Admissible monomials in $(\mathscr{C}^{\otimes
5}_{32})^{\omega^{\prime}_{(2)}}$

We have $(\mathscr{C}^{\otimes
5}_{32})^{\omega^{\prime}_{(2)}}=(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(2)})^{0}}\cup(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(2)})^{>0}},$ where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt&(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(2)})^{0}}=A(\omega^{\prime}_{(2)})=\\{\widetilde{Z^{\prime}_{j}}:\,1\leq
j\leq 175\\},\\\ &(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(2)})^{>0}}=B(\omega^{\prime}_{(2)})\cup
C(\omega^{\prime}_{(2)})=\\{Z^{\prime}_{j}:\,1\leq j\leq 310\\}.\end{array}$

The admissible monomials $\widetilde{Z^{\prime}_{j}}$ and $Z^{\prime}_{j}$ are
determined as follows:

$\widetilde{Z^{\prime}_{1}}=x_{2}x_{3}x_{4}^{15}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{2}}=x_{2}x_{3}^{3}x_{4}^{13}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{3}}=x_{2}x_{3}^{3}x_{4}^{15}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{4}}=x_{2}x_{3}^{7}x_{4}^{11}x_{5}^{13}$,
---|---|---|---
$\widetilde{Z^{\prime}_{5}}=x_{2}x_{3}^{15}x_{4}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{6}}=x_{2}x_{3}^{15}x_{4}^{3}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{7}}=x_{2}x_{3}^{15}x_{4}^{15}x_{5}$, |  $\widetilde{Z^{\prime}_{8}}=x_{2}^{3}x_{3}x_{4}^{13}x_{5}^{15}$,
$\widetilde{Z^{\prime}_{9}}=x_{2}^{3}x_{3}x_{4}^{15}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{10}}=x_{2}^{3}x_{3}^{3}x_{4}^{13}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{11}}=x_{2}^{3}x_{3}^{5}x_{4}^{9}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{12}}=x_{2}^{3}x_{3}^{5}x_{4}^{11}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{13}}=x_{2}^{3}x_{3}^{5}x_{4}^{15}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{14}}=x_{2}^{3}x_{3}^{7}x_{4}^{9}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{15}}=x_{2}^{3}x_{3}^{7}x_{4}^{13}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{16}}=x_{2}^{3}x_{3}^{13}x_{4}x_{5}^{15}$,
$\widetilde{Z^{\prime}_{17}}=x_{2}^{3}x_{3}^{13}x_{4}^{3}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{18}}=x_{2}^{3}x_{3}^{13}x_{4}^{15}x_{5}$, |  $\widetilde{Z^{\prime}_{19}}=x_{2}^{3}x_{3}^{15}x_{4}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{20}}=x_{2}^{3}x_{3}^{15}x_{4}^{5}x_{5}^{9}$,
$\widetilde{Z^{\prime}_{21}}=x_{2}^{3}x_{3}^{15}x_{4}^{13}x_{5}$, |  $\widetilde{Z^{\prime}_{22}}=x_{2}^{7}x_{3}x_{4}^{11}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{23}}=x_{2}^{7}x_{3}^{3}x_{4}^{9}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{24}}=x_{2}^{7}x_{3}^{3}x_{4}^{13}x_{5}^{9}$,
$\widetilde{Z^{\prime}_{25}}=x_{2}^{7}x_{3}^{7}x_{4}^{9}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{26}}=x_{2}^{7}x_{3}^{11}x_{4}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{27}}=x_{2}^{7}x_{3}^{11}x_{4}^{5}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{28}}=x_{2}^{7}x_{3}^{11}x_{4}^{13}x_{5}$,
$\widetilde{Z^{\prime}_{29}}=x_{2}^{15}x_{3}x_{4}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{30}}=x_{2}^{15}x_{3}x_{4}^{3}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{31}}=x_{2}^{15}x_{3}x_{4}^{15}x_{5}$, |  $\widetilde{Z^{\prime}_{32}}=x_{2}^{15}x_{3}^{3}x_{4}x_{5}^{13}$,

$\widetilde{Z^{\prime}_{33}}=x_{2}^{15}x_{3}^{3}x_{4}^{5}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{34}}=x_{2}^{15}x_{3}^{3}x_{4}^{13}x_{5}$, |  $\widetilde{Z^{\prime}_{35}}=x_{2}^{15}x_{3}^{15}x_{4}x_{5}$, |  $\widetilde{Z^{\prime}_{36}}=x_{1}x_{3}x_{4}^{15}x_{5}^{15}$,
---|---|---|---
$\widetilde{Z^{\prime}_{37}}=x_{1}x_{3}^{3}x_{4}^{13}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{38}}=x_{1}x_{3}^{3}x_{4}^{15}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{39}}=x_{1}x_{3}^{7}x_{4}^{11}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{40}}=x_{1}x_{3}^{15}x_{4}x_{5}^{15}$,
$\widetilde{Z^{\prime}_{41}}=x_{1}x_{3}^{15}x_{4}^{3}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{42}}=x_{1}x_{3}^{15}x_{4}^{15}x_{5}$, |  $\widetilde{Z^{\prime}_{43}}=x_{1}x_{2}x_{4}^{15}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{44}}=x_{1}x_{2}x_{3}^{15}x_{5}^{15}$,
$\widetilde{Z^{\prime}_{45}}=x_{1}x_{2}x_{3}^{15}x_{4}^{15}$, |  $\widetilde{Z^{\prime}_{46}}=x_{1}x_{2}^{3}x_{4}^{13}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{47}}=x_{1}x_{2}^{3}x_{4}^{15}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{48}}=x_{1}x_{2}^{3}x_{3}^{13}x_{5}^{15}$,
$\widetilde{Z^{\prime}_{49}}=x_{1}x_{2}^{3}x_{3}^{13}x_{4}^{15}$, |  $\widetilde{Z^{\prime}_{50}}=x_{1}x_{2}^{3}x_{3}^{15}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{51}}=x_{1}x_{2}^{3}x_{3}^{15}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{52}}=x_{1}x_{2}^{7}x_{4}^{11}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{53}}=x_{1}x_{2}^{7}x_{3}^{11}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{54}}=x_{1}x_{2}^{7}x_{3}^{11}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{55}}=x_{1}x_{2}^{15}x_{4}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{56}}=x_{1}x_{2}^{15}x_{4}^{3}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{57}}=x_{1}x_{2}^{15}x_{4}^{15}x_{5}$, |  $\widetilde{Z^{\prime}_{58}}=x_{1}x_{2}^{15}x_{3}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{59}}=x_{1}x_{2}^{15}x_{3}x_{4}^{15}$, |  $\widetilde{Z^{\prime}_{60}}=x_{1}x_{2}^{15}x_{3}^{3}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{61}}=x_{1}x_{2}^{15}x_{3}^{3}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{62}}=x_{1}x_{2}^{15}x_{3}^{15}x_{5}$, |  $\widetilde{Z^{\prime}_{63}}=x_{1}x_{2}^{15}x_{3}^{15}x_{4}$, |  $\widetilde{Z^{\prime}_{64}}=x_{1}^{3}x_{3}x_{4}^{13}x_{5}^{15}$,
$\widetilde{Z^{\prime}_{65}}=x_{1}^{3}x_{3}x_{4}^{15}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{66}}=x_{1}^{3}x_{3}^{3}x_{4}^{13}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{67}}=x_{1}^{3}x_{3}^{5}x_{4}^{9}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{68}}=x_{1}^{3}x_{3}^{5}x_{4}^{11}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{69}}=x_{1}^{3}x_{3}^{5}x_{4}^{15}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{70}}=x_{1}^{3}x_{3}^{7}x_{4}^{9}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{71}}=x_{1}^{3}x_{3}^{7}x_{4}^{13}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{72}}=x_{1}^{3}x_{3}^{13}x_{4}x_{5}^{15}$,
$\widetilde{Z^{\prime}_{73}}=x_{1}^{3}x_{3}^{13}x_{4}^{3}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{74}}=x_{1}^{3}x_{3}^{13}x_{4}^{15}x_{5}$, |  $\widetilde{Z^{\prime}_{75}}=x_{1}^{3}x_{3}^{15}x_{4}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{76}}=x_{1}^{3}x_{3}^{15}x_{4}^{5}x_{5}^{9}$,
$\widetilde{Z^{\prime}_{77}}=x_{1}^{3}x_{3}^{15}x_{4}^{13}x_{5}$, |  $\widetilde{Z^{\prime}_{78}}=x_{1}^{3}x_{2}x_{4}^{13}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{79}}=x_{1}^{3}x_{2}x_{4}^{15}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{80}}=x_{1}^{3}x_{2}x_{3}^{13}x_{5}^{15}$,
$\widetilde{Z^{\prime}_{81}}=x_{1}^{3}x_{2}x_{3}^{13}x_{4}^{15}$, |  $\widetilde{Z^{\prime}_{82}}=x_{1}^{3}x_{2}x_{3}^{15}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{83}}=x_{1}^{3}x_{2}x_{3}^{15}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{84}}=x_{1}^{3}x_{2}^{3}x_{4}^{13}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{85}}=x_{1}^{3}x_{2}^{3}x_{3}^{13}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{86}}=x_{1}^{3}x_{2}^{3}x_{3}^{13}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{87}}=x_{1}^{3}x_{2}^{5}x_{4}^{9}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{88}}=x_{1}^{3}x_{2}^{5}x_{4}^{11}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{89}}=x_{1}^{3}x_{2}^{5}x_{4}^{15}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{90}}=x_{1}^{3}x_{2}^{5}x_{3}^{9}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{91}}=x_{1}^{3}x_{2}^{5}x_{3}^{9}x_{4}^{15}$, |  $\widetilde{Z^{\prime}_{92}}=x_{1}^{3}x_{2}^{5}x_{3}^{11}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{93}}=x_{1}^{3}x_{2}^{5}x_{3}^{11}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{94}}=x_{1}^{3}x_{2}^{5}x_{3}^{15}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{95}}=x_{1}^{3}x_{2}^{5}x_{3}^{15}x_{4}^{9}$, |  $\widetilde{Z^{\prime}_{96}}=x_{1}^{3}x_{2}^{7}x_{4}^{9}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{97}}=x_{1}^{3}x_{2}^{7}x_{4}^{13}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{98}}=x_{1}^{3}x_{2}^{7}x_{3}^{9}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{99}}=x_{1}^{3}x_{2}^{7}x_{3}^{9}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{100}}=x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{5}^{9}$,
$\widetilde{Z^{\prime}_{101}}=x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{4}^{9}$, |  $\widetilde{Z^{\prime}_{102}}=x_{1}^{3}x_{2}^{13}x_{4}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{103}}=x_{1}^{3}x_{2}^{13}x_{4}^{3}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{104}}=x_{1}^{3}x_{2}^{13}x_{4}^{15}x_{5}$,
$\widetilde{Z^{\prime}_{105}}=x_{1}^{3}x_{2}^{13}x_{3}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{106}}=x_{1}^{3}x_{2}^{13}x_{3}x_{4}^{15}$, |  $\widetilde{Z^{\prime}_{107}}=x_{1}^{3}x_{2}^{13}x_{3}^{3}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{108}}=x_{1}^{3}x_{2}^{13}x_{3}^{3}x_{4}^{13}$,
$\widetilde{Z^{\prime}_{109}}=x_{1}^{3}x_{2}^{13}x_{3}^{15}x_{5}$, |  $\widetilde{Z^{\prime}_{110}}=x_{1}^{3}x_{2}^{13}x_{3}^{15}x_{4}$, |  $\widetilde{Z^{\prime}_{111}}=x_{1}^{3}x_{2}^{15}x_{4}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{112}}=x_{1}^{3}x_{2}^{15}x_{4}^{5}x_{5}^{9}$,
$\widetilde{Z^{\prime}_{113}}=x_{1}^{3}x_{2}^{15}x_{4}^{13}x_{5}$, |  $\widetilde{Z^{\prime}_{114}}=x_{1}^{3}x_{2}^{15}x_{3}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{115}}=x_{1}^{3}x_{2}^{15}x_{3}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{116}}=x_{1}^{3}x_{2}^{15}x_{3}^{5}x_{5}^{9}$,
$\widetilde{Z^{\prime}_{117}}=x_{1}^{3}x_{2}^{15}x_{3}^{5}x_{4}^{9}$, |  $\widetilde{Z^{\prime}_{118}}=x_{1}^{3}x_{2}^{15}x_{3}^{13}x_{5}$, |  $\widetilde{Z^{\prime}_{119}}=x_{1}^{3}x_{2}^{15}x_{3}^{13}x_{4}$, |  $\widetilde{Z^{\prime}_{120}}=x_{1}^{7}x_{3}x_{4}^{11}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{121}}=x_{1}^{7}x_{3}^{3}x_{4}^{9}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{122}}=x_{1}^{7}x_{3}^{3}x_{4}^{13}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{123}}=x_{1}^{7}x_{3}^{7}x_{4}^{9}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{124}}=x_{1}^{7}x_{3}^{11}x_{4}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{125}}=x_{1}^{7}x_{3}^{11}x_{4}^{5}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{126}}=x_{1}^{7}x_{3}^{11}x_{4}^{13}x_{5}$, |  $\widetilde{Z^{\prime}_{127}}=x_{1}^{7}x_{2}x_{4}^{11}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{128}}=x_{1}^{7}x_{2}x_{3}^{11}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{129}}=x_{1}^{7}x_{2}x_{3}^{11}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{130}}=x_{1}^{7}x_{2}^{3}x_{4}^{9}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{131}}=x_{1}^{7}x_{2}^{3}x_{4}^{13}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{132}}=x_{1}^{7}x_{2}^{3}x_{3}^{9}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{133}}=x_{1}^{7}x_{2}^{3}x_{3}^{9}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{134}}=x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{135}}=x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{4}^{9}$, |  $\widetilde{Z^{\prime}_{136}}=x_{1}^{7}x_{2}^{7}x_{4}^{9}x_{5}^{9}$,
$\widetilde{Z^{\prime}_{137}}=x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{138}}=x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{9}$, |  $\widetilde{Z^{\prime}_{139}}=x_{1}^{7}x_{2}^{11}x_{4}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{140}}=x_{1}^{7}x_{2}^{11}x_{4}^{5}x_{5}^{9}$,
$\widetilde{Z^{\prime}_{141}}=x_{1}^{7}x_{2}^{11}x_{4}^{13}x_{5}$, |  $\widetilde{Z^{\prime}_{142}}=x_{1}^{7}x_{2}^{11}x_{3}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{143}}=x_{1}^{7}x_{2}^{11}x_{3}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{144}}=x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{5}^{9}$,
$\widetilde{Z^{\prime}_{145}}=x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{4}^{9}$, |  $\widetilde{Z^{\prime}_{146}}=x_{1}^{7}x_{2}^{11}x_{3}^{13}x_{5}$, |  $\widetilde{Z^{\prime}_{147}}=x_{1}^{7}x_{2}^{11}x_{3}^{13}x_{4}$, |  $\widetilde{Z^{\prime}_{148}}=x_{1}^{15}x_{3}x_{4}x_{5}^{15}$,
$\widetilde{Z^{\prime}_{149}}=x_{1}^{15}x_{3}x_{4}^{3}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{150}}=x_{1}^{15}x_{3}x_{4}^{15}x_{5}$, |  $\widetilde{Z^{\prime}_{151}}=x_{1}^{15}x_{3}^{3}x_{4}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{152}}=x_{1}^{15}x_{3}^{3}x_{4}^{5}x_{5}^{9}$,
$\widetilde{Z^{\prime}_{153}}=x_{1}^{15}x_{3}^{3}x_{4}^{13}x_{5}$, |  $\widetilde{Z^{\prime}_{154}}=x_{1}^{15}x_{3}^{15}x_{4}x_{5}$, |  $\widetilde{Z^{\prime}_{155}}=x_{1}^{15}x_{2}x_{4}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{156}}=x_{1}^{15}x_{2}x_{4}^{3}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{157}}=x_{1}^{15}x_{2}x_{4}^{15}x_{5}$, |  $\widetilde{Z^{\prime}_{158}}=x_{1}^{15}x_{2}x_{3}x_{5}^{15}$, |  $\widetilde{Z^{\prime}_{159}}=x_{1}^{15}x_{2}x_{3}x_{4}^{15}$, |  $\widetilde{Z^{\prime}_{160}}=x_{1}^{15}x_{2}x_{3}^{3}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{161}}=x_{1}^{15}x_{2}x_{3}^{3}x_{4}^{13}$, |  $\widetilde{Z^{\prime}_{162}}=x_{1}^{15}x_{2}x_{3}^{15}x_{5}$, |  $\widetilde{Z^{\prime}_{163}}=x_{1}^{15}x_{2}x_{3}^{15}x_{4}$, |  $\widetilde{Z^{\prime}_{164}}=x_{1}^{15}x_{2}^{3}x_{4}x_{5}^{13}$,
$\widetilde{Z^{\prime}_{165}}=x_{1}^{15}x_{2}^{3}x_{4}^{5}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{166}}=x_{1}^{15}x_{2}^{3}x_{4}^{13}x_{5}$, |  $\widetilde{Z^{\prime}_{167}}=x_{1}^{15}x_{2}^{3}x_{3}x_{5}^{13}$, |  $\widetilde{Z^{\prime}_{168}}=x_{1}^{15}x_{2}^{3}x_{3}x_{4}^{13}$,
$\widetilde{Z^{\prime}_{169}}=x_{1}^{15}x_{2}^{3}x_{3}^{5}x_{5}^{9}$, |  $\widetilde{Z^{\prime}_{170}}=x_{1}^{15}x_{2}^{3}x_{3}^{5}x_{4}^{9}$, |  $\widetilde{Z^{\prime}_{171}}=x_{1}^{15}x_{2}^{3}x_{3}^{13}x_{5}$, |  $\widetilde{Z^{\prime}_{172}}=x_{1}^{15}x_{2}^{3}x_{3}^{13}x_{4}$,
$\widetilde{Z^{\prime}_{173}}=x_{1}^{15}x_{2}^{15}x_{4}x_{5}$, |  $\widetilde{Z^{\prime}_{174}}=x_{1}^{15}x_{2}^{15}x_{3}x_{5}$, |  $\widetilde{Z^{\prime}_{175}}=x_{1}^{15}x_{2}^{15}x_{3}x_{4}$, |  $Z^{\prime}_{1}=x_{1}x_{2}x_{3}x_{4}^{14}x_{5}^{15}$,
$Z^{\prime}_{2}=x_{1}x_{2}x_{3}x_{4}^{15}x_{5}^{14}$, |  $Z^{\prime}_{3}=x_{1}x_{2}x_{3}^{2}x_{4}^{13}x_{5}^{15}$, |  $Z^{\prime}_{4}=x_{1}x_{2}x_{3}^{2}x_{4}^{15}x_{5}^{13}$, |  $Z^{\prime}_{5}=x_{1}x_{2}x_{3}^{3}x_{4}^{12}x_{5}^{15}$,
$Z^{\prime}_{6}=x_{1}x_{2}x_{3}^{3}x_{4}^{13}x_{5}^{14}$, |  $Z^{\prime}_{7}=x_{1}x_{2}x_{3}^{3}x_{4}^{14}x_{5}^{13}$, |  $Z^{\prime}_{8}=x_{1}x_{2}x_{3}^{3}x_{4}^{15}x_{5}^{12}$, |  $Z^{\prime}_{9}=x_{1}x_{2}x_{3}^{6}x_{4}^{11}x_{5}^{13}$,
$Z^{\prime}_{10}=x_{1}x_{2}x_{3}^{7}x_{4}^{10}x_{5}^{13}$, |  $Z^{\prime}_{11}=x_{1}x_{2}x_{3}^{7}x_{4}^{11}x_{5}^{12}$, |  $Z^{\prime}_{12}=x_{1}x_{2}x_{3}^{14}x_{4}x_{5}^{15}$, |  $Z^{\prime}_{13}=x_{1}x_{2}x_{3}^{14}x_{4}^{3}x_{5}^{13}$,
$Z^{\prime}_{14}=x_{1}x_{2}x_{3}^{14}x_{4}^{15}x_{5}$, |  $Z^{\prime}_{15}=x_{1}x_{2}x_{3}^{15}x_{4}x_{5}^{14}$, |  $Z^{\prime}_{16}=x_{1}x_{2}x_{3}^{15}x_{4}^{2}x_{5}^{13}$, |  $Z^{\prime}_{17}=x_{1}x_{2}x_{3}^{15}x_{4}^{3}x_{5}^{12}$,
$Z^{\prime}_{18}=x_{1}x_{2}x_{3}^{15}x_{4}^{14}x_{5}$, |  $Z^{\prime}_{19}=x_{1}x_{2}^{2}x_{3}x_{4}^{13}x_{5}^{15}$, |  $Z^{\prime}_{20}=x_{1}x_{2}^{2}x_{3}x_{4}^{15}x_{5}^{13}$, |  $Z^{\prime}_{21}=x_{1}x_{2}^{2}x_{3}^{3}x_{4}^{13}x_{5}^{13}$,
$Z^{\prime}_{22}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{9}x_{5}^{15}$, |  $Z^{\prime}_{23}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{11}x_{5}^{13}$, |  $Z^{\prime}_{24}=x_{1}x_{2}^{2}x_{3}^{5}x_{4}^{15}x_{5}^{9}$, |  $Z^{\prime}_{25}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{9}x_{5}^{13}$,

$Z^{\prime}_{26}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{13}x_{5}^{9}$, |  $Z^{\prime}_{27}=x_{1}x_{2}^{2}x_{3}^{13}x_{4}x_{5}^{15}$, |  $Z^{\prime}_{28}=x_{1}x_{2}^{2}x_{3}^{13}x_{4}^{3}x_{5}^{13}$, |  $Z^{\prime}_{29}=x_{1}x_{2}^{2}x_{3}^{13}x_{4}^{15}x_{5}$,
---|---|---|---
$Z^{\prime}_{30}=x_{1}x_{2}^{2}x_{3}^{15}x_{4}x_{5}^{13}$, |  $Z^{\prime}_{31}=x_{1}x_{2}^{2}x_{3}^{15}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{32}=x_{1}x_{2}^{2}x_{3}^{15}x_{4}^{13}x_{5}$, |  $Z^{\prime}_{33}=x_{1}x_{2}^{3}x_{3}x_{4}^{12}x_{5}^{15}$,
$Z^{\prime}_{34}=x_{1}x_{2}^{3}x_{3}x_{4}^{13}x_{5}^{14}$, |  $Z^{\prime}_{35}=x_{1}x_{2}^{3}x_{3}x_{4}^{14}x_{5}^{13}$, |  $Z^{\prime}_{36}=x_{1}x_{2}^{3}x_{3}x_{4}^{15}x_{5}^{12}$, |  $Z^{\prime}_{37}=x_{1}x_{2}^{3}x_{3}^{2}x_{4}^{13}x_{5}^{13}$,
$Z^{\prime}_{38}=x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{12}x_{5}^{13}$, |  $Z^{\prime}_{39}=x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{13}x_{5}^{12}$, |  $Z^{\prime}_{40}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{9}x_{5}^{15}$, |  $Z^{\prime}_{41}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{11}x_{5}^{13}$,
$Z^{\prime}_{42}=x_{1}x_{2}^{3}x_{3}^{4}x_{4}^{15}x_{5}^{9}$, |  $Z^{\prime}_{43}=x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{8}x_{5}^{15}$, |  $Z^{\prime}_{44}=x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{9}x_{5}^{14}$, |  $Z^{\prime}_{45}=x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{10}x_{5}^{13}$,
$Z^{\prime}_{46}=x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{11}x_{5}^{12}$, |  $Z^{\prime}_{47}=x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{14}x_{5}^{9}$, |  $Z^{\prime}_{48}=x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{15}x_{5}^{8}$, |  $Z^{\prime}_{49}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{9}x_{5}^{13}$,
$Z^{\prime}_{50}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{13}x_{5}^{9}$, |  $Z^{\prime}_{51}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{8}x_{5}^{13}$, |  $Z^{\prime}_{52}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{9}x_{5}^{12}$, |  $Z^{\prime}_{53}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{12}x_{5}^{9}$,
$Z^{\prime}_{54}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{13}x_{5}^{8}$, |  $Z^{\prime}_{55}=x_{1}x_{2}^{3}x_{3}^{12}x_{4}x_{5}^{15}$, |  $Z^{\prime}_{56}=x_{1}x_{2}^{3}x_{3}^{12}x_{4}^{3}x_{5}^{13}$, |  $Z^{\prime}_{57}=x_{1}x_{2}^{3}x_{3}^{12}x_{4}^{15}x_{5}$,
$Z^{\prime}_{58}=x_{1}x_{2}^{3}x_{3}^{13}x_{4}x_{5}^{14}$, |  $Z^{\prime}_{59}=x_{1}x_{2}^{3}x_{3}^{13}x_{4}^{2}x_{5}^{13}$, |  $Z^{\prime}_{60}=x_{1}x_{2}^{3}x_{3}^{13}x_{4}^{3}x_{5}^{12}$, |  $Z^{\prime}_{61}=x_{1}x_{2}^{3}x_{3}^{13}x_{4}^{14}x_{5}$,
$Z^{\prime}_{62}=x_{1}x_{2}^{3}x_{3}^{14}x_{4}x_{5}^{13}$, |  $Z^{\prime}_{63}=x_{1}x_{2}^{3}x_{3}^{14}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{64}=x_{1}x_{2}^{3}x_{3}^{14}x_{4}^{13}x_{5}$, |  $Z^{\prime}_{65}=x_{1}x_{2}^{3}x_{3}^{15}x_{4}x_{5}^{12}$,
$Z^{\prime}_{66}=x_{1}x_{2}^{3}x_{3}^{15}x_{4}^{4}x_{5}^{9}$, |  $Z^{\prime}_{67}=x_{1}x_{2}^{3}x_{3}^{15}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{68}=x_{1}x_{2}^{3}x_{3}^{15}x_{4}^{12}x_{5}$, |  $Z^{\prime}_{69}=x_{1}x_{2}^{6}x_{3}x_{4}^{11}x_{5}^{13}$,
$Z^{\prime}_{70}=x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{9}x_{5}^{13}$, |  $Z^{\prime}_{71}=x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{13}x_{5}^{9}$, |  $Z^{\prime}_{72}=x_{1}x_{2}^{6}x_{3}^{7}x_{4}^{9}x_{5}^{9}$, |  $Z^{\prime}_{73}=x_{1}x_{2}^{6}x_{3}^{11}x_{4}x_{5}^{13}$,
$Z^{\prime}_{74}=x_{1}x_{2}^{6}x_{3}^{11}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{75}=x_{1}x_{2}^{6}x_{3}^{11}x_{4}^{13}x_{5}$, |  $Z^{\prime}_{76}=x_{1}x_{2}^{7}x_{3}x_{4}^{10}x_{5}^{13}$, |  $Z^{\prime}_{77}=x_{1}x_{2}^{7}x_{3}x_{4}^{11}x_{5}^{12}$,
$Z^{\prime}_{78}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{9}x_{5}^{13}$, |  $Z^{\prime}_{79}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{13}x_{5}^{9}$, |  $Z^{\prime}_{80}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{8}x_{5}^{13}$, |  $Z^{\prime}_{81}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{9}x_{5}^{12}$,
$Z^{\prime}_{82}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{12}x_{5}^{9}$, |  $Z^{\prime}_{83}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{13}x_{5}^{8}$, |  $Z^{\prime}_{84}=x_{1}x_{2}^{7}x_{3}^{6}x_{4}^{9}x_{5}^{9}$, |  $Z^{\prime}_{85}=x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{8}x_{5}^{9}$,
$Z^{\prime}_{86}=x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{9}x_{5}^{8}$, |  $Z^{\prime}_{87}=x_{1}x_{2}^{7}x_{3}^{10}x_{4}x_{5}^{13}$, |  $Z^{\prime}_{88}=x_{1}x_{2}^{7}x_{3}^{10}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{89}=x_{1}x_{2}^{7}x_{3}^{10}x_{4}^{13}x_{5}$,
$Z^{\prime}_{90}=x_{1}x_{2}^{7}x_{3}^{11}x_{4}x_{5}^{12}$, |  $Z^{\prime}_{91}=x_{1}x_{2}^{7}x_{3}^{11}x_{4}^{4}x_{5}^{9}$, |  $Z^{\prime}_{92}=x_{1}x_{2}^{7}x_{3}^{11}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{93}=x_{1}x_{2}^{7}x_{3}^{11}x_{4}^{12}x_{5}$,
$Z^{\prime}_{94}=x_{1}x_{2}^{14}x_{3}x_{4}x_{5}^{15}$, |  $Z^{\prime}_{95}=x_{1}x_{2}^{14}x_{3}x_{4}^{3}x_{5}^{13}$, |  $Z^{\prime}_{96}=x_{1}x_{2}^{14}x_{3}x_{4}^{15}x_{5}$, |  $Z^{\prime}_{97}=x_{1}x_{2}^{14}x_{3}^{3}x_{4}x_{5}^{13}$,
$Z^{\prime}_{98}=x_{1}x_{2}^{14}x_{3}^{3}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{99}=x_{1}x_{2}^{14}x_{3}^{3}x_{4}^{13}x_{5}$, |  $Z^{\prime}_{100}=x_{1}x_{2}^{14}x_{3}^{15}x_{4}x_{5}$, |  $Z^{\prime}_{101}=x_{1}x_{2}^{15}x_{3}x_{4}x_{5}^{14}$,
$Z^{\prime}_{102}=x_{1}x_{2}^{15}x_{3}x_{4}^{2}x_{5}^{13}$, |  $Z^{\prime}_{103}=x_{1}x_{2}^{15}x_{3}x_{4}^{3}x_{5}^{12}$, |  $Z^{\prime}_{104}=x_{1}x_{2}^{15}x_{3}x_{4}^{14}x_{5}$, |  $Z^{\prime}_{105}=x_{1}x_{2}^{15}x_{3}^{2}x_{4}x_{5}^{13}$,
$Z^{\prime}_{106}=x_{1}x_{2}^{15}x_{3}^{2}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{107}=x_{1}x_{2}^{15}x_{3}^{2}x_{4}^{13}x_{5}$, |  $Z^{\prime}_{108}=x_{1}x_{2}^{15}x_{3}^{3}x_{4}x_{5}^{12}$, |  $Z^{\prime}_{109}=x_{1}x_{2}^{15}x_{3}^{3}x_{4}^{4}x_{5}^{9}$,
$Z^{\prime}_{110}=x_{1}x_{2}^{15}x_{3}^{3}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{111}=x_{1}x_{2}^{15}x_{3}^{3}x_{4}^{12}x_{5}$, |  $Z^{\prime}_{112}=x_{1}x_{2}^{15}x_{3}^{14}x_{4}x_{5}$, |  $Z^{\prime}_{113}=x_{1}^{3}x_{2}x_{3}x_{4}^{12}x_{5}^{15}$,
$Z^{\prime}_{114}=x_{1}^{3}x_{2}x_{3}x_{4}^{13}x_{5}^{14}$, |  $Z^{\prime}_{115}=x_{1}^{3}x_{2}x_{3}x_{4}^{14}x_{5}^{13}$, |  $Z^{\prime}_{116}=x_{1}^{3}x_{2}x_{3}x_{4}^{15}x_{5}^{12}$, |  $Z^{\prime}_{117}=x_{1}^{3}x_{2}x_{3}^{2}x_{4}^{13}x_{5}^{13}$,
$Z^{\prime}_{118}=x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{12}x_{5}^{13}$, |  $Z^{\prime}_{119}=x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{13}x_{5}^{12}$, |  $Z^{\prime}_{120}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{9}x_{5}^{15}$, |  $Z^{\prime}_{121}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{11}x_{5}^{13}$,
$Z^{\prime}_{122}=x_{1}^{3}x_{2}x_{3}^{4}x_{4}^{15}x_{5}^{9}$, |  $Z^{\prime}_{123}=x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{8}x_{5}^{15}$, |  $Z^{\prime}_{124}=x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{9}x_{5}^{14}$, |  $Z^{\prime}_{125}=x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{10}x_{5}^{13}$,
$Z^{\prime}_{126}=x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{11}x_{5}^{12}$, |  $Z^{\prime}_{127}=x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{14}x_{5}^{9}$, |  $Z^{\prime}_{128}=x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{15}x_{5}^{8}$, |  $Z^{\prime}_{129}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{9}x_{5}^{13}$,
$Z^{\prime}_{130}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{13}x_{5}^{9}$, |  $Z^{\prime}_{131}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{8}x_{5}^{13}$, |  $Z^{\prime}_{132}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{9}x_{5}^{12}$, |  $Z^{\prime}_{133}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{12}x_{5}^{9}$,
$Z^{\prime}_{134}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{13}x_{5}^{8}$, |  $Z^{\prime}_{135}=x_{1}^{3}x_{2}x_{3}^{12}x_{4}x_{5}^{15}$, |  $Z^{\prime}_{136}=x_{1}^{3}x_{2}x_{3}^{12}x_{4}^{3}x_{5}^{13}$, |  $Z^{\prime}_{137}=x_{1}^{3}x_{2}x_{3}^{12}x_{4}^{15}x_{5}$,
$Z^{\prime}_{138}=x_{1}^{3}x_{2}x_{3}^{13}x_{4}x_{5}^{14}$, |  $Z^{\prime}_{139}=x_{1}^{3}x_{2}x_{3}^{13}x_{4}^{2}x_{5}^{13}$, |  $Z^{\prime}_{140}=x_{1}^{3}x_{2}x_{3}^{13}x_{4}^{3}x_{5}^{12}$, |  $Z^{\prime}_{141}=x_{1}^{3}x_{2}x_{3}^{13}x_{4}^{14}x_{5}$,
$Z^{\prime}_{142}=x_{1}^{3}x_{2}x_{3}^{14}x_{4}x_{5}^{13}$, |  $Z^{\prime}_{143}=x_{1}^{3}x_{2}x_{3}^{14}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{144}=x_{1}^{3}x_{2}x_{3}^{14}x_{4}^{13}x_{5}$, |  $Z^{\prime}_{145}=x_{1}^{3}x_{2}x_{3}^{15}x_{4}x_{5}^{12}$,
$Z^{\prime}_{146}=x_{1}^{3}x_{2}x_{3}^{15}x_{4}^{4}x_{5}^{9}$, |  $Z^{\prime}_{147}=x_{1}^{3}x_{2}x_{3}^{15}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{148}=x_{1}^{3}x_{2}x_{3}^{15}x_{4}^{12}x_{5}$, |  $Z^{\prime}_{149}=x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{12}x_{5}^{13}$,
$Z^{\prime}_{150}=x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{13}x_{5}^{12}$, |  $Z^{\prime}_{151}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{9}x_{5}^{13}$, |  $Z^{\prime}_{152}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{13}x_{5}^{9}$, |  $Z^{\prime}_{153}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{8}x_{5}^{13}$,
$Z^{\prime}_{154}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{9}x_{5}^{12}$, |  $Z^{\prime}_{155}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{12}x_{5}^{9}$, |  $Z^{\prime}_{156}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{13}x_{5}^{8}$, |  $Z^{\prime}_{157}=x_{1}^{3}x_{2}^{3}x_{3}^{12}x_{4}x_{5}^{13}$,
$Z^{\prime}_{158}=x_{1}^{3}x_{2}^{3}x_{3}^{12}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{159}=x_{1}^{3}x_{2}^{3}x_{3}^{12}x_{4}^{13}x_{5}$, |  $Z^{\prime}_{160}=x_{1}^{3}x_{2}^{3}x_{3}^{13}x_{4}x_{5}^{12}$, |  $Z^{\prime}_{161}=x_{1}^{3}x_{2}^{3}x_{3}^{13}x_{4}^{4}x_{5}^{9}$,
$Z^{\prime}_{162}=x_{1}^{3}x_{2}^{3}x_{3}^{13}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{163}=x_{1}^{3}x_{2}^{3}x_{3}^{13}x_{4}^{12}x_{5}$, |  $Z^{\prime}_{164}=x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{8}x_{5}^{15}$, |  $Z^{\prime}_{165}=x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{9}x_{5}^{14}$,
$Z^{\prime}_{166}=x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{10}x_{5}^{13}$, |  $Z^{\prime}_{167}=x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{11}x_{5}^{12}$, |  $Z^{\prime}_{168}=x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{14}x_{5}^{9}$, |  $Z^{\prime}_{169}=x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{15}x_{5}^{8}$,
$Z^{\prime}_{170}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{9}x_{5}^{13}$, |  $Z^{\prime}_{171}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{13}x_{5}^{9}$, |  $Z^{\prime}_{172}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{8}x_{5}^{13}$, |  $Z^{\prime}_{173}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{9}x_{5}^{12}$,
$Z^{\prime}_{174}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{12}x_{5}^{9}$, |  $Z^{\prime}_{175}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{13}x_{5}^{8}$, |  $Z^{\prime}_{176}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{9}x_{5}^{9}$, |  $Z^{\prime}_{177}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{8}x_{5}^{9}$,
$Z^{\prime}_{178}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{9}x_{5}^{8}$, |  $Z^{\prime}_{179}=x_{1}^{3}x_{2}^{5}x_{3}^{9}x_{4}x_{5}^{14}$, |  $Z^{\prime}_{180}=x_{1}^{3}x_{2}^{5}x_{3}^{10}x_{4}x_{5}^{13}$, |  $Z^{\prime}_{181}=x_{1}^{3}x_{2}^{5}x_{3}^{10}x_{4}^{5}x_{5}^{9}$,
$Z^{\prime}_{182}=x_{1}^{3}x_{2}^{5}x_{3}^{10}x_{4}^{13}x_{5}$, |  $Z^{\prime}_{183}=x_{1}^{3}x_{2}^{5}x_{3}^{11}x_{4}x_{5}^{12}$, |  $Z^{\prime}_{184}=x_{1}^{3}x_{2}^{5}x_{3}^{11}x_{4}^{4}x_{5}^{9}$, |  $Z^{\prime}_{185}=x_{1}^{3}x_{2}^{5}x_{3}^{11}x_{4}^{5}x_{5}^{8}$,
$Z^{\prime}_{186}=x_{1}^{3}x_{2}^{5}x_{3}^{11}x_{4}^{12}x_{5}$, |  $Z^{\prime}_{187}=x_{1}^{3}x_{2}^{5}x_{3}^{15}x_{4}x_{5}^{8}$, |  $Z^{\prime}_{188}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{8}x_{5}^{13}$, |  $Z^{\prime}_{189}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{9}x_{5}^{12}$,
$Z^{\prime}_{190}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{12}x_{5}^{9}$, |  $Z^{\prime}_{191}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{13}x_{5}^{8}$, |  $Z^{\prime}_{192}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{8}x_{5}^{9}$, |  $Z^{\prime}_{193}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{9}x_{5}^{8}$,
$Z^{\prime}_{194}=x_{1}^{3}x_{2}^{7}x_{3}^{9}x_{4}x_{5}^{12}$, |  $Z^{\prime}_{195}=x_{1}^{3}x_{2}^{7}x_{3}^{9}x_{4}^{4}x_{5}^{9}$, |  $Z^{\prime}_{196}=x_{1}^{3}x_{2}^{7}x_{3}^{9}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{197}=x_{1}^{3}x_{2}^{7}x_{3}^{9}x_{4}^{12}x_{5}$,
$Z^{\prime}_{198}=x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{4}x_{5}^{8}$, |  $Z^{\prime}_{199}=x_{1}^{3}x_{2}^{13}x_{3}x_{4}x_{5}^{14}$, |  $Z^{\prime}_{200}=x_{1}^{3}x_{2}^{13}x_{3}x_{4}^{2}x_{5}^{13}$, |  $Z^{\prime}_{201}=x_{1}^{3}x_{2}^{13}x_{3}x_{4}^{3}x_{5}^{12}$,
$Z^{\prime}_{202}=x_{1}^{3}x_{2}^{13}x_{3}x_{4}^{14}x_{5}$, |  $Z^{\prime}_{203}=x_{1}^{3}x_{2}^{13}x_{3}^{2}x_{4}x_{5}^{13}$, |  $Z^{\prime}_{204}=x_{1}^{3}x_{2}^{13}x_{3}^{2}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{205}=x_{1}^{3}x_{2}^{13}x_{3}^{2}x_{4}^{13}x_{5}$,
$Z^{\prime}_{206}=x_{1}^{3}x_{2}^{13}x_{3}^{3}x_{4}x_{5}^{12}$, |  $Z^{\prime}_{207}=x_{1}^{3}x_{2}^{13}x_{3}^{3}x_{4}^{4}x_{5}^{9}$, |  $Z^{\prime}_{208}=x_{1}^{3}x_{2}^{13}x_{3}^{3}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{209}=x_{1}^{3}x_{2}^{13}x_{3}^{3}x_{4}^{12}x_{5}$,
$Z^{\prime}_{210}=x_{1}^{3}x_{2}^{13}x_{3}^{14}x_{4}x_{5}$, |  $Z^{\prime}_{211}=x_{1}^{3}x_{2}^{15}x_{3}x_{4}x_{5}^{12}$, |  $Z^{\prime}_{212}=x_{1}^{3}x_{2}^{15}x_{3}x_{4}^{4}x_{5}^{9}$, |  $Z^{\prime}_{213}=x_{1}^{3}x_{2}^{15}x_{3}x_{4}^{5}x_{5}^{8}$,
$Z^{\prime}_{214}=x_{1}^{3}x_{2}^{15}x_{3}x_{4}^{12}x_{5}$, |  $Z^{\prime}_{215}=x_{1}^{3}x_{2}^{15}x_{3}^{5}x_{4}x_{5}^{8}$, |  $Z^{\prime}_{216}=x_{1}^{7}x_{2}x_{3}x_{4}^{10}x_{5}^{13}$, |  $Z^{\prime}_{217}=x_{1}^{7}x_{2}x_{3}x_{4}^{11}x_{5}^{12}$,

$Z^{\prime}_{218}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{9}x_{5}^{13}$, |  $Z^{\prime}_{219}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{13}x_{5}^{9}$, |  $Z^{\prime}_{220}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{8}x_{5}^{13}$, |  $Z^{\prime}_{221}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{9}x_{5}^{12}$,
---|---|---|---
$Z^{\prime}_{222}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{12}x_{5}^{9}$, |  $Z^{\prime}_{223}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{13}x_{5}^{8}$, |  $Z^{\prime}_{224}=x_{1}^{7}x_{2}x_{3}^{6}x_{4}^{9}x_{5}^{9}$, |  $Z^{\prime}_{225}=x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{8}x_{5}^{9}$,
$Z^{\prime}_{226}=x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{9}x_{5}^{8}$, |  $Z^{\prime}_{227}=x_{1}^{7}x_{2}x_{3}^{10}x_{4}x_{5}^{13}$, |  $Z^{\prime}_{228}=x_{1}^{7}x_{2}x_{3}^{10}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{229}=x_{1}^{7}x_{2}x_{3}^{10}x_{4}^{13}x_{5}$,
$Z^{\prime}_{230}=x_{1}^{7}x_{2}x_{3}^{11}x_{4}x_{5}^{12}$, |  $Z^{\prime}_{231}=x_{1}^{7}x_{2}x_{3}^{11}x_{4}^{4}x_{5}^{9}$, |  $Z^{\prime}_{232}=x_{1}^{7}x_{2}x_{3}^{11}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{233}=x_{1}^{7}x_{2}x_{3}^{11}x_{4}^{12}x_{5}$,
$Z^{\prime}_{234}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{8}x_{5}^{13}$, |  $Z^{\prime}_{235}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{9}x_{5}^{12}$, |  $Z^{\prime}_{236}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{12}x_{5}^{9}$, |  $Z^{\prime}_{237}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{13}x_{5}^{8}$,
$Z^{\prime}_{238}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{8}x_{5}^{9}$, |  $Z^{\prime}_{239}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{9}x_{5}^{8}$, |  $Z^{\prime}_{240}=x_{1}^{7}x_{2}^{3}x_{3}^{9}x_{4}x_{5}^{12}$, |  $Z^{\prime}_{241}=x_{1}^{7}x_{2}^{3}x_{3}^{9}x_{4}^{4}x_{5}^{9}$,
$Z^{\prime}_{242}=x_{1}^{7}x_{2}^{3}x_{3}^{9}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{243}=x_{1}^{7}x_{2}^{3}x_{3}^{9}x_{4}^{12}x_{5}$, |  $Z^{\prime}_{244}=x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{4}x_{5}^{8}$, |  $Z^{\prime}_{245}=x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{8}x_{5}^{9}$,
$Z^{\prime}_{246}=x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{9}x_{5}^{8}$, |  $Z^{\prime}_{247}=x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}x_{5}^{8}$, |  $Z^{\prime}_{248}=x_{1}^{7}x_{2}^{11}x_{3}x_{4}x_{5}^{12}$, |  $Z^{\prime}_{249}=x_{1}^{7}x_{2}^{11}x_{3}x_{4}^{4}x_{5}^{9}$,
$Z^{\prime}_{250}=x_{1}^{7}x_{2}^{11}x_{3}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{251}=x_{1}^{7}x_{2}^{11}x_{3}x_{4}^{12}x_{5}$, |  $Z^{\prime}_{252}=x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{4}x_{5}^{8}$, |  $Z^{\prime}_{253}=x_{1}^{15}x_{2}x_{3}x_{4}x_{5}^{14}$,
$Z^{\prime}_{254}=x_{1}^{15}x_{2}x_{3}x_{4}^{2}x_{5}^{13}$, |  $Z^{\prime}_{255}=x_{1}^{15}x_{2}x_{3}x_{4}^{3}x_{5}^{12}$, |  $Z^{\prime}_{256}=x_{1}^{15}x_{2}x_{3}x_{4}^{14}x_{5}$, |  $Z^{\prime}_{257}=x_{1}^{15}x_{2}x_{3}^{2}x_{4}x_{5}^{13}$,
$Z^{\prime}_{258}=x_{1}^{15}x_{2}x_{3}^{2}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{259}=x_{1}^{15}x_{2}x_{3}^{2}x_{4}^{13}x_{5}$, |  $Z^{\prime}_{260}=x_{1}^{15}x_{2}x_{3}^{3}x_{4}x_{5}^{12}$, |  $Z^{\prime}_{261}=x_{1}^{15}x_{2}x_{3}^{3}x_{4}^{4}x_{5}^{9}$,
$Z^{\prime}_{262}=x_{1}^{15}x_{2}x_{3}^{3}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{263}=x_{1}^{15}x_{2}x_{3}^{3}x_{4}^{12}x_{5}$, |  $Z^{\prime}_{264}=x_{1}^{15}x_{2}x_{3}^{14}x_{4}x_{5}$, |  $Z^{\prime}_{265}=x_{1}^{15}x_{2}^{3}x_{3}x_{4}x_{5}^{12}$,
$Z^{\prime}_{266}=x_{1}^{15}x_{2}^{3}x_{3}x_{4}^{4}x_{5}^{9}$, |  $Z^{\prime}_{267}=x_{1}^{15}x_{2}^{3}x_{3}x_{4}^{5}x_{5}^{8}$, |  $Z^{\prime}_{268}=x_{1}^{15}x_{2}^{3}x_{3}x_{4}^{12}x_{5}$, |  $Z^{\prime}_{269}=x_{1}^{15}x_{2}^{3}x_{3}^{5}x_{4}x_{5}^{8}$,
$Z^{\prime}_{270}=x_{1}^{3}x_{2}^{15}x_{3}^{5}x_{4}^{8}x_{5}$, |  $Z^{\prime}_{271}=x_{1}^{3}x_{2}^{4}x_{3}x_{4}^{11}x_{5}^{13}$, |  $Z^{\prime}_{272}=x_{1}^{3}x_{2}^{5}x_{3}^{15}x_{4}^{8}x_{5}$, |  $Z^{\prime}_{273}=x_{1}^{3}x_{2}^{4}x_{3}^{11}x_{4}^{13}x_{5}$,
$Z^{\prime}_{274}=x_{1}^{3}x_{2}^{7}x_{3}^{8}x_{4}^{13}x_{5}$, |  $Z^{\prime}_{275}=x_{1}^{3}x_{2}^{5}x_{3}^{8}x_{4}x_{5}^{15}$, |  $Z^{\prime}_{276}=x_{1}^{3}x_{2}^{4}x_{3}^{11}x_{4}x_{5}^{13}$, |  $Z^{\prime}_{277}=x_{1}^{3}x_{2}^{7}x_{3}^{8}x_{4}x_{5}^{13}$,
$Z^{\prime}_{278}=x_{1}^{3}x_{2}^{5}x_{3}^{8}x_{4}^{15}x_{5}$, |  $Z^{\prime}_{279}=x_{1}^{7}x_{2}^{3}x_{3}^{8}x_{4}^{13}x_{5}$, |  $Z^{\prime}_{280}=x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}^{9}x_{5}$, |  $Z^{\prime}_{281}=x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{8}x_{5}$,
$Z^{\prime}_{282}=x_{1}^{7}x_{2}^{3}x_{3}^{8}x_{4}x_{5}^{13}$, |  $Z^{\prime}_{283}=x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}x_{5}^{9}$, |  $Z^{\prime}_{284}=x_{1}^{15}x_{2}^{3}x_{3}^{5}x_{4}^{8}x_{5}$, |  $Z^{\prime}_{285}=x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}^{9}x_{5}^{13}$,
$Z^{\prime}_{286}=x_{1}^{3}x_{2}^{4}x_{3}^{3}x_{4}^{13}x_{5}^{9}$, |  $Z^{\prime}_{287}=x_{1}^{3}x_{2}^{4}x_{3}^{7}x_{4}^{9}x_{5}^{9}$, |  $Z^{\prime}_{288}=x_{1}^{3}x_{2}^{4}x_{3}^{11}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{289}=x_{1}^{3}x_{2}^{5}x_{3}^{8}x_{4}^{3}x_{5}^{13}$,
$Z^{\prime}_{290}=x_{1}^{3}x_{2}^{5}x_{3}^{9}x_{4}^{2}x_{5}^{13}$, |  $Z^{\prime}_{291}=x_{1}^{3}x_{2}^{5}x_{3}^{9}x_{4}^{3}x_{5}^{12}$, |  $Z^{\prime}_{292}=x_{1}^{3}x_{2}^{5}x_{3}^{9}x_{4}^{6}x_{5}^{9}$, |  $Z^{\prime}_{293}=x_{1}^{3}x_{2}^{5}x_{3}^{9}x_{4}^{7}x_{5}^{8}$,
$Z^{\prime}_{294}=x_{1}^{3}x_{2}^{5}x_{3}^{9}x_{4}^{14}x_{5}$, |  $Z^{\prime}_{295}=x_{1}^{3}x_{2}^{7}x_{3}^{4}x_{4}^{9}x_{5}^{9}$, |  $Z^{\prime}_{296}=x_{1}^{3}x_{2}^{7}x_{3}^{8}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{297}=x_{1}^{3}x_{2}^{7}x_{3}^{12}x_{4}x_{5}^{9}$,
$Z^{\prime}_{298}=x_{1}^{3}x_{2}^{7}x_{3}^{12}x_{4}^{9}x_{5}$, |  $Z^{\prime}_{299}=x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{4}^{8}x_{5}$, |  $Z^{\prime}_{300}=x_{1}^{3}x_{2}^{12}x_{3}^{3}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{301}=x_{1}^{7}x_{2}^{3}x_{3}^{4}x_{4}^{9}x_{5}^{9}$,
$Z^{\prime}_{302}=x_{1}^{7}x_{2}^{3}x_{3}^{8}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{303}=x_{1}^{7}x_{2}^{3}x_{3}^{12}x_{4}x_{5}^{9}$, |  $Z^{\prime}_{304}=x_{1}^{7}x_{2}^{3}x_{3}^{12}x_{4}^{9}x_{5}$, |  $Z^{\prime}_{305}=x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{4}^{8}x_{5}$,
$Z^{\prime}_{306}=x_{1}^{7}x_{2}^{8}x_{3}^{3}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{307}=x_{1}^{7}x_{2}^{9}x_{3}^{2}x_{4}^{5}x_{5}^{9}$, |  $Z^{\prime}_{308}=x_{1}^{7}x_{2}^{9}x_{3}^{3}x_{4}^{4}x_{5}^{9}$, |  $Z^{\prime}_{309}=x_{1}^{7}x_{2}^{9}x_{3}^{3}x_{4}^{5}x_{5}^{8}$,
$Z^{\prime}_{310}=x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{4}^{8}x_{5}$. |  |  |

###  3.8 Admissible monomials in $(\mathscr{C}^{\otimes
5}_{32})^{\omega^{\prime}_{(3)}}$

We have $(\mathscr{C}^{\otimes
5}_{32})^{\omega^{\prime}_{(3)}}=(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(3)})^{0}}\cup(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(3)})^{>0}},$ where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt&(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(3)})^{0}}=A(\omega^{\prime}_{(3)})=\\{\widetilde{Z^{\prime\prime}_{j}}:\,1\leq
j\leq 75\\},\\\ &(\mathscr{C}^{\otimes
5}_{32})^{(\omega^{\prime}_{(3)})^{>0}}=B(\omega^{\prime}_{(3)})\cup
C(\omega^{\prime}_{(3)})=\\{Z^{\prime\prime}_{j}:\,1\leq j\leq
320\\}.\end{array}$

The admissible monomials $\widetilde{Z^{\prime\prime}_{j}}$ and
$Z^{\prime\prime}_{j}$ are determined as follows:

$\widetilde{Z^{\prime\prime}_{1}}=x_{2}^{3}x_{3}^{7}x_{4}^{7}x_{5}^{15}$, |  $\widetilde{Z^{\prime\prime}_{2}}=x_{2}^{3}x_{3}^{7}x_{4}^{15}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{3}}=x_{2}^{3}x_{3}^{15}x_{4}^{7}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{4}}=x_{2}^{7}x_{3}^{3}x_{4}^{7}x_{5}^{15}$,
---|---|---|---
$\widetilde{Z^{\prime\prime}_{5}}=x_{2}^{7}x_{3}^{3}x_{4}^{15}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{6}}=x_{2}^{7}x_{3}^{7}x_{4}^{3}x_{5}^{15}$, |  $\widetilde{Z^{\prime\prime}_{7}}=x_{2}^{7}x_{3}^{7}x_{4}^{7}x_{5}^{11}$, |  $\widetilde{Z^{\prime\prime}_{8}}=x_{2}^{7}x_{3}^{7}x_{4}^{11}x_{5}^{7}$,
$\widetilde{Z^{\prime\prime}_{9}}=x_{2}^{7}x_{3}^{7}x_{4}^{15}x_{5}^{3}$, |  $\widetilde{Z^{\prime\prime}_{10}}=x_{2}^{7}x_{3}^{11}x_{4}^{7}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{11}}=x_{2}^{7}x_{3}^{15}x_{4}^{3}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{12}}=x_{2}^{7}x_{3}^{15}x_{4}^{7}x_{5}^{3}$,
$\widetilde{Z^{\prime\prime}_{13}}=x_{2}^{15}x_{3}^{3}x_{4}^{7}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{14}}=x_{2}^{15}x_{3}^{7}x_{4}^{3}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{15}}=x_{2}^{15}x_{3}^{7}x_{4}^{7}x_{5}^{3}$, |  $\widetilde{Z^{\prime\prime}_{16}}=x_{1}^{3}x_{3}^{7}x_{4}^{7}x_{5}^{15}$,
$\widetilde{Z^{\prime\prime}_{17}}=x_{1}^{3}x_{3}^{7}x_{4}^{15}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{18}}=x_{1}^{3}x_{3}^{15}x_{4}^{7}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{19}}=x_{1}^{3}x_{2}^{7}x_{4}^{7}x_{5}^{15}$, |  $\widetilde{Z^{\prime\prime}_{20}}=x_{1}^{3}x_{2}^{7}x_{4}^{15}x_{5}^{7}$,
$\widetilde{Z^{\prime\prime}_{21}}=x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{5}^{15}$, |  $\widetilde{Z^{\prime\prime}_{22}}=x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{4}^{15}$, |  $\widetilde{Z^{\prime\prime}_{23}}=x_{1}^{3}x_{2}^{7}x_{3}^{15}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{24}}=x_{1}^{3}x_{2}^{7}x_{3}^{15}x_{4}^{7}$,
$\widetilde{Z^{\prime\prime}_{25}}=x_{1}^{3}x_{2}^{15}x_{4}^{7}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{26}}=x_{1}^{3}x_{2}^{15}x_{3}^{7}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{27}}=x_{1}^{3}x_{2}^{15}x_{3}^{7}x_{4}^{7}$, |  $\widetilde{Z^{\prime\prime}_{28}}=x_{1}^{7}x_{3}^{3}x_{4}^{7}x_{5}^{15}$,
$\widetilde{Z^{\prime\prime}_{29}}=x_{1}^{7}x_{3}^{3}x_{4}^{15}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{30}}=x_{1}^{7}x_{3}^{7}x_{4}^{3}x_{5}^{15}$, |  $\widetilde{Z^{\prime\prime}_{31}}=x_{1}^{7}x_{3}^{7}x_{4}^{7}x_{5}^{11}$, |  $\widetilde{Z^{\prime\prime}_{32}}=x_{1}^{7}x_{3}^{7}x_{4}^{11}x_{5}^{7}$,
$\widetilde{Z^{\prime\prime}_{33}}=x_{1}^{7}x_{3}^{7}x_{4}^{15}x_{5}^{3}$, |  $\widetilde{Z^{\prime\prime}_{34}}=x_{1}^{7}x_{3}^{11}x_{4}^{7}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{35}}=x_{1}^{7}x_{3}^{15}x_{4}^{3}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{36}}=x_{1}^{7}x_{3}^{15}x_{4}^{7}x_{5}^{3}$,
$\widetilde{Z^{\prime\prime}_{37}}=x_{1}^{7}x_{2}^{3}x_{4}^{7}x_{5}^{15}$, |  $\widetilde{Z^{\prime\prime}_{38}}=x_{1}^{7}x_{2}^{3}x_{4}^{15}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{39}}=x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{5}^{15}$, |  $\widetilde{Z^{\prime\prime}_{40}}=x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{4}^{15}$,
$\widetilde{Z^{\prime\prime}_{41}}=x_{1}^{7}x_{2}^{3}x_{3}^{15}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{42}}=x_{1}^{7}x_{2}^{3}x_{3}^{15}x_{4}^{7}$, |  $\widetilde{Z^{\prime\prime}_{43}}=x_{1}^{7}x_{2}^{7}x_{4}^{3}x_{5}^{15}$, |  $\widetilde{Z^{\prime\prime}_{44}}=x_{1}^{7}x_{2}^{7}x_{4}^{7}x_{5}^{11}$,
$\widetilde{Z^{\prime\prime}_{45}}=x_{1}^{7}x_{2}^{7}x_{4}^{11}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{46}}=x_{1}^{7}x_{2}^{7}x_{4}^{15}x_{5}^{3}$, |  $\widetilde{Z^{\prime\prime}_{47}}=x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{5}^{15}$, |  $\widetilde{Z^{\prime\prime}_{48}}=x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{4}^{15}$,

$\widetilde{Z^{\prime\prime}_{49}}=x_{1}^{7}x_{2}^{7}x_{3}^{7}x_{5}^{11}$, |  $\widetilde{Z^{\prime\prime}_{50}}=x_{1}^{7}x_{2}^{7}x_{3}^{7}x_{4}^{11}$, |  $\widetilde{Z^{\prime\prime}_{51}}=x_{1}^{7}x_{2}^{7}x_{3}^{11}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{52}}=x_{1}^{7}x_{2}^{7}x_{3}^{11}x_{4}^{7}$,
---|---|---|---
$\widetilde{Z^{\prime\prime}_{53}}=x_{1}^{7}x_{2}^{7}x_{3}^{15}x_{5}^{3}$, |  $\widetilde{Z^{\prime\prime}_{54}}=x_{1}^{7}x_{2}^{7}x_{3}^{15}x_{4}^{3}$, |  $\widetilde{Z^{\prime\prime}_{55}}=x_{1}^{7}x_{2}^{11}x_{4}^{7}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{56}}=x_{1}^{7}x_{2}^{11}x_{3}^{7}x_{5}^{7}$,
$\widetilde{Z^{\prime\prime}_{57}}=x_{1}^{7}x_{2}^{11}x_{3}^{7}x_{4}^{7}$, |  $\widetilde{Z^{\prime\prime}_{58}}=x_{1}^{7}x_{2}^{15}x_{4}^{3}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{59}}=x_{1}^{7}x_{2}^{15}x_{4}^{7}x_{5}^{3}$, |  $\widetilde{Z^{\prime\prime}_{60}}=x_{1}^{7}x_{2}^{15}x_{3}^{3}x_{5}^{7}$,
$\widetilde{Z^{\prime\prime}_{61}}=x_{1}^{7}x_{2}^{15}x_{3}^{3}x_{4}^{7}$, |  $\widetilde{Z^{\prime\prime}_{62}}=x_{1}^{7}x_{2}^{15}x_{3}^{7}x_{5}^{3}$, |  $\widetilde{Z^{\prime\prime}_{63}}=x_{1}^{7}x_{2}^{15}x_{3}^{7}x_{4}^{3}$, |  $\widetilde{Z^{\prime\prime}_{64}}=x_{1}^{15}x_{3}^{3}x_{4}^{7}x_{5}^{7}$,
$\widetilde{Z^{\prime\prime}_{65}}=x_{1}^{15}x_{3}^{7}x_{4}^{3}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{66}}=x_{1}^{15}x_{3}^{7}x_{4}^{7}x_{5}^{3}$, |  $\widetilde{Z^{\prime\prime}_{67}}=x_{1}^{15}x_{2}^{3}x_{4}^{7}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{68}}=x_{1}^{15}x_{2}^{3}x_{3}^{7}x_{5}^{7}$,
$\widetilde{Z^{\prime\prime}_{69}}=x_{1}^{15}x_{2}^{3}x_{3}^{7}x_{4}^{7}$, |  $\widetilde{Z^{\prime\prime}_{70}}=x_{1}^{15}x_{2}^{7}x_{4}^{3}x_{5}^{7}$, |  $\widetilde{Z^{\prime\prime}_{71}}=x_{1}^{15}x_{2}^{7}x_{4}^{7}x_{5}^{3}$, |  $\widetilde{Z^{\prime\prime}_{72}}=x_{1}^{15}x_{2}^{7}x_{3}^{3}x_{5}^{7}$,
$\widetilde{Z^{\prime\prime}_{73}}=x_{1}^{15}x_{2}^{7}x_{3}^{3}x_{4}^{7}$, |  $\widetilde{Z^{\prime\prime}_{74}}=x_{1}^{15}x_{2}^{7}x_{3}^{7}x_{5}^{3}$, |  $\widetilde{Z^{\prime\prime}_{75}}=x_{1}^{15}x_{2}^{7}x_{3}^{7}x_{4}^{3}$, |  $Z^{\prime\prime}_{1}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{7}x_{5}^{15}$,
$Z^{\prime\prime}_{2}=x_{1}x_{2}^{2}x_{3}^{7}x_{4}^{15}x_{5}^{7}$, |  $Z^{\prime\prime}_{3}=x_{1}x_{2}^{2}x_{3}^{15}x_{4}^{7}x_{5}^{7}$, |  $Z^{\prime\prime}_{4}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{7}x_{5}^{15}$, |  $Z^{\prime\prime}_{5}=x_{1}x_{2}^{3}x_{3}^{6}x_{4}^{15}x_{5}^{7}$,
$Z^{\prime\prime}_{6}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{6}x_{5}^{15}$, |  $Z^{\prime\prime}_{7}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{7}x_{5}^{14}$, |  $Z^{\prime\prime}_{8}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{14}x_{5}^{7}$, |  $Z^{\prime\prime}_{9}=x_{1}x_{2}^{3}x_{3}^{7}x_{4}^{15}x_{5}^{6}$,
$Z^{\prime\prime}_{10}=x_{1}x_{2}^{3}x_{3}^{14}x_{4}^{7}x_{5}^{7}$, |  $Z^{\prime\prime}_{11}=x_{1}x_{2}^{3}x_{3}^{15}x_{4}^{6}x_{5}^{7}$, |  $Z^{\prime\prime}_{12}=x_{1}x_{2}^{3}x_{3}^{15}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{13}=x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{7}x_{5}^{15}$,
$Z^{\prime\prime}_{14}=x_{1}x_{2}^{6}x_{3}^{3}x_{4}^{15}x_{5}^{7}$, |  $Z^{\prime\prime}_{15}=x_{1}x_{2}^{6}x_{3}^{7}x_{4}^{3}x_{5}^{15}$, |  $Z^{\prime\prime}_{16}=x_{1}x_{2}^{6}x_{3}^{7}x_{4}^{7}x_{5}^{11}$, |  $Z^{\prime\prime}_{17}=x_{1}x_{2}^{6}x_{3}^{7}x_{4}^{11}x_{5}^{7}$,
$Z^{\prime\prime}_{18}=x_{1}x_{2}^{6}x_{3}^{7}x_{4}^{15}x_{5}^{3}$, |  $Z^{\prime\prime}_{19}=x_{1}x_{2}^{6}x_{3}^{11}x_{4}^{7}x_{5}^{7}$, |  $Z^{\prime\prime}_{20}=x_{1}x_{2}^{6}x_{3}^{15}x_{4}^{3}x_{5}^{7}$, |  $Z^{\prime\prime}_{21}=x_{1}x_{2}^{6}x_{3}^{15}x_{4}^{7}x_{5}^{3}$,
$Z^{\prime\prime}_{22}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{7}x_{5}^{15}$, |  $Z^{\prime\prime}_{23}=x_{1}x_{2}^{7}x_{3}^{2}x_{4}^{15}x_{5}^{7}$, |  $Z^{\prime\prime}_{24}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{6}x_{5}^{15}$, |  $Z^{\prime\prime}_{25}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{7}x_{5}^{14}$,
$Z^{\prime\prime}_{26}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{14}x_{5}^{7}$, |  $Z^{\prime\prime}_{27}=x_{1}x_{2}^{7}x_{3}^{3}x_{4}^{15}x_{5}^{6}$, |  $Z^{\prime\prime}_{28}=x_{1}x_{2}^{7}x_{3}^{6}x_{4}^{3}x_{5}^{15}$, |  $Z^{\prime\prime}_{29}=x_{1}x_{2}^{7}x_{3}^{6}x_{4}^{7}x_{5}^{11}$,
$Z^{\prime\prime}_{30}=x_{1}x_{2}^{7}x_{3}^{6}x_{4}^{11}x_{5}^{7}$, |  $Z^{\prime\prime}_{31}=x_{1}x_{2}^{7}x_{3}^{6}x_{4}^{15}x_{5}^{3}$, |  $Z^{\prime\prime}_{32}=x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{2}x_{5}^{15}$, |  $Z^{\prime\prime}_{33}=x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{3}x_{5}^{14}$,
$Z^{\prime\prime}_{34}=x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{6}x_{5}^{11}$, |  $Z^{\prime\prime}_{35}=x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{7}x_{5}^{10}$, |  $Z^{\prime\prime}_{36}=x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{10}x_{5}^{7}$, |  $Z^{\prime\prime}_{37}=x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{11}x_{5}^{6}$,
$Z^{\prime\prime}_{38}=x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{14}x_{5}^{3}$, |  $Z^{\prime\prime}_{39}=x_{1}x_{2}^{7}x_{3}^{7}x_{4}^{15}x_{5}^{2}$, |  $Z^{\prime\prime}_{40}=x_{1}x_{2}^{7}x_{3}^{10}x_{4}^{7}x_{5}^{7}$, |  $Z^{\prime\prime}_{41}=x_{1}x_{2}^{7}x_{3}^{11}x_{4}^{6}x_{5}^{7}$,
$Z^{\prime\prime}_{42}=x_{1}x_{2}^{7}x_{3}^{11}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{43}=x_{1}x_{2}^{7}x_{3}^{14}x_{4}^{3}x_{5}^{7}$, |  $Z^{\prime\prime}_{44}=x_{1}x_{2}^{7}x_{3}^{14}x_{4}^{7}x_{5}^{3}$, |  $Z^{\prime\prime}_{45}=x_{1}x_{2}^{7}x_{3}^{15}x_{4}^{2}x_{5}^{7}$,
$Z^{\prime\prime}_{46}=x_{1}x_{2}^{7}x_{3}^{15}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{47}=x_{1}x_{2}^{7}x_{3}^{15}x_{4}^{6}x_{5}^{3}$, |  $Z^{\prime\prime}_{48}=x_{1}x_{2}^{7}x_{3}^{15}x_{4}^{7}x_{5}^{2}$, |  $Z^{\prime\prime}_{49}=x_{1}x_{2}^{14}x_{3}^{3}x_{4}^{7}x_{5}^{7}$,
$Z^{\prime\prime}_{50}=x_{1}x_{2}^{14}x_{3}^{7}x_{4}^{3}x_{5}^{7}$, |  $Z^{\prime\prime}_{51}=x_{1}x_{2}^{14}x_{3}^{7}x_{4}^{7}x_{5}^{3}$, |  $Z^{\prime\prime}_{52}=x_{1}x_{2}^{15}x_{3}^{2}x_{4}^{7}x_{5}^{7}$, |  $Z^{\prime\prime}_{53}=x_{1}x_{2}^{15}x_{3}^{3}x_{4}^{6}x_{5}^{7}$,
$Z^{\prime\prime}_{54}=x_{1}x_{2}^{15}x_{3}^{3}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{55}=x_{1}x_{2}^{15}x_{3}^{6}x_{4}^{3}x_{5}^{7}$, |  $Z^{\prime\prime}_{56}=x_{1}x_{2}^{15}x_{3}^{6}x_{4}^{7}x_{5}^{3}$, |  $Z^{\prime\prime}_{57}=x_{1}x_{2}^{15}x_{3}^{7}x_{4}^{2}x_{5}^{7}$,
$Z^{\prime\prime}_{58}=x_{1}x_{2}^{15}x_{3}^{7}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{59}=x_{1}x_{2}^{15}x_{3}^{7}x_{4}^{6}x_{5}^{3}$, |  $Z^{\prime\prime}_{60}=x_{1}x_{2}^{15}x_{3}^{7}x_{4}^{7}x_{5}^{2}$, |  $Z^{\prime\prime}_{61}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{7}x_{5}^{15}$,
$Z^{\prime\prime}_{62}=x_{1}^{3}x_{2}x_{3}^{6}x_{4}^{15}x_{5}^{7}$, |  $Z^{\prime\prime}_{63}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{6}x_{5}^{15}$, |  $Z^{\prime\prime}_{64}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{7}x_{5}^{14}$, |  $Z^{\prime\prime}_{65}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{14}x_{5}^{7}$,
$Z^{\prime\prime}_{66}=x_{1}^{3}x_{2}x_{3}^{7}x_{4}^{15}x_{5}^{6}$, |  $Z^{\prime\prime}_{67}=x_{1}^{3}x_{2}x_{3}^{14}x_{4}^{7}x_{5}^{7}$, |  $Z^{\prime\prime}_{68}=x_{1}^{3}x_{2}x_{3}^{15}x_{4}^{6}x_{5}^{7}$, |  $Z^{\prime\prime}_{69}=x_{1}^{3}x_{2}x_{3}^{15}x_{4}^{7}x_{5}^{6}$,
$Z^{\prime\prime}_{70}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{7}x_{5}^{15}$, |  $Z^{\prime\prime}_{71}=x_{1}^{3}x_{2}^{3}x_{3}^{4}x_{4}^{15}x_{5}^{7}$, |  $Z^{\prime\prime}_{72}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{15}$, |  $Z^{\prime\prime}_{73}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{7}x_{5}^{14}$,
$Z^{\prime\prime}_{74}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{14}x_{5}^{7}$, |  $Z^{\prime\prime}_{75}=x_{1}^{3}x_{2}^{3}x_{3}^{5}x_{4}^{15}x_{5}^{6}$, |  $Z^{\prime\prime}_{76}=x_{1}^{3}x_{2}^{3}x_{3}^{7}x_{4}^{4}x_{5}^{15}$, |  $Z^{\prime\prime}_{77}=x_{1}^{3}x_{2}^{3}x_{3}^{7}x_{4}^{5}x_{5}^{14}$,
$Z^{\prime\prime}_{78}=x_{1}^{3}x_{2}^{3}x_{3}^{7}x_{4}^{7}x_{5}^{12}$, |  $Z^{\prime\prime}_{79}=x_{1}^{3}x_{2}^{3}x_{3}^{7}x_{4}^{12}x_{5}^{7}$, |  $Z^{\prime\prime}_{80}=x_{1}^{3}x_{2}^{3}x_{3}^{7}x_{4}^{13}x_{5}^{6}$, |  $Z^{\prime\prime}_{81}=x_{1}^{3}x_{2}^{3}x_{3}^{7}x_{4}^{15}x_{5}^{4}$,
$Z^{\prime\prime}_{82}=x_{1}^{3}x_{2}^{3}x_{3}^{12}x_{4}^{7}x_{5}^{7}$, |  $Z^{\prime\prime}_{83}=x_{1}^{3}x_{2}^{3}x_{3}^{13}x_{4}^{6}x_{5}^{7}$, |  $Z^{\prime\prime}_{84}=x_{1}^{3}x_{2}^{3}x_{3}^{13}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{85}=x_{1}^{3}x_{2}^{3}x_{3}^{15}x_{4}^{4}x_{5}^{7}$,
$Z^{\prime\prime}_{86}=x_{1}^{3}x_{2}^{3}x_{3}^{15}x_{4}^{5}x_{5}^{6}$, |  $Z^{\prime\prime}_{87}=x_{1}^{3}x_{2}^{3}x_{3}^{15}x_{4}^{7}x_{5}^{4}$, |  $Z^{\prime\prime}_{88}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{7}x_{5}^{15}$, |  $Z^{\prime\prime}_{89}=x_{1}^{3}x_{2}^{5}x_{3}^{2}x_{4}^{15}x_{5}^{7}$,
$Z^{\prime\prime}_{90}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{6}x_{5}^{15}$, |  $Z^{\prime\prime}_{91}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{7}x_{5}^{14}$, |  $Z^{\prime\prime}_{92}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{14}x_{5}^{7}$, |  $Z^{\prime\prime}_{93}=x_{1}^{3}x_{2}^{5}x_{3}^{3}x_{4}^{15}x_{5}^{6}$,
$Z^{\prime\prime}_{94}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{3}x_{5}^{15}$, |  $Z^{\prime\prime}_{95}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{7}x_{5}^{11}$, |  $Z^{\prime\prime}_{96}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{11}x_{5}^{7}$, |  $Z^{\prime\prime}_{97}=x_{1}^{3}x_{2}^{5}x_{3}^{6}x_{4}^{15}x_{5}^{3}$,
$Z^{\prime\prime}_{98}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{2}x_{5}^{15}$, |  $Z^{\prime\prime}_{99}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{3}x_{5}^{14}$, |  $Z^{\prime\prime}_{100}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{6}x_{5}^{11}$, |  $Z^{\prime\prime}_{101}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{7}x_{5}^{10}$,
$Z^{\prime\prime}_{102}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{10}x_{5}^{7}$, |  $Z^{\prime\prime}_{103}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{11}x_{5}^{6}$, |  $Z^{\prime\prime}_{104}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{14}x_{5}^{3}$, |  $Z^{\prime\prime}_{105}=x_{1}^{3}x_{2}^{5}x_{3}^{7}x_{4}^{15}x_{5}^{2}$,
$Z^{\prime\prime}_{106}=x_{1}^{3}x_{2}^{5}x_{3}^{10}x_{4}^{7}x_{5}^{7}$, |  $Z^{\prime\prime}_{107}=x_{1}^{3}x_{2}^{5}x_{3}^{11}x_{4}^{6}x_{5}^{7}$, |  $Z^{\prime\prime}_{108}=x_{1}^{3}x_{2}^{5}x_{3}^{11}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{109}=x_{1}^{3}x_{2}^{5}x_{3}^{14}x_{4}^{3}x_{5}^{7}$,
$Z^{\prime\prime}_{110}=x_{1}^{3}x_{2}^{5}x_{3}^{14}x_{4}^{7}x_{5}^{3}$, |  $Z^{\prime\prime}_{111}=x_{1}^{3}x_{2}^{5}x_{3}^{15}x_{4}^{2}x_{5}^{7}$, |  $Z^{\prime\prime}_{112}=x_{1}^{3}x_{2}^{5}x_{3}^{15}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{113}=x_{1}^{3}x_{2}^{5}x_{3}^{15}x_{4}^{6}x_{5}^{3}$,
$Z^{\prime\prime}_{114}=x_{1}^{3}x_{2}^{5}x_{3}^{15}x_{4}^{7}x_{5}^{2}$, |  $Z^{\prime\prime}_{115}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{6}x_{5}^{15}$, |  $Z^{\prime\prime}_{116}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{7}x_{5}^{14}$, |  $Z^{\prime\prime}_{117}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{14}x_{5}^{7}$,
$Z^{\prime\prime}_{118}=x_{1}^{3}x_{2}^{7}x_{3}x_{4}^{15}x_{5}^{6}$, |  $Z^{\prime\prime}_{119}=x_{1}^{3}x_{2}^{7}x_{3}^{3}x_{4}^{4}x_{5}^{15}$, |  $Z^{\prime\prime}_{120}=x_{1}^{3}x_{2}^{7}x_{3}^{3}x_{4}^{5}x_{5}^{14}$, |  $Z^{\prime\prime}_{121}=x_{1}^{3}x_{2}^{7}x_{3}^{3}x_{4}^{7}x_{5}^{12}$,
$Z^{\prime\prime}_{122}=x_{1}^{3}x_{2}^{7}x_{3}^{3}x_{4}^{12}x_{5}^{7}$, |  $Z^{\prime\prime}_{123}=x_{1}^{3}x_{2}^{7}x_{3}^{3}x_{4}^{13}x_{5}^{6}$, |  $Z^{\prime\prime}_{124}=x_{1}^{3}x_{2}^{7}x_{3}^{3}x_{4}^{15}x_{5}^{4}$, |  $Z^{\prime\prime}_{125}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{2}x_{5}^{15}$,
$Z^{\prime\prime}_{126}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{3}x_{5}^{14}$, |  $Z^{\prime\prime}_{127}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{6}x_{5}^{11}$, |  $Z^{\prime\prime}_{128}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{7}x_{5}^{10}$, |  $Z^{\prime\prime}_{129}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{10}x_{5}^{7}$,
$Z^{\prime\prime}_{130}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{11}x_{5}^{6}$, |  $Z^{\prime\prime}_{131}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{14}x_{5}^{3}$, |  $Z^{\prime\prime}_{132}=x_{1}^{3}x_{2}^{7}x_{3}^{5}x_{4}^{15}x_{5}^{2}$, |  $Z^{\prime\prime}_{133}=x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{4}x_{5}^{14}$,
$Z^{\prime\prime}_{134}=x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{4}^{3}x_{5}^{12}$, |  $Z^{\prime\prime}_{135}=x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{4}^{5}x_{5}^{10}$, |  $Z^{\prime\prime}_{136}=x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{4}^{7}x_{5}^{8}$, |  $Z^{\prime\prime}_{137}=x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{4}^{9}x_{5}^{6}$,
$Z^{\prime\prime}_{138}=x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{4}^{13}x_{5}^{2}$, |  $Z^{\prime\prime}_{139}=x_{1}^{3}x_{2}^{7}x_{3}^{9}x_{4}^{6}x_{5}^{7}$, |  $Z^{\prime\prime}_{140}=x_{1}^{3}x_{2}^{7}x_{3}^{9}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{141}=x_{1}^{3}x_{2}^{7}x_{3}^{11}x_{4}^{5}x_{5}^{6}$,
$Z^{\prime\prime}_{142}=x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{4}^{2}x_{5}^{7}$, |  $Z^{\prime\prime}_{143}=x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{144}=x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{4}^{6}x_{5}^{3}$, |  $Z^{\prime\prime}_{145}=x_{1}^{3}x_{2}^{7}x_{3}^{13}x_{4}^{7}x_{5}^{2}$,
$Z^{\prime\prime}_{146}=x_{1}^{3}x_{2}^{7}x_{3}^{15}x_{4}x_{5}^{6}$, |  $Z^{\prime\prime}_{147}=x_{1}^{3}x_{2}^{7}x_{3}^{15}x_{4}^{3}x_{5}^{4}$, |  $Z^{\prime\prime}_{148}=x_{1}^{3}x_{2}^{7}x_{3}^{15}x_{4}^{5}x_{5}^{2}$, |  $Z^{\prime\prime}_{149}=x_{1}^{3}x_{2}^{13}x_{3}^{2}x_{4}^{7}x_{5}^{7}$,
$Z^{\prime\prime}_{150}=x_{1}^{3}x_{2}^{13}x_{3}^{3}x_{4}^{6}x_{5}^{7}$, |  $Z^{\prime\prime}_{151}=x_{1}^{3}x_{2}^{13}x_{3}^{3}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{152}=x_{1}^{3}x_{2}^{13}x_{3}^{6}x_{4}^{3}x_{5}^{7}$, |  $Z^{\prime\prime}_{153}=x_{1}^{3}x_{2}^{13}x_{3}^{6}x_{4}^{7}x_{5}^{3}$,
$Z^{\prime\prime}_{154}=x_{1}^{3}x_{2}^{13}x_{3}^{7}x_{4}^{2}x_{5}^{7}$, |  $Z^{\prime\prime}_{155}=x_{1}^{3}x_{2}^{13}x_{3}^{7}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{156}=x_{1}^{3}x_{2}^{13}x_{3}^{7}x_{4}^{6}x_{5}^{3}$, |  $Z^{\prime\prime}_{157}=x_{1}^{3}x_{2}^{13}x_{3}^{7}x_{4}^{7}x_{5}^{2}$,

$Z^{\prime\prime}_{158}=x_{1}^{3}x_{2}^{15}x_{3}x_{4}^{6}x_{5}^{7}$, |  $Z^{\prime\prime}_{159}=x_{1}^{3}x_{2}^{15}x_{3}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{160}=x_{1}^{3}x_{2}^{15}x_{3}^{3}x_{4}^{4}x_{5}^{7}$, |  $Z^{\prime\prime}_{161}=x_{1}^{3}x_{2}^{15}x_{3}^{3}x_{4}^{5}x_{5}^{6}$,
---|---|---|---
$Z^{\prime\prime}_{162}=x_{1}^{3}x_{2}^{15}x_{3}^{3}x_{4}^{7}x_{5}^{4}$, |  $Z^{\prime\prime}_{163}=x_{1}^{3}x_{2}^{15}x_{3}^{5}x_{4}^{2}x_{5}^{7}$, |  $Z^{\prime\prime}_{164}=x_{1}^{3}x_{2}^{15}x_{3}^{5}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{165}=x_{1}^{3}x_{2}^{15}x_{3}^{5}x_{4}^{6}x_{5}^{3}$,
$Z^{\prime\prime}_{166}=x_{1}^{3}x_{2}^{15}x_{3}^{5}x_{4}^{7}x_{5}^{2}$, |  $Z^{\prime\prime}_{167}=x_{1}^{3}x_{2}^{15}x_{3}^{7}x_{4}x_{5}^{6}$, |  $Z^{\prime\prime}_{168}=x_{1}^{3}x_{2}^{15}x_{3}^{7}x_{4}^{3}x_{5}^{4}$, |  $Z^{\prime\prime}_{169}=x_{1}^{3}x_{2}^{15}x_{3}^{7}x_{4}^{5}x_{5}^{2}$,
$Z^{\prime\prime}_{170}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{7}x_{5}^{15}$, |  $Z^{\prime\prime}_{171}=x_{1}^{7}x_{2}x_{3}^{2}x_{4}^{15}x_{5}^{7}$, |  $Z^{\prime\prime}_{172}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{6}x_{5}^{15}$, |  $Z^{\prime\prime}_{173}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{7}x_{5}^{14}$,
$Z^{\prime\prime}_{174}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{14}x_{5}^{7}$, |  $Z^{\prime\prime}_{175}=x_{1}^{7}x_{2}x_{3}^{3}x_{4}^{15}x_{5}^{6}$, |  $Z^{\prime\prime}_{176}=x_{1}^{7}x_{2}x_{3}^{6}x_{4}^{3}x_{5}^{15}$, |  $Z^{\prime\prime}_{177}=x_{1}^{7}x_{2}x_{3}^{6}x_{4}^{7}x_{5}^{11}$,
$Z^{\prime\prime}_{178}=x_{1}^{7}x_{2}x_{3}^{6}x_{4}^{11}x_{5}^{7}$, |  $Z^{\prime\prime}_{179}=x_{1}^{7}x_{2}x_{3}^{6}x_{4}^{15}x_{5}^{3}$, |  $Z^{\prime\prime}_{180}=x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{2}x_{5}^{15}$, |  $Z^{\prime\prime}_{181}=x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{3}x_{5}^{14}$,
$Z^{\prime\prime}_{182}=x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{6}x_{5}^{11}$, |  $Z^{\prime\prime}_{183}=x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{7}x_{5}^{10}$, |  $Z^{\prime\prime}_{184}=x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{10}x_{5}^{7}$, |  $Z^{\prime\prime}_{185}=x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{11}x_{5}^{6}$,
$Z^{\prime\prime}_{186}=x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{14}x_{5}^{3}$, |  $Z^{\prime\prime}_{187}=x_{1}^{7}x_{2}x_{3}^{7}x_{4}^{15}x_{5}^{2}$, |  $Z^{\prime\prime}_{188}=x_{1}^{7}x_{2}x_{3}^{10}x_{4}^{7}x_{5}^{7}$, |  $Z^{\prime\prime}_{189}=x_{1}^{7}x_{2}x_{3}^{11}x_{4}^{6}x_{5}^{7}$,
$Z^{\prime\prime}_{190}=x_{1}^{7}x_{2}x_{3}^{11}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{191}=x_{1}^{7}x_{2}x_{3}^{14}x_{4}^{3}x_{5}^{7}$, |  $Z^{\prime\prime}_{192}=x_{1}^{7}x_{2}x_{3}^{14}x_{4}^{7}x_{5}^{3}$, |  $Z^{\prime\prime}_{193}=x_{1}^{7}x_{2}x_{3}^{15}x_{4}^{2}x_{5}^{7}$,
$Z^{\prime\prime}_{194}=x_{1}^{7}x_{2}x_{3}^{15}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{195}=x_{1}^{7}x_{2}x_{3}^{15}x_{4}^{6}x_{5}^{3}$, |  $Z^{\prime\prime}_{196}=x_{1}^{7}x_{2}x_{3}^{15}x_{4}^{7}x_{5}^{2}$, |  $Z^{\prime\prime}_{197}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{6}x_{5}^{15}$,
$Z^{\prime\prime}_{198}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{7}x_{5}^{14}$, |  $Z^{\prime\prime}_{199}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{14}x_{5}^{7}$, |  $Z^{\prime\prime}_{200}=x_{1}^{7}x_{2}^{3}x_{3}x_{4}^{15}x_{5}^{6}$, |  $Z^{\prime\prime}_{201}=x_{1}^{7}x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{15}$,
$Z^{\prime\prime}_{202}=x_{1}^{7}x_{2}^{3}x_{3}^{3}x_{4}^{5}x_{5}^{14}$, |  $Z^{\prime\prime}_{203}=x_{1}^{7}x_{2}^{3}x_{3}^{3}x_{4}^{7}x_{5}^{12}$, |  $Z^{\prime\prime}_{204}=x_{1}^{7}x_{2}^{3}x_{3}^{3}x_{4}^{12}x_{5}^{7}$, |  $Z^{\prime\prime}_{205}=x_{1}^{7}x_{2}^{3}x_{3}^{3}x_{4}^{13}x_{5}^{6}$,
$Z^{\prime\prime}_{206}=x_{1}^{7}x_{2}^{3}x_{3}^{3}x_{4}^{15}x_{5}^{4}$, |  $Z^{\prime\prime}_{207}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{15}$, |  $Z^{\prime\prime}_{208}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{3}x_{5}^{14}$, |  $Z^{\prime\prime}_{209}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{11}$,
$Z^{\prime\prime}_{210}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{7}x_{5}^{10}$, |  $Z^{\prime\prime}_{211}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{10}x_{5}^{7}$, |  $Z^{\prime\prime}_{212}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{11}x_{5}^{6}$, |  $Z^{\prime\prime}_{213}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{14}x_{5}^{3}$,
$Z^{\prime\prime}_{214}=x_{1}^{7}x_{2}^{3}x_{3}^{5}x_{4}^{15}x_{5}^{2}$, |  $Z^{\prime\prime}_{215}=x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{4}x_{5}^{14}$, |  $Z^{\prime\prime}_{216}=x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{4}^{3}x_{5}^{12}$, |  $Z^{\prime\prime}_{217}=x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{4}^{5}x_{5}^{10}$,
$Z^{\prime\prime}_{218}=x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{4}^{7}x_{5}^{8}$, |  $Z^{\prime\prime}_{219}=x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{4}^{9}x_{5}^{6}$, |  $Z^{\prime\prime}_{220}=x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{4}^{13}x_{5}^{2}$, |  $Z^{\prime\prime}_{221}=x_{1}^{7}x_{2}^{3}x_{3}^{9}x_{4}^{6}x_{5}^{7}$,
$Z^{\prime\prime}_{222}=x_{1}^{7}x_{2}^{3}x_{3}^{9}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{223}=x_{1}^{7}x_{2}^{3}x_{3}^{11}x_{4}^{5}x_{5}^{6}$, |  $Z^{\prime\prime}_{224}=x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{4}^{2}x_{5}^{7}$, |  $Z^{\prime\prime}_{225}=x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{4}^{3}x_{5}^{6}$,
$Z^{\prime\prime}_{226}=x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{4}^{6}x_{5}^{3}$, |  $Z^{\prime\prime}_{227}=x_{1}^{7}x_{2}^{3}x_{3}^{13}x_{4}^{7}x_{5}^{2}$, |  $Z^{\prime\prime}_{228}=x_{1}^{7}x_{2}^{3}x_{3}^{15}x_{4}x_{5}^{6}$, |  $Z^{\prime\prime}_{229}=x_{1}^{7}x_{2}^{3}x_{3}^{15}x_{4}^{3}x_{5}^{4}$,
$Z^{\prime\prime}_{230}=x_{1}^{7}x_{2}^{3}x_{3}^{15}x_{4}^{5}x_{5}^{2}$, |  $Z^{\prime\prime}_{231}=x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{2}x_{5}^{15}$, |  $Z^{\prime\prime}_{232}=x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{3}x_{5}^{14}$, |  $Z^{\prime\prime}_{233}=x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{6}x_{5}^{11}$,
$Z^{\prime\prime}_{234}=x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{7}x_{5}^{10}$, |  $Z^{\prime\prime}_{235}=x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{10}x_{5}^{7}$, |  $Z^{\prime\prime}_{236}=x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{11}x_{5}^{6}$, |  $Z^{\prime\prime}_{237}=x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{14}x_{5}^{3}$,
$Z^{\prime\prime}_{238}=x_{1}^{7}x_{2}^{7}x_{3}x_{4}^{15}x_{5}^{2}$, |  $Z^{\prime\prime}_{239}=x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{4}x_{5}^{14}$, |  $Z^{\prime\prime}_{240}=x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{4}^{3}x_{5}^{12}$, |  $Z^{\prime\prime}_{241}=x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{4}^{5}x_{5}^{10}$,
$Z^{\prime\prime}_{242}=x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{4}^{7}x_{5}^{8}$, |  $Z^{\prime\prime}_{243}=x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{4}^{9}x_{5}^{6}$, |  $Z^{\prime\prime}_{244}=x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{4}^{13}x_{5}^{2}$, |  $Z^{\prime\prime}_{245}=x_{1}^{7}x_{2}^{7}x_{3}^{7}x_{4}x_{5}^{10}$,
$Z^{\prime\prime}_{246}=x_{1}^{7}x_{2}^{7}x_{3}^{7}x_{4}^{3}x_{5}^{8}$, |  $Z^{\prime\prime}_{247}=x_{1}^{7}x_{2}^{7}x_{3}^{7}x_{4}^{8}x_{5}^{3}$, |  $Z^{\prime\prime}_{248}=x_{1}^{7}x_{2}^{7}x_{3}^{7}x_{4}^{9}x_{5}^{2}$, |  $Z^{\prime\prime}_{249}=x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{2}x_{5}^{7}$,
$Z^{\prime\prime}_{250}=x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{251}=x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{6}x_{5}^{3}$, |  $Z^{\prime\prime}_{252}=x_{1}^{7}x_{2}^{7}x_{3}^{9}x_{4}^{7}x_{5}^{2}$, |  $Z^{\prime\prime}_{253}=x_{1}^{7}x_{2}^{7}x_{3}^{11}x_{4}x_{5}^{6}$,
$Z^{\prime\prime}_{254}=x_{1}^{7}x_{2}^{7}x_{3}^{11}x_{4}^{5}x_{5}^{2}$, |  $Z^{\prime\prime}_{255}=x_{1}^{7}x_{2}^{7}x_{3}^{15}x_{4}x_{5}^{2}$, |  $Z^{\prime\prime}_{256}=x_{1}^{7}x_{2}^{11}x_{3}x_{4}^{6}x_{5}^{7}$, |  $Z^{\prime\prime}_{257}=x_{1}^{7}x_{2}^{11}x_{3}x_{4}^{7}x_{5}^{6}$,
$Z^{\prime\prime}_{258}=x_{1}^{7}x_{2}^{11}x_{3}^{3}x_{4}^{5}x_{5}^{6}$, |  $Z^{\prime\prime}_{259}=x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{4}^{2}x_{5}^{7}$, |  $Z^{\prime\prime}_{260}=x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{261}=x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{4}^{6}x_{5}^{3}$,
$Z^{\prime\prime}_{262}=x_{1}^{7}x_{2}^{11}x_{3}^{5}x_{4}^{7}x_{5}^{2}$, |  $Z^{\prime\prime}_{263}=x_{1}^{7}x_{2}^{11}x_{3}^{7}x_{4}x_{5}^{6}$, |  $Z^{\prime\prime}_{264}=x_{1}^{7}x_{2}^{11}x_{3}^{7}x_{4}^{5}x_{5}^{2}$, |  $Z^{\prime\prime}_{265}=x_{1}^{7}x_{2}^{15}x_{3}x_{4}^{2}x_{5}^{7}$,
$Z^{\prime\prime}_{266}=x_{1}^{7}x_{2}^{15}x_{3}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{267}=x_{1}^{7}x_{2}^{15}x_{3}x_{4}^{6}x_{5}^{3}$, |  $Z^{\prime\prime}_{268}=x_{1}^{7}x_{2}^{15}x_{3}x_{4}^{7}x_{5}^{2}$, |  $Z^{\prime\prime}_{269}=x_{1}^{7}x_{2}^{15}x_{3}^{3}x_{4}x_{5}^{6}$,
$Z^{\prime\prime}_{270}=x_{1}^{7}x_{2}^{15}x_{3}^{3}x_{4}^{3}x_{5}^{4}$, |  $Z^{\prime\prime}_{271}=x_{1}^{7}x_{2}^{15}x_{3}^{3}x_{4}^{5}x_{5}^{2}$, |  $Z^{\prime\prime}_{272}=x_{1}^{7}x_{2}^{15}x_{3}^{7}x_{4}x_{5}^{2}$, |  $Z^{\prime\prime}_{273}=x_{1}^{15}x_{2}x_{3}^{2}x_{4}^{7}x_{5}^{7}$,
$Z^{\prime\prime}_{274}=x_{1}^{15}x_{2}x_{3}^{3}x_{4}^{6}x_{5}^{7}$, |  $Z^{\prime\prime}_{275}=x_{1}^{15}x_{2}x_{3}^{3}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{276}=x_{1}^{15}x_{2}x_{3}^{6}x_{4}^{3}x_{5}^{7}$, |  $Z^{\prime\prime}_{277}=x_{1}^{15}x_{2}x_{3}^{6}x_{4}^{7}x_{5}^{3}$,
$Z^{\prime\prime}_{278}=x_{1}^{15}x_{2}x_{3}^{7}x_{4}^{2}x_{5}^{7}$, |  $Z^{\prime\prime}_{279}=x_{1}^{15}x_{2}x_{3}^{7}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{280}=x_{1}^{15}x_{2}x_{3}^{7}x_{4}^{6}x_{5}^{3}$, |  $Z^{\prime\prime}_{281}=x_{1}^{15}x_{2}x_{3}^{7}x_{4}^{7}x_{5}^{2}$,
$Z^{\prime\prime}_{282}=x_{1}^{15}x_{2}^{3}x_{3}x_{4}^{6}x_{5}^{7}$, |  $Z^{\prime\prime}_{283}=x_{1}^{15}x_{2}^{3}x_{3}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{284}=x_{1}^{15}x_{2}^{3}x_{3}^{3}x_{4}^{4}x_{5}^{7}$, |  $Z^{\prime\prime}_{285}=x_{1}^{15}x_{2}^{3}x_{3}^{3}x_{4}^{5}x_{5}^{6}$,
$Z^{\prime\prime}_{286}=x_{1}^{15}x_{2}^{3}x_{3}^{3}x_{4}^{7}x_{5}^{4}$, |  $Z^{\prime\prime}_{287}=x_{1}^{15}x_{2}^{3}x_{3}^{5}x_{4}^{2}x_{5}^{7}$, |  $Z^{\prime\prime}_{288}=x_{1}^{15}x_{2}^{3}x_{3}^{5}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{289}=x_{1}^{15}x_{2}^{3}x_{3}^{5}x_{4}^{6}x_{5}^{3}$,
$Z^{\prime\prime}_{290}=x_{1}^{15}x_{2}^{3}x_{3}^{5}x_{4}^{7}x_{5}^{2}$, |  $Z^{\prime\prime}_{291}=x_{1}^{15}x_{2}^{3}x_{3}^{7}x_{4}x_{5}^{6}$, |  $Z^{\prime\prime}_{292}=x_{1}^{15}x_{2}^{3}x_{3}^{7}x_{4}^{3}x_{5}^{4}$, |  $Z^{\prime\prime}_{293}=x_{1}^{15}x_{2}^{3}x_{3}^{7}x_{4}^{5}x_{5}^{2}$,
$Z^{\prime\prime}_{294}=x_{1}^{15}x_{2}^{7}x_{3}x_{4}^{2}x_{5}^{7}$, |  $Z^{\prime\prime}_{295}=x_{1}^{15}x_{2}^{7}x_{3}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{296}=x_{1}^{15}x_{2}^{7}x_{3}x_{4}^{6}x_{5}^{3}$, |  $Z^{\prime\prime}_{297}=x_{1}^{15}x_{2}^{7}x_{3}x_{4}^{7}x_{5}^{2}$,
$Z^{\prime\prime}_{298}=x_{1}^{15}x_{2}^{7}x_{3}^{3}x_{4}x_{5}^{6}$, |  $Z^{\prime\prime}_{299}=x_{1}^{15}x_{2}^{7}x_{3}^{3}x_{4}^{3}x_{5}^{4}$, |  $Z^{\prime\prime}_{300}=x_{1}^{15}x_{2}^{7}x_{3}^{3}x_{4}^{5}x_{5}^{2}$, |  $Z^{\prime\prime}_{301}=x_{1}^{15}x_{2}^{7}x_{3}^{7}x_{4}x_{5}^{2}$,
$Z^{\prime\prime}_{302}=x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{4}^{11}x_{5}^{4}$, |  $Z^{\prime\prime}_{303}=x_{1}^{3}x_{2}^{7}x_{3}^{11}x_{4}^{4}x_{5}^{7}$, |  $Z^{\prime\prime}_{304}=x_{1}^{3}x_{2}^{7}x_{3}^{7}x_{4}^{8}x_{5}^{7}$, |  $Z^{\prime\prime}_{305}=x_{1}^{3}x_{2}^{7}x_{3}^{11}x_{4}^{7}x_{5}^{4}$,
$Z^{\prime\prime}_{306}=x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{4}^{11}x_{5}^{4}$, |  $Z^{\prime\prime}_{307}=x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{4}^{11}x_{5}^{4}$, |  $Z^{\prime\prime}_{308}=x_{1}^{7}x_{2}^{7}x_{3}^{11}x_{4}^{3}x_{5}^{4}$, |  $Z^{\prime\prime}_{309}=x_{1}^{7}x_{2}^{11}x_{3}^{7}x_{4}^{3}x_{5}^{4}$,
$Z^{\prime\prime}_{310}=x_{1}^{7}x_{2}^{9}x_{3}^{7}x_{4}^{3}x_{5}^{6}$, |  $Z^{\prime\prime}_{311}=x_{1}^{7}x_{2}^{3}x_{3}^{11}x_{4}^{4}x_{5}^{7}$, |  $Z^{\prime\prime}_{312}=x_{1}^{7}x_{2}^{11}x_{3}^{3}x_{4}^{4}x_{5}^{7}$, |  $Z^{\prime\prime}_{313}=x_{1}^{7}x_{2}^{9}x_{3}^{3}x_{4}^{6}x_{5}^{7}$,
$Z^{\prime\prime}_{314}=x_{1}^{7}x_{2}^{7}x_{3}^{3}x_{4}^{8}x_{5}^{7}$, |  $Z^{\prime\prime}_{315}=x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}^{3}x_{5}^{7}$, |  $Z^{\prime\prime}_{316}=x_{1}^{7}x_{2}^{3}x_{3}^{7}x_{4}^{8}x_{5}^{7}$, |  $Z^{\prime\prime}_{317}=x_{1}^{7}x_{2}^{3}x_{3}^{11}x_{4}^{7}x_{5}^{4}$,
$Z^{\prime\prime}_{318}=x_{1}^{7}x_{2}^{11}x_{3}^{3}x_{4}^{7}x_{5}^{4}$, |  $Z^{\prime\prime}_{319}=x_{1}^{7}x_{2}^{9}x_{3}^{3}x_{4}^{7}x_{5}^{6}$, |  $Z^{\prime\prime}_{320}=x_{1}^{7}x_{2}^{7}x_{3}^{8}x_{4}^{7}x_{5}^{3}$. |

###  3.9 An algorithm in MAGMA

In this subsection, to verify the dimension of the space $Q^{\otimes 5}$ in
degree $N=31$ and degree $N=32,$ we provide an algorithm in MAGMA, which is
established as follows:

⬇

N := 31;

R<x1,x2,x3,x4,x5> := PolynomialRing(GF(2),5);

P<t> := PolynomialRing(R);

Sqhom := hom<R->P | [x + t*x^2 : x in [x1,x2,x3,x4,x5]]>;

function Sq(j,x)

c := Coefficients(Sqhom(x));

if j+1 gt #c then

return R!0;

else

return c[j+1];

end if;

end function;

M := MonomialsOfDegree(R,N);

V := VectorSpace(GF(2),#M);

function MtoV(m)

if IsZero(m) then

return V!0;

else

cv := [Index(M,mm) : mm in Terms(m)];

return CharacteristicVector(V,cv);

end if;

end function;

M1 := MonomialsOfDegree(R,N-1);

M2 := MonomialsOfDegree(R,N-2);

M4 := MonomialsOfDegree(R,N-4);

M8 := MonomialsOfDegree(R,N-8);

M16 := MonomialsOfDegree(R,N-16);

print [#m : m in [M1,M2,M4,M8, M16]];

¿ output: [46376, 40920, 31465, 17550, 3876]

⬇

S1 := [Sq(1,x) : x in M1];

print "S1";

H1 := sub<V | [MtoV(x) : x in S1]>;

print Dimension(H1);

¿ output: 24615

⬇

S2 := [Sq(2,x) : x in M2];

print "S2";

H2 := sub<V | [MtoV(x) : x in S2]>;

print Dimension(H2);

¿ output: 28665

⬇

H := H1 + H2;

print Dimension(H);

¿ output: 43334

⬇

S4 := [Sq(4,x) : x in M4];

print "S4";

H4 := sub<V | [MtoV(x): x in S4]>;

print Dimension(H4);

¿ output: 26520

⬇

H := H + H4;

print Dimension(H);

¿ output: 49530

⬇

S8 := [Sq(8,x) : x in M8];

print "S8";

H8 := sub<V | [MtoV(x) : x in S8]>;

print Dimension(H8);

¿ output: 15900

⬇

H := H + H8;

print Dimension(H);

¿ output: 51494

⬇

S16 := [Sq(16,x) : x in M16];

print "S16";

H16 := sub<V | [MtoV(x) : x in S16]>;

print Dimension(H16);

¿ output: 0

⬇

H := H + H16;

print Dimension(H);

¿ output: 51494

⬇

print "\n",Dimension(V);

¿ output: 52360

⬇

print Dimension(H);

¿ output: 51494

⬇

print Dimension(V) - Dimension(H);

The calculations indicated that

$\dim Q^{\otimes 5}_{31}=\dim V-\dim H=52360-51494=866.$

Replacing $N=31$ by $N=32$ in the above algorithm, we obtain

$\dim Q^{\otimes 5}_{32}=\dim V-\dim H=58905-57901=1004.$

Remark. To compute the hit monomials and the inadmissible monomials, we only
need to consider the effects of the Steenrod squaring operations
$Sq^{j}:\mathbb{P}^{\otimes 5}_{N-j}\to\mathbb{P}^{\otimes 5}_{N}$ for
$j=2^{t},\,t\geq 0,\,2j\leq N.$ This can be explained as follows: Because the
set of all these squares are a minimal generating set for $\mathscr{A}_{2}$ as
an algebra over $\mathbb{Z}_{2}$. Further, as well known, for any
$X\in\mathbb{P}_{N-j}^{\otimes 5}$, $Sq^{2^{t}}(X)=0$ if $\deg(X)=N-j<2^{t}.$
And therefore we need only to compute the action of the squares $Sq^{2^{t}}$
on $\mathbb{P}_{N}^{\otimes 5}$ such that $2^{t}\leq\deg(X)$. More
importantly, by using Kameko’s homomorphism and changing the number of
variables in the polynomial ring ”$R<x_{1},x_{2},\ldots,x_{5}>$” and degree
”$N$” in the above algorithm, we can verify all the results of Peterson [20],
Kameko [13] and Sum [34] on the dimension of the space (1.2) for $t\leq 4.$

## References

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