$T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{T}_{2,4,4}$. Then there exist a unique
non-crossing pairing $\sigma\in NC_{2}(m_{1},m_{2},m_{3})$ such that
$\gamma_{m_{1},m_{2},m_{3}}\sigma=\pi$.

###### Proof.

We proceed as before, it is enough to prove that there exist a unique pairing
$\sigma\in\mathcal{P}_{2}(m)$ satisfying,

  1. $(i)$

$\sigma\vee\gamma_{m_{1},m_{2},m_{3}}=1_{m}$

  2. $(ii)$

if $\\{u,v\\}$ is a block of $\sigma$ then $e_{u}$ and $e_{v}$ connect the
same pair of vertices and have the opposite orientation in
$T_{m_{1},m_{2},m_{3}}^{\pi}$.

As pointed in Theorem 8.10 the edges of multiplicity $2$ of
$\overline{T_{m_{1},m_{2},m_{3}}^{\pi}}$ determines uniquely the blocks of any
$\sigma$ satisfying $(ii)$. Let
$\overline{e}=\\{e_{i_{1}},e_{i_{2}},e_{i_{3}},e_{i_{4}}\\}$ and
$\overline{e^{\prime}}=\\{e_{j_{1}},e_{j_{2}},e_{j_{3}},e_{j_{4}}\\}$ be the
edges of multiplicity $4$ of $\overline{T_{m_{1},m_{2},m_{3}}^{\pi}}$. Without
loss of generality assume $\overline{e}$ is a $(1,2)$-edge and
$\overline{e^{\prime}}$ is a $(1,3)$-edge such that
$e_{i_{1}},e_{i_{2}},e_{j_{1}},e_{j_{2}}\in E_{1}$, $e_{i_{3}},e_{i_{4}}\in
E_{2}$, $e_{j_{3}},e_{j_{4}}\in E_{3}$ and with $e_{i_{1}}$ and $e_{i_{3}}$
having the same orientation and $e_{j_{1}}$ and $e_{j_{3}}$ having the same
orientation. Any $\sigma$ satisfying condition $(ii)$ must pair only elements
in the same equivalence class and with opposite orientations, so the possible
$\sigma_{i}$ restricted to $\\{i_{1},\dots,i_{4},j_{1},\dots,j_{4}\\}$ are
given in next table.

$\sigma_{i}$ | Blocks of $\sigma_{i}$ | $\gamma_{m_{1},m_{2},m_{3}}\vee\sigma_{i}$
---|---|---
$\sigma_{1}$ | $\\{i_{1},i_{2}\\},\\{i_{3},i_{4}\\},\\{j_{1},j_{2}\\},\\{j_{3},j_{4}\\}$ | $\gamma_{m_{1},m_{2},m_{3}}$
$\sigma_{2}$ | $\\{i_{1},i_{2}\\},\\{i_{3},i_{4}\\},\\{j_{1},j_{4}\\},\\{j_{2},j_{3}\\}$ | $[\\![m_{1}]\\!]\cup[\\![m_{3}]\\!],[\\![m_{2}]\\!]$
$\sigma_{3}$ | $\\{i_{1},i_{4}\\},\\{i_{2},i_{3}\\},\\{j_{1},j_{2}\\},\\{j_{3},j_{4}\\}$ | $[\\![m_{1}]\\!]\cup[\\![m_{2}]\\!],[\\![m_{3}]\\!]$
$\sigma_{4}$ | $\\{i_{1},i_{4}\\},\\{i_{2},i_{3}\\},\\{j_{1},j_{4}\\},\\{j_{2},j_{3}\\}$ | $1_{m}$

Last column shows that only $\sigma_{4}$ satisfies condition $(i)$ and so that
is the only pairing. ∎

######  Lemma 8.12.

Let $\pi\in\mathcal{P}(m)$ and let
$T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{UL}_{2,4}$. Then there exist exactly
two non-crossing pairings $\sigma_{1},\sigma_{2}\in NC_{2}(m_{1},m_{2},m_{3})$
such that $\gamma_{m_{1},m_{2},m_{3}}\sigma_{i}=\pi$ for $i=1,2$.

###### Proof.

It suffices to prove that there exist exactly two pairings
$\sigma_{1},\sigma_{2}\in\mathcal{P}_{2}(m)$ satisfying,

  1. $(i)$

$\sigma_{i}\vee\gamma_{m_{1},m_{2},m_{3}}=1_{m}$

  2. $(ii)$

if $\\{u,v\\}$ is a block of $\sigma_{i}$ then $e_{u}$ and $e_{v}$ connect the
same pair of vertices and have the opposite orientation in
$T_{m_{1},m_{2},m_{3}}^{\pi}$.

We know that edges of multiplicity $2$ of
$\overline{T_{m_{1},m_{2},m_{3}}^{\pi}}$ determines uniquely the blocks of any
$\sigma$ satisfying $(ii)$. Let
$\overline{e}=\\{e_{i_{1}},e_{i_{2}},e_{i_{3}},e_{i_{4}}\\}$ be the edge of
multiplicity $4$ of $\overline{T_{m_{1},m_{2},m_{3}}^{\pi}}$. Without loss of
generality assume $e_{i_{1}},e_{i_{2}}\in E_{1}$, $e_{i_{3}}\in E_{2}$,
$e_{i_{4}}\in E_{3}$. As this edge of multiplicity $4$ is a loop then any
pairing of $\\{i_{1}.i_{2},i_{3},i_{4}\\}$ satisfies $(ii)$, so the possible
pairings $\sigma_{i}$ restricted to $\\{i_{1},\dots,i_{4}\\}$ are given in
next table,

$\sigma_{i}$ | Blocks of $\sigma_{i}$ | $\gamma_{m_{1},m_{2},m_{3}}\vee\sigma_{i}$
---|---|---
$\sigma_{1}$ | $\\{i_{1},i_{2}\\},\\{i_{3},i_{4}\\}$ | $[\\![m_{1}]\\!]\cup[\\![m_{2}]\\!],[\\![m_{3}]\\!]$
$\sigma_{2}$ | $\\{i_{1},i_{3}\\},\\{i_{2},i_{4}\\}$ | $1_{m}$
$\sigma_{3}$ | $\\{i_{1},i_{4}\\},\\{i_{2},i_{3}\\}$ | $1_{m}$

Last column shows that only $\sigma_{2}$ and $\sigma_{3}$ satisfies condition
$(i)$ and so those are the only pairings satisfying both conditions. ∎

######  Lemma 8.13.

Let $\pi\in\mathcal{P}(m)$ and let
$T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{UC}_{2,4}$, then there exist a unique
non-crossing pairings $\sigma\in NC_{2}(m_{1},m_{2},m_{3})$ such that
$\gamma_{m_{1},m_{2},m_{3}}\sigma=\pi$.

###### Proof.

It suffices to prove that there exist a unique pairings
$\sigma\in\mathcal{P}_{2}(m)$ satisfying,

  1. $(i)$

$\sigma\vee\gamma_{m_{1},m_{2},m_{3}}=1_{m}$

  2. $(ii)$

if $\\{u,v\\}$ is a block of $\sigma$ then $e_{u}$ and $e_{v}$ connect the
same pair of vertices and have the opposite orientation in
$T_{m_{1},m_{2},m_{3}}^{\pi}$.

The edges of multiplicity $2$ of $\overline{T_{m_{1},m_{2},m_{3}}^{\pi}}$
determines uniquely the blocks of any $\sigma$ satisfying $(ii)$. It remains
considering the edge of multiplicity $4$. In this case we have two
possibilities.

Case 1. The edge of multiplicity $4$ is not in the circuit of
$\overline{T_{m_{1},m_{2},m_{3}}^{\pi}}$. In this case any edge in the circuit
is connecting. Suppose without loss of generality any edge in the circuit is a
$(2,3)$-edge. This means the edge of multiplicity $4$ is either a $(1,2)$-edge
or a $(1,3)$-edge. Assume it is a $(1,2)$-edge. Let
$\overline{e}=\\{e_{i_{1}},e_{i_{2}},e_{i_{3}},e_{i_{4}}\\}$ be the edge of
multiplicity $4$ with $e_{i_{1}},e_{i_{2}}\in E_{1}$, $e_{i_{3}},e_{i_{4}}\in
E_{2}$ and $e_{i_{1}}$ and $e_{i_{3}}$ having the same orientation. Any
$\sigma$ satisfying $(ii)$ must pair edges in opposite orientations, so the
pairings $\sigma_{i}$ restricted to $\\{i_{1},\dots,i_{4}\\}$ are given in
next table.

$\sigma_{i}$ | Blocks of $\sigma_{i}$ | $\gamma_{m_{1},m_{2},m_{3}}\vee\sigma_{i}$
---|---|---
$\sigma_{1}$ | $\\{i_{1},i_{2}\\},\\{i_{3},i_{4}\\}$ | $[\\![m_{1}]\\!],[\\![m_{2}]\\!]\cup[\\![m_{3}]\\!]$
$\sigma_{2}$ | $\\{i_{1},i_{4}\\},\\{i_{2},i_{3}\\}$ | $1_{m}$

Last column shows that only $\sigma_{1}$ satisfies condition $(i)$ and so that
is the only pairing.
Case 1. The edge of multiplicity $4$ is in the circuit. of
$\overline{T_{m_{1},m_{2},m_{3}}^{\pi}}$. In this case the edge of
multiplicity $4$ consist of two edges from one basic cycle, say $E_{1}$, and
one edge from each $E_{2}$ and $E_{3}$, let
$\overline{e}=\\{e_{i_{1}},e_{i_{2}},e_{i_{3}},e_{i_{4}}\\}$ be the edge of
multiplicity $4$ with $e_{i_{1}},e_{i_{2}}\in E_{1}$, $e_{i_{3}}\in E_{2}$,
$e_{i_{4}}\in E_{3}$ and such that $e_{i_{1}}$ and $e_{i_{3}}$ have the same
orientation. Any $\sigma$ satisfying $(ii)$ must pair edges in opposite
orientations, so the possible pairings $\sigma_{i}$ restricted to
$\\{i_{1},\dots,i_{4}\\}$ are given in next table,

$\sigma_{i}$ | Blocks of $\sigma_{i}$ | $\gamma_{m_{1},m_{2},m_{3}}\vee\sigma_{i}$
---|---|---
$\sigma_{1}$ | $\\{i_{1},i_{2}\\},\\{i_{3},i_{4}\\}$ | $[\\![m_{1}]\\!],[\\![m_{2}]\\!]\cup[\\![m_{3}]\\!]$
$\sigma_{2}$ | $\\{i_{1},i_{4}\\},\\{i_{2},i_{3}\\}$ | $1_{m}$

Last column shows that only $\sigma_{1}$ satisfies condition $(i)$ and so that
is the only pairing satisfying both conditions, in any case we get that there
exist a unique pairing satisfying $(i)$ and $(ii)$. ∎

######  Lemma 8.14.

Let $\pi\in\mathcal{P}(m)$ and let
$T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{DB}$, then there exist a unique non-
crossing pairings $\sigma\in NC_{2}(m_{1},m_{2},m_{3})$ such that
$\gamma_{m_{1},m_{2},m_{3}}\sigma=\pi$.

###### Proof.

It suffices to prove that there exist a unique pairings
$\sigma\in\mathcal{P}_{2}(m)$ satisfying,

  1. $(i)$

$\sigma\vee\gamma_{m_{1},m_{2},m_{3}}=1_{m}$

  2. $(ii)$

if $\\{u,v\\}$ is a block of $\sigma$ then $e_{u}$ and $e_{v}$ connect the
same pair of vertices and have the opposite orientation in
$T_{m_{1},m_{2},m_{3}}^{\pi}$.

Since $T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{DB}$ then any edge has
multiplicity $2$, so we let $\sigma=\overline{\pi}$, note that any edge of
multiplicity $2$ of $\overline{T_{m_{1},m_{2},m_{3}}^{\pi}}$ consist of $2$
edges in opposite orientation (Theorem 6.5), so $\sigma$ satisfies $(ii)$. As
we proved in Theorem 8.10 the edges of multiplicity $2$ of
$\overline{T_{m_{1},m_{2},m_{3}}^{\pi}}$ determines uniquely the blocks of any
$\sigma$ satisfying $(ii)$, so in this case such a $\sigma$ satisfying $(ii)$
is unique and is given by $\sigma=\overline{\pi}$. It remains proving
$\sigma=\overline{\pi}$ satisfies $(i)$, however, this is an immediate
consequence of Theorem 6.5 as $C_{\pi}\neq 0$. ∎

######  Corollary 8.15.

Let $\pi\in\mathcal{P}(m)$ be such that
$T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{LG}$ then there exist $\sigma\in
NC_{2}(m_{1},m_{2},m_{3})$ such that $\pi=\gamma_{m_{1},m_{2},m_{3}}\sigma$,
consequently
$T_{m_{1},m_{2},m_{3}}^{\pi}=T_{m_{1},m_{2},m_{3}}^{\gamma_{m_{1},m_{2},m_{3}}\sigma}$.
Furthermore,

  1. $(i)$

If $T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{T}_{2,6}\cup\mathcal{UL}_{2,4}$
there exist exactly two non-crossing pairings $\sigma_{1},\sigma_{2}\in
NC_{2}(m_{1},m_{2},m_{3})$ such that

$T_{m_{1},m_{2},m_{3}}^{\pi}=T_{m_{1},m_{2},m_{3}}^{\gamma_{m_{1},m_{2},m_{3}}\sigma_{1}}=T_{m_{1},m_{2},m_{3}}^{\gamma_{m_{1},m_{2},m_{3}}\sigma_{2}}.$

  2. $(ii)$

If
$T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{T}_{2,4,4}\cup\mathcal{UC}_{2,4}\cup\mathcal{DB}$
there exist a unique non-crossing pairings $\sigma\in
NC_{2}(m_{1},m_{2},m_{3})$ such that

$T_{m_{1},m_{2},m_{3}}^{\pi}=T_{m_{1},m_{2},m_{3}}^{\gamma_{m_{1},m_{2},m_{3}}\sigma}.$

Corollary 8.15 together with Lemma 8.7 determines the relation between non-
crossing pairing and limit graphs. This is given as follows.

######  Lemma 8.16.

Let $\pi\in\mathcal{P}(m)$ then $T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{LG}$
if and only if there exist $\sigma\in NC_{2}(m_{1},m_{2},m_{3})$ such that
$\pi=\gamma_{m_{1},m_{2},m_{3}}\sigma$, furthermore,

$\displaystyle|NC_{2}(m_{1},m_{2},m_{3})|=|\mathcal{LG}|+|\mathcal{UL}_{2,4}|+|\mathcal{T}_{2,6}|$

###### Proof.

Corollary 8.15 and Lemma 8.7 proves that the mapping,

$T:NC_{2}(m_{1},m_{2},m_{3})\rightarrow\mathcal{LG},$

given by,

$T(\sigma)=T_{m_{1},m_{2},m_{3}}^{\gamma_{m_{1},m_{2},m_{3}}\sigma},$

is surjective which proves the first part of Lemma. For the second part note
that Corollary 8.15 proves that only if
$T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{UL}_{2,4}\cup\mathcal{T}_{2,6}$ there
exist exactly two non-crossing pairings being mapped under $T$ to the same
quotient graph $T_{m_{1},m_{2},m_{3}}^{\pi}$ which proves the second part. ∎

######  Corollary 8.17.

$|\mathcal{DB}|=|NC_{2}(m_{1},m_{2},m_{3})|-|\mathcal{UC}_{2,4}|-|\mathcal{T}_{2,4,4}|-2|\mathcal{UL}_{2,4}|-2|\mathcal{T}_{2,6}|$

Corollary 8.17 reduces the problem of counting all limit graphs to only
counting $\mathcal{T}_{2,6},\mathcal{T}_{2,4,4},\mathcal{UL}_{2,4}$ and
$\mathcal{UC}_{2,4}$, so, for the rest of the paper we will work in counting
each type of these.

###  8.3. Counting double trees and double unicircuit graphs

Our motivation for counting double trees and double unicircuit graphs is that
most of the limit graphs can be expressed in terms of these; in fact double
trees and double unicircuit graphs appear when computing the second order
moments as seen in [8], where they provide a way of counting certain graphs
using the set of non-crossing pairings in an $(m_{1},m_{2})$-annulus. In this
subsection we provide our alternative proof of that result.

######  Lemma 8.18.

Let $m\in\mathbb{N}$, $\pi\in\mathcal{P}(m)$ and $\gamma_{m}=(1,\dots,m)\in
S_{m}$. Let $T_{m}=(V,E)$ be the graph consisting of a single basic cycle. Let
$\pi\in\mathcal{P}(m)$. $T_{m}^{\pi}$ is a double tree if and only if
$\pi=\gamma_{m}\sigma$ for some $\sigma\in NC_{2}(m)$. Moreover if
$\sigma_{1}\neq\sigma_{2}$ then $T_{m}^{\gamma_{m}\sigma_{1}}$ and
$T_{m}^{\gamma_{m}\sigma_{2}}$ are distinct quotient graphs, consequently,

$|\\{\pi\in\mathcal{P}(m):T_{m}^{\pi}\text{ is a double
tree}\\}|=|NC_{2}(m)|.$

###### Proof.

Suppose $T_{m}^{\pi}=(V,E)$ is a double tree, we let
$\sigma=\overline{\pi}\in\mathcal{P}_{2}(m)$. Theorem 8.6 says,
$\gamma_{m}\sigma\leq\pi$, therefore,

$m+1=|V|+|E|=\\#(\pi)+\\#(\sigma)\leq\\#(\gamma_{m}\sigma)+\\#(\sigma)\leq
m+1$

which forces $\pi=\gamma_{m}\sigma$ and $\sigma\in NC_{2}(m)$. Conversely let
$\sigma\in\mathit{NC}_{2}(m)$ and $\pi=\gamma_{m}\sigma$. Note that
$G_{\sigma}^{\gamma_{m}}$ has $\\#(\gamma_{m}\sigma)$ vertices and
$\\#(\sigma)$ edges and it is connected because $T_{m}^{\pi}$ is connected.
Thus,

$1\geq\\#(\gamma_{m}\sigma)-\\#(\sigma)=\\#(\gamma_{m}\sigma)+\\#(\sigma)-m=1,$

therefore all above must be equality which means $G_{\sigma}^{\gamma_{m}}$ is
a tree. On the other hand Theorem 8.4 says
$T(G_{\sigma}^{\gamma_{m}})=T_{m}^{\pi}$, therefore $T_{m}^{\pi}$ is a double
tree, moreover if $\\{u,v\\}$ is a block of $\sigma$ then $e_{u}$ and $e_{v}$
are joining the same pair or vertices of $T_{m}^{\pi}$, i.e.
$\sigma=\overline{\pi}$. The latest observation means that for
$\sigma_{1}\neq\sigma_{2}$ the partitions $\overline{\gamma_{m}\sigma_{1}}$
and $\overline{\gamma_{m}\sigma_{2}}$ are distinct and therefore the quotient
graphs $T_{m}^{\gamma\sigma_{1}}$ and $T_{m}^{\gamma\sigma_{2}}$ must be
different. ∎

To count the double unicyclic graphs we will use the set of non-crossing
pairings on the $(m_{1},m_{2})$-annulus.

######  Lemma 8.19.

Let $m_{1},m_{2}\in\mathbb{N}$, $m=m_{1}+m_{2}$, $\pi\in\mathcal{P}(m)$ and
$\gamma_{m_{1},m_{2}}=(1,\dots,m_{1})(m_{1}+1,\dots,m)\in S_{m}$. Let
$T_{m_{1},m_{2}}$ be defined as in Section 5. Let $k\in\mathbb{N}$ with $k\neq
2$. $T^{\pi}_{m_{1},m_{2}}$ is a double unicircuit graph where the unique
circuit of $\overline{T_{m_{1},m_{2}}^{\pi}}$ has length $k$ if and only if
there exist $\sigma\in NC_{2}^{(k)}(m_{1},m_{2})$ such that
$\pi=\gamma_{m_{1},m_{2}}\sigma$. Moreover for $\sigma_{1}\neq\sigma_{2}$ the
quotient graphs $T_{m_{1},m_{2}}^{\gamma_{m_{1},m_{2}}\sigma_{1}}$ and
$T_{m_{1},m_{2}}^{\gamma_{m_{1},m_{2}}\sigma_{2}}$ are distinct.

###### Proof.

Suppose $T^{\pi}_{m_{1},m_{2}}$ is a double unicircuit graph where the unique
circuit of $\overline{T_{m_{1},m_{2}}^{\pi}}$ has length $k$.
$\overline{T_{m_{1},m_{2}}^{\pi}}$ is a connected graph with $\\#(\pi)$
vertices and $m/2$ edges, therefore $q(\pi)=\\#(\pi)-m/2=0$. Let
$\sigma\in\mathcal{P}(m)$ be the partition defined by
$u\overset{\sigma}{\sim}v$ if $e_{u}$ and $e_{v}$ connect the same pair of
vertices of $T_{m_{1},m_{2}}^{\pi}$ then $\sigma$ is a pairing satisfying
$\sigma\vee\gamma_{m_{1},m_{2}}=1_{m}$ and if $\\{u,v\\}$ is a block of
$\sigma$ then $e_{u}$ and $e_{v}$ have the opposite orientation, Theorem 8.8
says that $\sigma\in NC_{2}(m_{1},m_{2})$ and
$\pi=\gamma_{m_{1},m_{2}}\sigma$. Moreover, by definition of $\sigma$ the
through strings of $\sigma$ correspond to the edges in the circuit of
$\overline{T_{m_{1},m_{2}}^{\pi}}$, so $\sigma\in NC_{2}^{(k)}(m_{1},m_{2})$.
Conversely, let $\sigma\in NC_{2}^{(k)}(m_{1},m_{2})$ and
$\pi=\gamma_{m_{1},m_{2}}\sigma$. Let $(V,E)$ be the graph
$G_{\sigma}^{\gamma_{m_{1},m_{2}}}$, Lemma 8.5 says
$\overline{T^{\gamma_{m_{1},m_{2}}\sigma}_{m_{1},m_{2}}}$ is a connected graph
and so is $G_{\sigma}^{\gamma_{m_{1},m_{2}}}$. Note,

$\displaystyle|V|-|E|$ $\displaystyle=$
$\displaystyle\\#(\gamma_{m_{1},m_{2}}\sigma)-\\#(\sigma)=\\#(\gamma_{m_{1},m_{2}}\sigma)+\\#(\sigma)-m=0,$

thus, $G_{\sigma}^{\gamma_{m_{1},m_{2}}}$ is a unicircuit graph. Let
$B=\\{u,v\\}$ be a block of $\sigma$ not in the circuit of
$G_{\sigma}^{\gamma_{m_{1},m_{2}}}$, then $B$ is a cutting edge of
$G_{\sigma}^{\gamma_{m_{1},m_{2}}}$ and so
$\overline{e_{u}}=\\{e_{u},e_{v}\\}$ is a cutting edge of
$\overline{T_{m_{1},m_{2}}^{\gamma_{m_{1},m_{2}}\sigma}}$, that forces to
$e_{u}$ and $e_{v}$ being from the same basic cycle, and so $B$ is a non-
through string of $\sigma$, that means that all through strings
$B_{1},\dots,B_{k}$ of $\sigma$ must correspond to edges in the circuit of
$G_{\sigma}^{\gamma_{m_{1},m_{2}}}$. Suppose one of the edges in the circuit
is not a through string, then there is a vertex of
$G_{\sigma}^{\gamma_{m_{1},m_{2}}}$, $V$, in its unique circuit whose two
adjacent edges in within the circuit correspond to one non-through string and
one through string. When doubling the edges of
$G_{\sigma}^{\gamma_{m_{1},m_{2}}}$ we obtain that $V$ will be adjacent to an
odd number of edges from $E_{1}$ (the trough string produces $1$ adjacent edge
from $E_{1}$ and any non-trough string produces either $0$ or $2$ adjacent
edges from $E_{1}$), this is not possible as $\deg(V)_{1}$ must be even, so
all edges in the circuit of $G_{\sigma}^{\gamma_{m_{1},m_{2}}}$ must
correspond to through strings of $\sigma$, i.e
$G_{\sigma}^{\gamma_{m_{1},m_{2}}}$ is a graph with a single circuit of length
$k$. Remember that doubling the edges of $G_{\sigma}^{\gamma_{m_{1},m_{2}}}$
produces the graph $T^{\gamma_{m_{1},m_{2}}\sigma}_{m_{1},m_{2}}$ and since
the there are no edges of $G_{\sigma}^{\gamma_{m_{1},m_{2}}}$ connecting the
same pair of vertices (that would mean having a circuit of length $2$ and we
assumed $k\neq 2$), then $T^{\gamma_{m_{1},m_{2}}\sigma}_{m_{1},m_{2}}$ is a
graph such that $\overline{T^{\gamma_{m_{1},m_{2}}\sigma}_{m_{1},m_{2}}}$ has
a unique circuit and all its edges have multiplicity $2$ and consist of two
edges in opposite orientation. Moreover, it must be that
$\sigma=\overline{\pi}$ because $\sigma\leq\overline{\pi}$ (Theorem 8.5) and
both have only blocks of size $2$. To verify $T_{m_{1},m_{2}}^{\pi}$ is a
double unicircuit graph it remains to prove that all edges in the circuit of
$\overline{T_{m_{1},m_{2}}^{\pi}}$ correspond to non-through strings, this
follows because all edges in the circuit of $\overline{T_{m_{1},m_{2}}^{\pi}}$
correspond to edges in the circuit of $G_{\sigma}^{\gamma_{m_{1},m_{2}}}$ and
$\sigma=\overline{\pi}$. Finally the condition $\sigma=\overline{\pi}$ proves
that if $\sigma_{1}\neq\sigma_{2}$ then
$T_{m_{1},m_{2}}^{\gamma_{m_{1},m_{2}}\sigma_{1}}$ and
$T_{m_{1},m_{2}}^{\gamma_{m_{1},m_{2}}\sigma_{2}}$ are distinct. ∎

The following is a consequence of Lemma 8.19.

######  Corollary 8.20.

Let $k\in\mathbb{N}$ with $k\neq 2$. The following are satisfied.

  1. $(i)$

$|NC_{2}^{(k)}(m_{1},m_{2})|=|\\{\pi\in\mathcal{P}(m):T_{m_{1},m_{2}}^{\pi}\text{
is a double unicircuit graph}\\\ \text{ with
}\overline{T_{m_{1},m_{2}}^{\pi}}\text{ having a \text{circuit }of length
}k\\}|$

  2. $(ii)$

$|NC_{2}(m_{1},m_{2})\setminus
NC_{2}^{(2)}(m_{1},m_{2})|=|NC_{2}(m_{1},m_{2})|-|NC_{2}^{(2)}(m_{1},m_{2})|\\\
=|\\{\pi\in\mathcal{P}(m):T_{m_{1},m_{2}}^{\pi}\text{ is a double unicircuit
graph}\\}|$

###  8.4. Counting $2$-$6$ and $2$-$4$-$4$-tree types

Counting $2$-$6$ and $2$-$4$-$4$ tree types is an immediate consequence of
counting double trees. For this section we set
$m_{1},m_{2},m_{3}\in\mathbb{N}$ and $m=m_{1}+m_{2}+m_{3}$. Let us introduce
the following set of partitioned permutations.

######  Notation 8.21.

For a partitioned permutation

$(\mathcal{V},\pi)\in\mathcal{PS}_{NC}^{(1,1,1)}(m_{1},m_{2},m_{3})\cup\mathcal{PS}_{NC}^{(2,1,1)}(m_{1},m_{2},m_{3}),$

we can write $\pi$ as $\pi=\pi_{1}\times\pi_{2}\times\pi_{3}$ with $\pi_{i}\in
NC(m_{i})$.

  1. $(i)$

We denote by $\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})$ to the set
of partitioned permutations
$(\mathcal{V},\pi)\in\mathcal{PS}_{NC}^{(1,1,1)}(m_{1},m_{2},m_{3})$ such that
$\pi_{i}\in NC_{2}(m_{i})$ $\forall i=1,2,3$.

  2. $(ii)$

We denote by $\mathcal{PS}_{NC_{2}}^{(2,1,1)}(m_{1},m_{2},m_{3})$ to the set
of partitioned permutations
$(\mathcal{V},\pi)\in\mathcal{PS}_{NC}^{(2,1,1)}(m_{1},m_{2},m_{3})$ such that
$\pi_{i}\in NC_{2}(m_{i})$ $\forall i=1,2,3$.

######  Remark 8.22.

The cardinality of $\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})$ can be
computed easily. Each element
$(\mathcal{V},\pi)\in\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})$ is
such that $\pi=\pi_{1}\times\pi_{2}\times\pi_{3}$ with $\pi_{i}\in
NC_{2}(m_{i})$ and each block of $\mathcal{V}$ is a cycle of $\pi$ except by
one block which is the union of three cycles of $\pi$, thus the number of
permutations $\pi$, can be counted by

$|NC_{2}(m_{1})||NC_{2}(m_{2})||NC_{2}(m_{3})|,$

while the number of partitions $\mathcal{V}$, can be counted by choosing a
cycle from each $\pi_{i}$, which can be done in
$\frac{m_{1}}{2}\frac{m_{2}}{2}\frac{m_{3}}{2}$ ways. Therefore,

$|\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})|=\frac{m_{1}}{2}\frac{m_{2}}{2}\frac{m_{3}}{2}|NC_{2}(m_{1})||NC_{2}(m_{2})||NC_{2}(m_{3})|.$

In a similar manner we can compute the cardinality of
$\mathcal{PS}_{NC}^{(2,1,1)}(m_{1},m_{2},\allowbreak m_{3})$, namely,

$|\mathcal{PS}_{NC_{2}}^{(2,1,1)}(m_{1},m_{2},m_{3})|\\\
=\frac{m_{1}}{2}\frac{m_{2}}{2}\frac{m_{3}}{2}(\frac{m}{2}-3)|NC_{2}(m_{1})||NC_{2}(m_{2})||NC_{2}(m_{3})|.$

######  Lemma 8.23.

$|\mathcal{T}_{2,6}|=4|\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})|$

###### Proof.

For $\pi\in\mathcal{P}(m)$ the quotient graph $T_{m_{1},m_{2},m_{3}}^{\pi}$ is
a $2$-$6$ tree type if all three graphs $T_{m_{1}}^{\pi}$,$T_{m_{2}}^{\pi}$
and $T_{m_{3}}^{\pi}$ are double trees, Lemma 8.18 says that each double tree
can be chosen in $|NC_{2}(m_{1})|$,$|NC_{2}(m_{2})|$ and $|NC_{2}(m_{3})|$
distinct ways respectively. For each choice of double trees we choose an edge
from each one, which can be done in
$\frac{m_{1}}{2}\frac{m_{2}}{2}\frac{m_{3}}{2}$ ways. Finally we can join the
graph along those edges in $4$ different ways depending on the orientation, as
each orientation correspond to a different partition $\pi$, then,

$|\mathcal{T}_{2,6}|=4\frac{m_{1}}{2}\frac{m_{2}}{2}\frac{m_{3}}{2}|NC_{2}(m_{1})||NC_{2}(m_{2})||NC_{2}(m_{3})|.$

As said in Remark 8.22 last expression is precisely fourth times the
cardinality of $\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})$. ∎

######  Lemma 8.24.

$|\mathcal{T}_{2,4,4}|=4|\mathcal{PS}_{NC_{2}}^{(2,1,1)}(m_{1},m_{2},m_{3})|$

###### Proof.

We proceed as in Lemma 8.23. The double trees can be chosen in
$|NC_{2}(m_{1})||NC_{2}(m_{2})||NC_{2}(m_{3})|$ ways. For a choice of the
double trees we choose two edges from one of them and one edge from each of
the other two. Suppose we chose two edges from $\overline{T_{m_{1}}^{\pi}}$.
Those can be chosen in $\frac{1}{2}\frac{m_{1}}{2}(\frac{m_{1}}{2}-1)$ ways.
The edges of the other two double trees can be chosen in
$\frac{m_{2}}{2}\frac{m_{3}}{2}$ ways. Once we selected the edges we have to
make the union along these edges, we can pair them in $2$ different ways and
for each pair there are two possible orientations of the edges, giving a total
of $4$ ways. So the total number of ways is given by,

$8\frac{1}{2}\frac{m_{1}}{2}(\frac{m_{1}}{2}-1)\frac{m_{2}}{2}\frac{m_{3}}{2}=4\frac{m_{1}}{2}(\frac{m_{1}}{2}-1)\frac{m_{2}}{2}\frac{m_{3}}{2}$

Similarly when choosing two edges from $\overline{T_{m_{2}}^{\pi}}$ and
$\overline{T_{m_{3}}^{\pi}}$ we get
$4\frac{m_{2}}{2}(\frac{m_{2}}{2}-1)\frac{m_{1}}{2}\frac{m_{3}}{2}$ and
$4\frac{m_{3}}{2}(\frac{m_{3}}{2}-1)\frac{m_{1}}{2}\frac{m_{2}}{2}$ ways
respectively, so the total number of $2$-$4$-$4$ tree types is given by,

$4\left[\frac{m_{1}}{2}(\frac{m_{1}}{2}-1)\frac{m_{2}}{2}\frac{m_{3}}{2}+\frac{m_{2}}{2}(\frac{m_{2}}{2}-1)\frac{m_{1}}{2}\frac{m_{3}}{2}+\frac{m_{3}}{2}(\frac{m_{3}}{2}-1)\frac{m_{1}}{2}\frac{m_{2}}{2}\right]\\\
|NC_{2}(m_{1})||NC_{2}(m_{2})||NC_{2}(m_{3})|\\\
=4\frac{m_{1}}{2}\frac{m_{2}}{2}\frac{m_{3}}{2}(\frac{m}{2}-3)|NC_{2}(m_{1})||NC_{2}(m_{2})||NC_{2}(m_{3})|.$

This last expression is fourth times the cardinality of
$\mathcal{PS}_{NC_{2}}^{(2,1,1)}(m_{1},m_{2},m_{3})$. ∎

###  8.5. Counting $2$-$4$ uniloop and $2$-$4$ unicircuit types

$2$-$4$-uniloop and $2$-$4$ unicircuit types are made of double unicircuit and
double tree graphs, this allow us to count them in a simple manner. For this
section we set $m_{1},m_{2},m_{3}\in\mathbb{N}$ and $m=m_{1}+m_{2}+m_{3}$
unless other is specified. Let us introduce the following sets of partitioned
permutations.

######  Notation 8.25.

For a partitioned permutation

$(\mathcal{V},\pi)\in\mathcal{PS}_{NC}^{(1,1,1)}(m_{1},m_{2},m_{3}),$

we write $\pi=\pi_{1}\times\pi_{2}\times\pi_{3}$ with $\pi_{i}\in NC(m_{i})$
and each block of $\mathcal{V}$ is a cycle of $\pi$ except by one block which
is the union of three cycles of $\pi$ one from each permutation $\pi_{i}$. We
denote by $\mathcal{PS}_{NC_{2,1,1}}^{(1,1,1)}(m_{1},m_{2},m_{3})$ to the set
of partitioned permutations
$(\mathcal{V},\pi)\in\mathcal{PS}_{NC}^{(1,1,1)}(m_{1},m_{2},m_{3})$
satisfying the following conditions,

  1. $(i)$

$\pi_{i_{1}}$ and $\pi_{i_{2}}$ have all cycles of size $2$ except by one
cycle of size $1$ while $\pi_{i_{3}}\in NC_{2}(m_{i_{3}})$ with
$(i_{1},i_{2},i_{3})$ a permutation of $(1,2,3)$.

  2. $(ii)$

The block of $\mathcal{V}$ which is the union of three cycles of $\pi$
consists of the cycles of size $1$ from $\pi_{i_{1}}$ and $\pi_{i_{2}}$ and
any cycle from $\pi_{i_{3}}$.

An example can be seen in Figure 13

Figure 13.  A partitioned permutation $(\mathcal{V},\pi)$ in the set
$\mathcal{PS}_{NC_{2,1,1}}^{(1,1,1)}(m_{1},m_{2},m_{3})$ corresponding to
$\pi=(1,6)(2,3)(4,5)(7)(8,9)(10)(11,12)(13,14)$
$\mathcal{V}=\\{\\{1,6,7,10\\},\\{2,3\\},\\{4,5\\},\\{8,9\\},\\{11,12\\},\\{13,14\\}\\}.$
Each block of $\mathcal{V}$ is a cycle of $\pi$ except by one block which is
the union of the two cycles of size $1$ of $\pi$ and one cycle of size $2$.

######  Notation 8.26.

For a partitioned permutation

$(\mathcal{V},\pi)\in\mathcal{PS}_{NC}^{(1,1)}(m_{1},m_{2},m_{3}),$

we write $\pi=\pi_{1}\times\pi_{2}$ with $\pi_{1}\in
S_{NC}(m_{i_{1}},m_{i_{2}})$ and $\pi_{2}\in NC(m_{i_{3}})$ for some
permutation $(i_{1},i_{2},i_{3})$ of $(1,2,3)$. Each block of $\mathcal{V}$ is
a cycle of $\pi$ except by one block which is the union of two cycles of $\pi$
one from each permutation $\pi_{i}$.

  1. $(i)$

We denote by $\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})$ to the set of
partitioned permutations
$(\mathcal{V},\pi)\in\mathcal{PS}_{NC}^{(1,1)}(m_{1},m_{2},m_{3})$ such that
$\pi_{1}\in NC_{2}(m_{i_{1}},m_{i_{2}})$ and $\pi_{3}\in NC_{2}(m_{i_{3}})$,

  2. $(ii)$

We denote by $\mathcal{PS}_{NC_{2}^{(2)}}^{(1,1)}(m_{1},m_{2},m_{3})$ to the
set of partitioned permutations
$(\mathcal{V},\pi)\in\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})$ such
that $\pi_{1}$ has exactly two through strings, i.e $\pi_{1}\in
NC_{2}^{(2)}(m_{i_{1}},m_{i_{2}})$.

  3. $(iii)$

We denote by $\mathcal{PS}_{NC_{2}}^{(1,1)(t)}(m_{1},m_{2},m_{3})$ to the set
of partitioned permutations
$(\mathcal{V},\pi)\in\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})$ such
that $\pi_{1}$ has exactly one through string, (i.e $\pi_{1}\in
NC_{2}^{(1)}(m_{i_{1}},m_{i_{2}})$) and the block of $\mathcal{V}$ which is
the union of two cycles of $\pi$ consist of the unique through string of
$\pi_{1}$ and any block of $\pi_{2}$.

Some examples can be seen in Figures 14, 15, and 16.

Figure 14.  $(\mathcal{V},\pi)$ in $\mathcal{P\\!S}_{NC}^{(1,1)}(9,5,4)$ where
$\pi=(1,8)\allowbreak(2,14)(3,4)(5,13)(6,12)(7,11)(9,10)(15,16)(17,18)$ and
$\mathcal{V}=\\{\\{1,8\\},\\{2,14,15,16\\},\\{3,4\\},\allowbreak\\{5,13\\},\\{6,12\\},\\{7,11\\},\\{9,10\\},\\{17,18\\}\\}$.

######  Definition 8.27.

Let $n\in\mathbb{N}$. We define the set of block-pairings on $[n]$, which we
denote by $NC_{2}^{block}(n)$, as the set of pairs $(B,\sigma)$ where
$\sigma\in NC_{2}(n)$ and $B$ is a block of $\sigma$.

######  Remark 8.28.

The cardinality of $NC_{2}^{block}(n)$ is given by $\frac{n}{2}|NC_{2}(n)|$ as
for each non-crossing pairing there are $\frac{n}{2}$ blocks to be chosen.

######  Definition 8.29.

Let $n\in\mathbb{N}$ and $T_{n}$ be the graph consisting of a unique basic
cycle defined as in Section 5. For a partition $\pi\in\mathcal{P}(n)$, we say
that the graph $T_{n}^{\pi}$ is a double uniloop graph if $T_{n}^{\pi}$
consists of a graph with two loops over the same vertex and such that removing
those loops results in a double tree. We denote the set of double uniloop
graphs by $\mathcal{DUL}(n)$.

For a block-pairing, $(B,\sigma)\in NC_{2}^{block}(n)$, we may think of the
block $B=\\{u,v\\}$ as a transposition of $S_{n}$ whose cycle decomposition is
$(u,v)$. Under this interpretation let us define the function
$\Psi:NC_{2}^{block}(n)\rightarrow\mathcal{Q}(T_{n})$ given by,

$\Psi(B,\sigma)=T_{n}^{\gamma_{n}\sigma B},$

where $\mathcal{Q}(T_{n})$ denotes the set of quotient graphs of $T_{n}$ and
$\gamma_{n}=(1,\dots,n)\allowbreak\in S_{n}$.

######  Lemma 8.30.

Let $n\in\mathbb{N}$ and
$\Psi:\mathit{NC}_{2}^{block}(n)\rightarrow\mathcal{Q}(T_{n})$ be defined as
above. The image, $Im(\Psi)$, equals to the set of double uniloop graphs;
$\mathcal{DUL}(n)$. Moreover, the function,

$\Psi:\mathit{NC}_{2}^{block}(n)\rightarrow\mathcal{DUL}(n)$

is injective. Therefore,

$\displaystyle|\\{\pi\in\mathcal{P}(n):T_{n}^{\pi}\text{ is a double uniloop
graph}\\}|$ $\displaystyle=$ $\displaystyle|NC_{2}^{block}(n)|$
$\displaystyle=$ $\displaystyle\frac{n}{2}|\mathit{NC}_{2}(n)|.$

###### Proof.

Let $(B,\sigma)\in\mathit{NC}_{2}^{set}(n)$ and let
$\pi^{\prime}=\gamma_{n}\sigma$ with $\gamma_{n}=(1,\dots,n)\in S_{n}$. Since
$\sigma\in\mathit{NC}_{2}(n)$ Lemma 8.18 says that $T_{n}^{\pi^{\prime}}$ is a
double tree and $\overline{\pi^{\prime}}=\sigma$. In other words, any block,
$\\{u,v\\}$ of $\sigma$, corresponds to the edges $e_{u}$ and $e_{v}$ of
$T_{m}^{\pi^{\prime}}$ connecting the same pair of vertices and with opposite
orientation. We let $B=\\{u^{\prime},v^{\prime}\\}$. Observe that
$\pi^{\prime}$ has two cycles: $A$ and $B$ (which we regard as blocks and
vertices of $T_{m}^{\pi^{\prime}}$), such that
$u^{\prime},\gamma_{n}(v^{\prime})\in A$ and
$v^{\prime},\gamma_{n}(u^{\prime})\in B$. Therefore, $\pi=\pi^{\prime}B$ have
exactly the same cycles of $\pi^{\prime}$ except by one which is obtained by
the union of $A$ and $B$. This means that the quotient graph $T_{n}^{\pi}$ is
obtained by identifying the vertices $A$ and $B$ of $T_{n}^{\pi^{\prime}}$
which results in $T_{n}^{\pi}$ being a double uniloop graph. This proves
$Im(\Psi)\subset\mathcal{DUL}$.

Conversely, let $T_{n}^{\pi}$ be a graph of double uniloop type. Then all
edges of $\overline{T_{n}^{\pi}}$ have multiplicity $2$. We let $\sigma$ be
the pairing obtained by $u\overset{\sigma}{\sim}v$ if $e_{u}$ and $e_{v}$
connect the same pair of vertices. Let $\tau=\sigma(u^{\prime},v^{\prime})$
where $e_{u^{\prime}}$ and $e_{v^{\prime}}$ correspond to the loops of
$T_{n}^{\pi}$. Then $\\#(\tau)=n/2+1$. By Theorem 8.6, we have that, as
partitions, $\gamma_{n}\sigma\leq\pi$; so
$\\#(\gamma_{n}\sigma)\geq\\#(\pi)=n/2$. By [10, 2.10], we have that
$\\#(\tau)+\\#(\gamma_{n}\tau)+\\#(\gamma_{n})\leq n+2$; so
$\\#(\gamma_{n}\tau)\leq n/2$. Thus

$\\#(\gamma_{n}\tau)\leq n/2\leq\\#(\gamma_{n}\sigma).$

If $u^{\prime}\sim_{\gamma_{n}\sigma}v^{\prime}$, then
$\\#(\gamma_{n}\tau)=\\#(\gamma_{n}\sigma(u^{\prime},v^{\prime}))=\\#(\gamma_{n}\sigma)+1$,
which is impossible. Thus $u^{\prime}\not\sim_{\gamma_{n}\sigma}v^{\prime}$
and hence $\gamma_{n}\sigma<\pi$. This implies that $\\#(\gamma_{n}\sigma)\geq
n/2+1$. Since $\sigma$ is a pairing we must have
$\\#(\gamma_{n}\sigma)=n/2+1$, and thus $\sigma\in\mathit{NC}_{2}(n)$, i.e.
$(\\{u^{\prime},v^{\prime}\\},\sigma)\in\mathit{NC}_{2}^{block}(n)$. Moreover,
note that $\\#(\pi)=n/2$ and $\\#(\gamma_{n}\sigma)=n/2+1$, since
$\gamma_{n}\sigma\leq\pi$ then $\pi$ is obtained by joining two blocks of
$\gamma_{n}\sigma$, these blocks must correspond to
$[u^{\prime}]_{\gamma_{n}\sigma}$ and $[v^{\prime}]_{\gamma_{n}\sigma}$ as we
proved that $u^{\prime}\not\sim_{\gamma_{n}\sigma}v^{\prime}$. On the other
hand, the cycles of the permutation $\gamma_{n}\sigma(u^{\prime},v^{\prime})$
are the same as the cycles of $\gamma_{n}\sigma$ except by one which is the
union of two cycles of $\gamma_{n}\sigma$: $[u^{\prime}]_{\gamma_{n}\sigma}$
and $[v^{\prime}]_{\gamma_{n}\sigma}$. This proves that, as partitions,
$\gamma_{n}\sigma(u^{\prime},v^{\prime})=\pi$ which proves that
$\Psi(\\{u^{\prime},v^{\prime}\\},\sigma)=T_{n}^{\pi}$, thus
$Im(\Psi)=\mathcal{DUL}(n)$.

To verify injectivity, let us remember that if
$(B,\sigma)\in\mathit{NC}_{2}^{block}(n)$ and $\pi=\gamma_{n}\sigma B$ then
$T_{n}^{\pi}$ pair the edges $e_{u}$ and $e_{v}$ whenever $\\{u,v\\}$ is a
block of $\sigma$, i.e. $\overline{\pi}=\sigma$. Moreover, the block
$B=\\{u^{\prime},v^{\prime}\\}$ correspond to the loops $e_{u^{\prime}}$ and
$e_{v^{\prime}}$ of $T_{n}^{\pi}$. Let $(B_{1},\sigma_{1})$ and
$(B_{2},\sigma_{2})$ be such that, as partitions,
$\pi_{1}=\gamma_{n}\sigma_{1}B_{1}$ and $\pi_{2}=\gamma_{n}\sigma_{2}B_{2}$
are the same. By the pointed before,
$\sigma_{1}=\overline{\pi_{1}}=\overline{\pi_{2}}=\sigma_{2}$. Finally observe
that since $T_{n}^{\pi_{1}}$ and $T_{n}^{\pi_{2}}$ are the same then, their
corresponding loops are the same, which means that the blocks $B_{1}$ and
$B_{2}$ must be the same. ∎

Figure 15. $(\mathcal{V},\pi)\in\mathcal{P\\!S}_{\mathit{NC}}^{(1,1)}(9,5,4)$
with $\pi=(1,8)(2,13)\allowbreak(3,6)(4,5)(7,10)(9,14)(11,12)(15,18)(16,17)$
and
$\mathcal{V}=\\{\\{1,8\\},\\{2,13\\},\\{3,6\\},\\{4,5\\},\\{7,10\\},\\{9,14\\},\\{11,12,16,17\\},\allowbreak\\{15,18\\}\\}.$
The permutation $\pi$ can be written as $\pi_{1}\times\pi_{2}$ with
$\pi_{1}=(1,8)(2,13)(3,6)(4,5)(7,10)(9,14)(11,12)$ and
$\pi_{2}=(15,18)(16,17)$, the permutation $\pi_{1}$ has only the two through
strings $(2,13)$ and $(7,10)$. Figure 16.
$(\mathcal{V},\pi)\in\mathcal{P\\!S}_{\mathit{NC}}^{(1,1)(t)}(9,5,4)$ given by
$\pi=(1,14)\allowbreak(2,3)(4,7)(5,6)(8,9)(10,11)(12,13)(15,16)(17,18)$ and
$\mathcal{V}=\\{\\{1,14,15,16\\},\\{2,3\\},\\{4,7\\},\\{5,6\\},\\{8,9\\},\\{10,11\\},\\{12,13\\},\allowbreak\\{17,18\\}\\}$.
We have $\pi=\pi_{1}\times\pi_{2}\in NC_{2}(9,5)\times NC_{2}(4)$ with
$\pi_{1}$ equal to $\\{(1,14)(2,3)(4,7)(5,6)(8,9)(10,11)(12,13)\\}$ and
$\pi_{2}=\allowbreak(15,16)(17,18)$. Note that $\pi_{1}$ has the unique
through string, $(1,14)$. The block $\\{1,14,15,16\\}$ of $\mathcal{V}$ is
obtained by joining the cycle $(1,14)$ of $\pi_{1}$ to the cycle $(15,16)$ of
$\pi_{2}$.

######  Lemma 8.31.

$|\mathcal{UL}_{2,4}|=|\mathcal{PS}_{NC_{2,1,1}}^{(1,1,1)}(m_{1},m_{2},m_{3})|=|\mathcal{PS}_{NC_{2}}^{(1,1)(t)}(m_{1},m_{2},m_{3})|$

###### Proof.

A quotient graph $T_{m_{1},m_{2},m_{3}}^{\pi}$ can be of $2$-$4$ uniloop type
only when one of $m_{1},m_{2}$ or $m_{3}$ is even and the other two are odd,
similarly $\mathcal{PS}_{NC_{2,1,1}}^{(1,1,1)}(m_{1},m_{2},m_{3})$ and
$\mathcal{PS}_{NC_{2}}^{(1,1)(t)}(m_{1},m_{2},m_{3})$ are non-empty only in
that case, so we may assume $m_{1}$ is even and $m_{2},m_{3}$ are both odd.
The graph $T_{m_{1}}^{\pi}$ is of double uniloop type and can be chosen in
$\frac{m_{1}}{2}|NC_{2}(m_{1})|$ ways as seen in Lemma 8.30. The loop of
$T_{m_{2}}^{\pi}$ can be chosen in $m_{2}$ ways. Then we quotient the rest to
get a double tree. This is basically doing the quotient of a basic cycle of
length $m_{2}-1$ which produces $|NC_{2}(m_{2}-1)|$ distinct double trees, so
$T_{m_{2}}^{\pi}$ is chosen in $m_{2}|NC_{2}(m_{2}-1)|$ ways. Similarly
$T_{m_{3}}^{\pi}$ is chosen in $m_{3}|NC_{2}(m_{3}-1)|$ ways, and since the
union of the graphs $T_{m_{1}}^{\pi}$, $T_{m_{2}}^{\pi}$ and $T_{m_{2}}^{\pi}$
is already determined by the loop then the number of ways of choosing the
graph $T_{m_{1},m_{2},m_{3}}^{\pi}$ is,

$\frac{m_{1}m_{2}m_{3}}{2}|NC_{2}(m_{1})||NC_{2}(m_{2}-1)||NC_{2}(m_{3}-1)|$

which is the cardinality of
$\mathcal{PS}_{NC_{2,1,1}}^{(1,1,1)}(m_{1},m_{2},m_{3})$. The cardinality of
$\mathcal{PS}_{NC_{2}}^{(1,1)(t)}(m_{1},m_{2},m_{3})$ can be computed easily.
We choose a non-crossing pairing on the $(m_{2},m_{3})$-annulus with a single
through string, By [2, Lemma 13] this can be done in
$\binom{m_{2}}{m_{2}-1/2}\binom{m_{3}}{m_{3}-1/2}$ ways. The last expression
can be rewritten as,

$m_{2}m_{3}|NC_{2}(m_{2}-1)||NC_{2}(m_{3}-1)|,$

by using the well know property
$|NC_{2}(n)|=Cat_{n/2}\vcentcolon=\frac{1}{n/2+1}\binom{n}{n/2}$. On the other
hand, the non-crossing pairing on $[m_{1}]$ points can be chosen in
$|NC_{2}(m_{1})|$ ways, and we select a block of that pairing which can be
chosen in $\frac{m_{1}}{2}$ ways. Therefore the cardinality of
$\mathcal{PS}_{NC_{2}}^{(1,1)(t)}(m_{1},m_{2},m_{3})$ is,

$\frac{m_{1}m_{2}m_{3}}{2}|NC_{2}(m_{1})||NC_{2}(m_{2}-1)||NC_{2}(m_{3}-1)|,$

as required. ∎

######  Lemma 8.32.

$|\mathcal{UC}_{2,4}|=2|\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})|-2|\mathcal{PS}_{NC_{2}^{(2)}}^{(1,1)}(m_{1},m_{2},m_{3})|\\\
-2|\mathcal{PS}_{NC_{2}}^{(1,1)(t)}(m_{1},m_{2},m_{3})|$

###### Proof.

We will count all graphs $T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{UC}_{2,4}$.
If $T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{UC}_{2,4}$ then one of the graphs,
$T_{m_{i_{1}}}^{\pi}$, is a double tree, $T_{m_{i_{2}},m_{i_{3}}}^{\pi}$ is a
double unicircuit graph with $(i_{1},i_{2},i_{3})$ a permutation of $(1,2,3)$
and $T_{m_{1},m_{2},m_{3}}^{\pi}$ results of joining $T_{m_{i_{1}}}^{\pi}$ and
$T_{m_{i_{2}},m_{i_{3}}}^{\pi}$ along some edge.

We count firstly the case where the unique circuit of
$\overline{T_{m_{i_{2}},m_{i_{3}}}^{\pi}}$ is not a loop. Lemma 8.18 says that
$T_{m_{i_{1}}}^{\pi}$ can be chosen in $|\mathit{NC}_{2}(m_{i_{1}})|$ ways.
Similarly, Corollary 8.20 says that $T_{m_{i_{2}},m_{i_{3}}}^{\pi}$ can be
chosen in

$|\mathit{NC}_{2}(m_{i_{2}},m_{i_{3}})|\allowbreak-|\mathit{NC}_{2}^{(1)}(m_{i_{2}},m_{i_{3}})|-|\mathit{NC}_{2}^{(2)}(m_{i_{2}},m_{i_{3}})|$

ways. Then we choose an edge from each graph $T_{m_{i_{1}}}^{\pi}$ and
$T_{m_{i_{2}},m_{i_{3}}}^{\pi}$. In the first graph there are
$\frac{m_{i_{1}}}{2}$ choices, and in the second there are
$\frac{m_{i_{2}}+m_{i_{3}}}{2}$ choices. Once selected the edges we make the
union of the graphs $T_{m_{i_{1}}}^{\pi}$ and $T_{m_{i_{2}},m_{i_{3}}}^{\pi}$
along these edges which can be done in $2$ ways depending the orientation of
the edges. Therefore, the total number of graphs is given by,

$2\frac{m_{i_{1}}(m_{i_{2}}+m_{i_{3}})}{4}|\mathit{NC}_{2}(m_{i_{1}})|\times\\\
\left[|\mathit{NC}_{2}(m_{i_{2}},m_{i_{3}})|-|\mathit{NC}_{2}^{(1)}(m_{i_{2}},m_{i_{3}})|-|\mathit{NC}_{2}^{(2)}(m_{i_{2}},m_{i_{3}})|\right].$

The last expression is twice the number of partitioned permutations
$(\mathcal{V},\pi_{1}\times\pi_{2})\in\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})$
with $\pi_{2}\in\mathit{NC}_{2}(m_{i_{1}})$ and,

$\pi_{1}\in\mathit{NC}_{2}(m_{i_{2}},m_{i_{3}})\setminus(\mathit{NC}_{2}^{(1)}(m_{i_{2}},m_{i_{3}})\cup\mathit{NC}_{2}^{(2)}(m_{2},m_{3})),$

and we choose a cycle from each $\pi_{1}$ and $\pi_{2}$ and join them together
to make a block of $\mathcal{V}$.

Similarly we count the case where the unique circuit of
$\overline{T_{m_{i_{2}},m_{i_{3}}}^{\pi}}$ is a loop. $T_{m_{i_{1}}}^{\pi}$
can be chosen in $|\mathit{NC}_{2}(m_{i_{1}})|$ ways, and
$T_{m_{i_{2}},m_{i_{3}}}^{\pi}$ can be chosen in
$|\mathit{NC}_{2}^{(1)}(m_{i_{2}},m_{i_{3}})|$ ways. Then we choose an edge
from each graph $T_{m_{i_{1}}}^{\pi}$ and $T_{m_{i_{2}},m_{i_{3}}}^{\pi}$. In
the first graph there are $\frac{m_{i_{1}}}{2}$ choices, and in the second
there are $\frac{m_{i_{2}}+m_{i_{3}}-2}{2}$ choices because the circuit cannot
be chosen as it is a loop. Once selected the edges we have two possible
orientations for the union along the edges. Thus the total number of graphs is
given by,

$2\frac{m_{i_{1}}(m_{i_{2}}+m_{i_{3}}-2)}{4}|\mathit{NC}_{2}(m_{i_{1}})||\mathit{NC}_{2}^{(1)}(m_{i_{2}},m_{i_{3}})|.$

The last expression is twice the number of partitioned permutations
$(\mathcal{V},\pi_{1}\times\pi_{2})\in\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})$
whit $\pi_{2}\in\mathit{NC}_{2}(m_{i_{1}})$,
$\pi_{1}\in\mathit{NC}_{2}^{(1)}(m_{i_{2}},\allowbreak m_{i_{3}})$, and we
choose a cycle from each $\pi_{1}$ and $\pi_{2}$ and join them together to
make a block of $\mathcal{V}$ with the restriction that the selected cycle of
$\pi_{1}$ cannot be the unique through string of $\pi_{1}$. Adding both cases
up says that the number of $2$-$4$ unicircuit graphs,
$T_{m_{1},m_{2},m_{3}}^{\pi}$, equals twice the number of partitioned
permutations
$(\mathcal{V},\pi_{1}\times\pi_{2})\in\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})$
with $\pi_{1}\in\mathit{NC}_{2}(m_{i_{2}},m_{i_{3}})$,
$\pi_{2}\in\mathit{NC}_{2}(m_{i_{1}})$ and such that $\pi_{1}$ doesn’t have
two through strings and if $\pi_{1}$ has a single through string then this
through string is never joined to another cycle of $\pi_{2}$ to make a block
of $\mathcal{V}$, i.e. $(\mathcal{V},\pi_{1}\times\pi_{2})$ belongs to,

$\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})\setminus(\mathcal{PS}_{NC_{2}^{(2)}}^{(1,1)}(m_{1},m_{2},m_{3})\cup\mathcal{PS}_{NC_{2}}^{(1,1)(t)}(m_{1},m_{2},m_{3})).$

Thus the number of all possible graphs
$T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{UC}_{2,4}$ is twice the cardinality of
the set,

$\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})\setminus(\mathcal{PS}_{NC_{2}^{(2)}}^{(1,1)}(m_{1},m_{2},m_{3})\cup\mathcal{PS}_{NC_{2}}^{(1,1)(t)}(m_{1},m_{2},m_{3})),$

as desired. ∎

##  9\. The third order moments and cumulants

We are ready to provide a proof to the main theorem, let us recall Corollary
7.12,

$\alpha_{m_{1},m_{2},m_{3}}=\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(m)\\\
T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{T}_{2,6}\end{subarray}}(k_{6}+6k_{4}+2)+\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(m)\\\
T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{T}_{2,4,4}\end{subarray}}(k_{4}+1)^{2}\\\
+\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(m)\\\
T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{UL}_{2,4}\end{subarray}}(\mathring{k_{4}}+2)+\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(m)\\\
T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{UC}_{2,4}\end{subarray}}(k_{4}+1)+\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(m)\\\
T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{DB}\end{subarray}}1.\\\ $

On the other hand, Corollary 8.17 says,

$|\mathcal{DB}|=|NC_{2}(m_{1},m_{2},m_{3})|-|\mathcal{UC}_{2,4}|-|\mathcal{T}_{2,4,4}|-2|\mathcal{UL}_{2,4}|-2|\mathcal{T}_{2,6}|.$

Combining these two expressions we get the following simpler expression,

(9.1)
$\alpha_{m_{1},m_{2},m_{3}}=\sum_{NC_{2}(m_{1},m_{2},m_{3})}1+\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(m)\\\
T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{T}_{2,6}\end{subarray}}(k_{6}+6k_{4})\\\
+\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(m)\\\
T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{T}_{2,4,4}\end{subarray}}(k_{4}^{2}+2k_{4})+\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(m)\\\
T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{UL}_{2,4}\end{subarray}}\mathring{k}_{4}+\sum_{\begin{subarray}{c}\pi\in\mathcal{P}(m)\\\
T_{m_{1},m_{2},m_{3}}^{\pi}\in\mathcal{UC}_{2,4}\end{subarray}}k_{4}$

######  Theorem 9.1.

Let $m_{1},m_{2},m_{3}\in\mathbb{N}$. Then,

$\alpha_{m_{1},m_{2},m_{3}}=|NC_{2}(m_{1},m_{2},m_{3})|+4k_{6}|\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})|\\\
+4k_{4}^{2}|\mathcal{PS}_{NC_{2}}^{(2,1,1)}(m_{1},m_{2},m_{3})|+2k_{4}|\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})|\\\
+(\mathring{k}_{4}-2k_{4})|\mathcal{PS}_{NC_{2,1,1}}^{(1,1,1)}(m_{1},m_{2},m_{3})|$

###### Proof.

Lemmas 8.23, 8.24, 8.32 and 8.31 let us count the sets
$\mathcal{T}_{2,6},\allowbreak\mathcal{T}_{2,4,4},\allowbreak\mathcal{UL}_{2,4}$
and $\mathcal{UC}_{2,4}$; combining this with Equation (9.1) gives,

(9.2)
$\alpha_{m_{1},m_{2},m_{3}}=|NC_{2}(m_{1},m_{2},m_{3})|+4k_{6}|\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})|\\\
+4k_{4}^{2}|\mathcal{PS}_{NC_{2}}^{(2,1,1)}(m_{1},m_{2},m_{3})|+2k_{4}|\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})|\\\
+(\mathring{k}_{4}-2k_{4})|\mathcal{PS}_{NC_{2}}^{(1,1)(t)}(m_{1},m_{2},m_{3})|+2k_{4}R,$

where $R$ is given by,

$12|\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})|+4|\mathcal{PS}_{NC_{2}}^{(2,1,1)}(m_{1},m_{2},m_{3})|\\\
\mbox{}-|\mathcal{PS}_{NC_{2}^{(2)}}^{(1,1)}(m_{1},m_{2},m_{3})|.$

We will prove that $R=0$. We know that,

$12|\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})|\\\
=12\frac{m_{1}}{2}\frac{m_{2}}{2}\frac{m_{3}}{2}|NC_{2}(m_{1})||NC_{2}(m_{2})||NC_{2}(m_{3})|,$

and,

$4|\mathcal{PS}_{NC_{2}}^{(2,1,1)}(m_{1},m_{2},m_{3})|=\\\
4\frac{m_{1}}{2}\frac{m_{2}}{2}\frac{m_{3}}{2}(\frac{m}{2}-3)|NC_{2}(m_{1})||NC_{2}(m_{2})||NC_{2}(m_{3})|,$

whit $m=m_{1}+m_{2}+m_{3}$. Thus,

(9.3)
$12|\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})|+4|\mathcal{PS}_{NC_{2}}^{(2,1,1)}(m_{1},m_{2},m_{3})|=\\\
\frac{mm_{1}m_{2}m_{3}}{4}|NC_{2}(m_{1})||NC_{2}(m_{2})||NC_{2}(m_{3})|$

Now we compute $|\mathcal{PS}_{NC_{2}^{(2)}}^{(1,1)}(m_{1},m_{2},m_{3})|$.
Each element
$(\mathcal{V},\pi_{1}\times\pi_{2})\in\mathcal{PS}_{NC_{2}^{(2)}}^{(1,1)}(m_{1},m_{2},m_{3})$
is such that we have $\pi_{1}\in NC_{2}^{(2)}(m_{i_{1}},m_{i_{1}})$ and
$\pi_{2}\in\allowbreak NC_{2}(m_{i_{3}})$ for some permutation
$(i_{1},i_{2},i_{3})$ of $(1,2,3)$. Assume $\pi_{1}\in
NC_{2}^{(2)}(m_{1},m_{2})$ and $\pi_{2}\in NC_{2}(m_{3})$, the partition
$\mathcal{V}$ is such that each block of $\mathcal{V}$ is a cycle of $\pi$
except by one block which is the union of two cycles of $\pi$ one from each
$\pi_{1}$ and $\pi_{2}$. Those cycles can be chosen in
$\frac{m_{1}+m_{2}}{2}\frac{m_{3}}{2}$ ways. Thus the number of those
partitioned permutation is,

$\frac{(m_{1}+m_{2})m_{3}}{4}|NC_{2}(m_{3})||NC_{2}^{(2)}(m_{1},m_{2})|.$

The set $NC_{2}^{(2)}(m_{1},m_{2})$ can be counted easily. On each circle we
choose two points corresponding to the through strings. By [2, Lemma 13] that
can be done in
$\binom{m_{1}}{\frac{m_{1}}{2}-1}\binom{m_{2}}{\frac{m_{2}}{2}-1}$ ways. Then
we join the points to make the two through strings, which can be done in two
ways. Thus,

$\displaystyle|NC_{2}^{(2)}(m_{1},m_{2})|$ $\displaystyle=$ $\displaystyle
2\binom{m_{1}}{\frac{m_{1}}{2}-1}\binom{m_{2}}{\frac{m_{2}}{2}-1}$
$\displaystyle=$ $\displaystyle
2\frac{m_{1}}{2}\frac{m_{2}}{2}\frac{1}{m_{1}/2}\binom{m_{1}}{m_{1}/2}\frac{1}{m_{2}/2}\binom{m_{2}}{m_{2}/2}$
$\displaystyle=$
$\displaystyle\frac{m_{1}m_{2}}{2}|NC_{2}(m_{1})||NC_{2}(m_{2})|,$

so the total number of elements
$(\mathcal{V},\pi_{1}\times\pi_{2})\in\mathcal{PS}_{NC_{2}^{(2)}}^{(1,1)}(m_{1},m_{2},m_{3})$
with $\pi_{1}\in NC_{2}^{(2)}(m_{1},m_{2})$ and $\pi_{2}\in NC_{2}(m_{3})$ is
given by,

$\frac{m_{1}m_{2}m_{3}(m_{1}+m_{2})}{8}|NC_{2}(m_{1})||NC_{2}(m_{2})||NC_{2}(m_{3})|.$

In the same way we count the other two cases corresponding to $\pi_{1}\in
NC_{2}^{(2)}(m_{1},m_{3})$ and $\pi_{2}\in NC_{2}(m_{2})$ and $\pi_{1}\in
NC_{2}^{(2)}(m_{2},m_{3})$ and $\pi_{2}\in NC_{2}(m_{1})$. Adding all together
up gives,

$\displaystyle|\mathcal{PS}_{NC_{2}^{(2)}}^{(1,1)}(m_{1},m_{2},m_{3})|$
$\displaystyle=$
$\displaystyle(2m_{1}+2m_{2}+2m_{3})\frac{m_{1}m_{2}m_{3}}{8}|NC_{2}(m_{1})|\,|NC_{2}(m_{2})|\,|NC_{2}(m_{3})|$
$\displaystyle=$
$\displaystyle\frac{mm_{1}m_{2}m_{3}}{4}|NC_{2}(m_{1})|\,|NC_{2}(m_{2})|\,|NC_{2}(m_{3})|.$

So Equation (9.3) implies $R=0$. This turns Equation (9.2) into,

$\alpha_{m_{1},m_{2},m_{3}}=|NC_{2}(m_{1},m_{2},m_{3})|+4k_{6}|\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})|\\\
+4k_{4}^{2}|\mathcal{PS}_{NC_{2}}^{(2,1,1)}(m_{1},m_{2},m_{3})|+2k_{4}|\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})|\\\
+(\mathring{k}_{4}-2k_{4})|\mathcal{PS}_{NC_{2}}^{(1,1)(t)}(m_{1},m_{2},m_{3})|.$

We conclude the proof by replacing
$|\mathcal{PS}_{NC_{2}}^{(1,1)(t)}(m_{1},m_{2},m_{3})|$ by
$|\mathcal{PS}_{NC_{2,1,1}}^{(1,1,1)}(m_{1},m_{2},m_{3})|$, as we proved in
Lemma 8.31 they are equal. ∎

Proof of main theorem. As we have seen the third order fluctuation moments
$\alpha_{m_{1},m_{2},m_{3}}$ are somewhat involved. However the higher order
cumulants are very simple.

######  Theorem 9.2 (Main Theorem).

The third order cumulants, $(\kappa_{p,q,r})_{p,q,r}$, of a Wigner Ensemble,
$X$, are given by,

$\kappa_{p,q,r}=\left\\{\begin{array}[]{lcc}4k_{6}&if&p=q=r=2\\\
\mathring{k}_{4}-2k_{4}&if&\\{p,q,r\\}=\\{2,1,1\\}\\\
0&otherwise\end{array}\right.$

###### Proof.

Let us recall that, up to order two, the free cumulants of $X$ are given by;
$\kappa_{2}=1,\kappa_{2,2}=2k_{4}$ and $0$ otherwise. Let
$(\kappa^{\prime}_{n})_{n},(\kappa^{\prime}_{p,q})_{p,q}$ and
$(\kappa^{\prime}_{p,q,r})_{p,q,r}$ be the sequences defined by
$\kappa_{n}^{\prime}=\kappa_{n}$ for all $n$,
$\kappa^{\prime}_{p,q}=\kappa_{p,q}$ for all $p,q$ and
$\kappa^{\prime}_{2,2,2}=4k_{6},\kappa^{\prime}_{2,1,1}=\mathring{k}_{4}-2k_{4}$
and $0$ otherwise. By definition $\kappa^{\prime}_{n}$ and
$\kappa^{\prime}_{p,q}$ coincide with the free cumulants of first and second
order $\kappa_{n}$ and $\kappa_{p,q}$. Therefore these sequences satisfy the
moment-cumulant relations:

(9.4)
$\alpha_{m}=\sum_{(\mathcal{V},\pi)\in\mathcal{PS}_{NC}(m)}\kappa^{\prime}_{(\mathcal{V},\pi)}$
(9.5)
$\alpha_{m_{1},m_{2}}=\sum_{(\mathcal{V},\pi)\in\mathcal{PS}_{NC}(m_{1},m_{2})}\kappa^{\prime}_{(\mathcal{V},\pi)}$

for all $m,m_{1},m_{2}$. For a non-crossing partitioned permutation

$(\mathcal{V},\pi)\in\mathcal{PS}_{NC}(m_{1},m_{2},m_{3}),$

let $\kappa^{\prime}_{(\mathcal{V},\pi)}$ be the multiplicative extension of
$(\kappa^{\prime}_{n})_{n},(\kappa^{\prime}_{p,q})_{p,q}$ and
$(\kappa^{\prime}_{p,q,r})_{p,q,r}$, i.e;

$\kappa_{(\mathcal{V},\pi)}=\prod_{\begin{subarray}{c}B\text{ block of
}\mathcal{V}\\\ V_{1},\dots,V_{i}\text{ cycles of }\pi\text{ with
}V_{i}\subset B\end{subarray}}\kappa^{\prime}_{|V_{1}|,\dots,|V_{i}|}.$

Observe that,

$\sum_{(\mathcal{V},\pi)\in\mathcal{PS}_{NC}(m_{1},m_{2},m_{3})}\kappa^{\prime}_{(\mathcal{V},\pi)}=|\mathit{NC}_{2}(m_{1},m_{2},m_{3})|\\\
+4k_{6}|\mathcal{PS}_{NC_{2}}^{(1,1,1)}(m_{1},m_{2},m_{3})|+4k_{4}^{2}|\mathcal{PS}_{NC_{2}}^{(2,1,1)}(m_{1},m_{2},m_{3})|\\\
+2k_{4}|\mathcal{PS}_{NC_{2}}^{(1,1)}(m_{1},m_{2},m_{3})|+(\mathring{k}_{4}-2k_{4})|\mathcal{PS}_{NC_{2,1,1}}^{(1,1,1)}(m_{1},m_{2},m_{3})|.$

According to Theorem 9.1 last expression equals $\alpha_{m_{1},m_{2},m_{3}}$,
so the sequences $(\kappa^{\prime}_{n})_{n},(\kappa^{\prime}_{p,q})_{p,q}$ and
$(\kappa^{\prime}_{p,q,r})_{p,q,r}$ satisfy the moment-cumulant relation of
order three, namely,

(9.6)
$\alpha_{m_{1},m_{2},m_{3}}=\sum_{(\mathcal{V},\pi)\in\mathcal{PS}_{NC}(m_{1},m_{2},m_{3})}\kappa^{\prime}_{(\mathcal{V},\pi)}.$

However the free cumulants $(\kappa_{n})_{n},(\kappa_{p,q})_{p,q}$ and
$(\kappa_{p,q,r})_{p,q,r}$ are the unique sequences satisfying Equations
(9.4), (9.5) and (9.6), so it must be $\kappa^{\prime}_{p,q,r}=\kappa_{p,q,r}$
as desired. ∎

## Acknowledgements

We would like to thank Roland Speicher for his comments and fruitful
discussions while preparing this paper.

## References

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M††footnotetext:  xxvii.viii.mmxxiv


# Elastic properties of a Sc-Zr-Nb-Ta-Rh-Pd high-entropy alloy superconductor

Yupeng Pana, Xiaobo Hea, Binjie Zhoua, Denver Strongb, Jian Zhanga, Hai-Bin
Yua, Yunfei Tana, Robert J. Cavab∗, and Yongkang Luoa† a Wuhan National High
Magnetic Field Center and School of Physics, Huazhong University of Science
and Technology, Wuhan 430074, China  b Department of Chemistry, Princeton
University, Princeton, New Jersey 08544, United States<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We report a comprehensive study on the elastic properties of a hexanary high-
entropy alloy superconductor (ScZrNbTa)0.685[RhPd]0.315 at room and cryogenic
temperatures, by Resonant Ultrasound Spectroscopy experiments. The derived
elastic constants are bulk modulus $K=132.7$ GPa, Young’s modulus $E=121.0$
GPa, shear modulus $G=44.9$ GPa, and Poisson’s ratio $\nu$=0.348 for room
temperature. The Young’s and shear moduli are $\sim 10\%$ larger than those in
NbTi superconductor with similar $T_{c}$, while the ductility is comparable.
Moreover, the mechanical performance is further enhanced at cryogenic
temperature. Our work confirms the advantageous mechanical properties of high-
entropy alloy superconductors and suggests the application prospects.

Keywords: High-entropy alloy, Superconductor, Elastic constants

###### pacs:

74.25.Ld, 62.20.DC, 43.35.Cg

##  1 Introduction

Superconductors (SCs) are a class of ever-green functional materials
discovered in 1911[1], with great application prospects in electrical current
transfer, ground transportation, nuclear magnetic resonance (NMR) and medical
resonance imaging (MRI), International Thermonuclear Experimental Reactor
(ITER) and new-generation quantum computation as well. Until now, the so-
called “low-$T_{c}$ superconductors” Nb-Ti and Nb3Sn based superconducting
wires still dominate most of the commercial applications for superconducting
eletromagnets. For industrial applications, the most crucial prerequisites of
SC are high critical temperature ($T_{c}$), high upper critical field
($H_{c2}$), and high critical current density ($j_{c}$), which guarantee the
strong magnetic field output. Apart from these factors, another technical
issue one has to confront with is the mechanical performance. On the one hand,
materials with good ductility are prone to make into wires; on the other hand,
the strain effect exerted by the electromagnetic force on the superconducting
wires reduces their superconducting properties[2], and this is particularly
the case for the A15-phase Nb3Sn[3, 4]. For these reasons, SCs with both high
strength and good ductility are of considerable interest.

High-entropy alloys (HEAs) refer to systems containing more than four metallic
elements in equimolar or near-equimolar ratios, offering a rich platform for
materials design. Earlier studies on HEAs have revealed a series of intriguing
properties, such as high hardness and strength[5, 6], simultaneous strength
and ductility[7], outstanding corrosion and oxidation resistance[6, 8, 9],
elegant strength-to-weight ratio[10], improved mechanical properties at
cryogenic temperatures[11], etc. A natural question concerns whether
applicable superconducting wires can be made of HEA. Indeed, HEA SCs present a
unique crossing point between novel superconductors and functional high-
entropy alloys that have attracted extensive interests in recent years. In
2014, Koželj et al reported the synthesis of the first HEA SC
Ta34Nb33Hf8Zr14Ti11 with $T_{c}\approx 7.3$ K[12]. One salient feature of this
SC is that it exhibits extraordinarily robust zero-resistance
superconductivity under pressure up to 190.6 GPa[13]. Later on,
superconductivity was also observed in other HEAs e.g. CsCl-type Sc-Zr-Nb-Rh-
Pd, Sc-Zr-Nb-Ta-Rh-Pd[14], $Tr$Zr2-type (Fe,Co,Ni,Rh,Ir)Zr2 [15, 16], and hcp-
structured (MoReRu)(1-2x)/33(PdPt)x [17]. Most interestingly, the pentanary
HEA SC (ScZrNb)0.65[RhPd]0.35 has $T_{c}\approx 9.7$ K and $H_{c2}\approx
10.7$ T, comparable to those in NbTi[18, 19], the superconducting alloy that
accounts for a majority of the global superconductivity market for prevalent
MRIs. Although HEA SCs have manifested themselves with excellent
superconductivity, little has been known about their mechanical
performance[17, 20], and in particular, a comprehensive study about their
elastic constants and moduli at cryogenic temperature is still lacking.

Herein, by employing the Resonant Ultrasound Spectroscopy (RUS) technique, we
performed a comprehensive study on the the second-rank elastic tensor of a
representative HEA SC (ScZrNbTa)0.685[RhPd]0.315, which becomes a SC below
$\sim$7 K. The derived elastic constants at room temperature are bulk modulus
$K=132.7$ GPa, Young’s modulus $E=121.0$ GPa, shear modulus $G=44.9$ GPa, and
Poisson’s ratio $\nu=0.348$. In particular, $E$ and $G$ are $\sim 10\%$ larger
than the NbTi SC, while their $\nu$s are at the same level. These parameters
suggest that this superconducting HEA possess both good strength and
ductility. Meanwhile, unlike Nb3Sn, the excellent mechanical performance
retains even at cryogenic temperature. Our work confirms the advantageous
mechanical properties of high-entropy alloy superconductors and suggests the
application prospects.

##  2 Experimental

The polycrystalline hexanary HEA (ScZrNbTa)0.685[RhPd]0.315 sample studied
here was grown by the arc-melting method as described elsewhere[14]. The
composition and structural characterizations were performd by X-ray
diffraction (Cu-$K_{\alpha}$) and energy-dispersive X-ray spectroscopy (EDS)
measurements. Electrical resistivity was measured as a function of temperature
by the standard four-lead method in a commercial Physical Property
Measurements System (PPMS-9, Quantum Design). For RUS measurements, the sample
was carefully polished into a parallelepiped with the dimensions $1.152\times
0.975\times 0.586$ mm3 and mass 6.33 mg. A schematic of the RUS experimental
set-up is shown in Fig. 1(a). The transducers are made of a Lead Zirconate
Titanate (PZT) plate and an Al2O3 hemisphere, and the latter is used for
electrical isolation and mechanical protection[21]. A pair of transducers were
used in RUS measurements, the bottom one as the ultrasound driving source, and
the top one as the signal pick-up. To reduce the damping of the vibration
modes, the sample is point-touch mounted between the two transducers. The
measurements were made by sweeping frequency at fixed temperatures. More
details about the RUS measurements can be found in Ref. [22]. To measure the
low-temperature elastic constants, a helium-flow cryostat (OptistatCF, Oxford)
was exploited to cool the sample down to $\sim$5.4 K.

Figure 1: (a) Schematic of RUS experimental set-up. (b) Temperature dependent
resistivity of (ScZrNbTa)0.685[RhPd]0.315 showing an onset of the
superconducting transition at $T_{c}^{on}=7$ K. The inset is the crystalline
structure.

##  3 Results and Discussion

The composition of the sample studied in this paper is
(ScZrNbTa)0.685[RhPd]0.315, which is a hexanary HEA superconductor with onset
superconducting transition $T_{c}^{on}\approx 7$ K, verified by resistivity
measurements shown in Fig. 1(b). Bulk nature of superconductivity was
confirmed by the Meissner effect in our previous work[14]. This material has
CsCl-type structure and mixed-site occupancies[14]. The advantage of RUS
experiment is that it can extract the full elastic tensor $\\{C_{ij}\\}$
($i,j$=1-6) in a single frequency sweep. Because the sample is an isotropic
polycrystal, the elastic tensor has only two independent elements, viz.
$C_{11}$ and $C_{44}$ in Voigt notation, whereas $C_{12}$ can be retrieved by
$C_{12}=C_{11}-2C_{44}$.

Figure 2: Bottom panel, vibrational spectrum of (ScZrNbTa)0.685[RhPd]0.315 at
room temperature. Top panel shows the first 59 resonances, red circles -
experimental data, and black crosses - calculated data.

For a sample with given elastic constants, density and dimensions,
theoretically, all the normal resonance modes can be computed directly by
solving a three dimensional elastic wave function[22, 23, 24]. The way we did
for RUS measurements is opposite. We swept frequency from 0.7 to 3.5 MHz, and
collected the full spectrum at different temperatures. A representative
spectrum is displayed in the bottom panel of Fig. 2. About 59 resonant peaks
can be recognized in this range, and each stands for a specific vibration
mode. The elastic constants are derived by a computerized fitting algorithm
with a least-square criterion, in which $C_{11}$ and $C_{44}$ are set as free
fitting parameters. The iteration continues until
$\chi^{2}\equiv\sum_{n}[(F_{n}^{expt}-F_{n}^{cal})/F_{n}^{expt}]^{2}$
minimizes, where $F_{n}^{cal}$ and $F_{n}^{expt}$ are the $n$th calculated and
experimental frequencies, respectively. The fitting yields $C_{11}=192.6$ GPa,
$C_{12}=102.8$ GPa, and $C_{44}=44.9$ GPa for $T=300$ K. The fitting pattern
is presented in the top panel of Fig. 2. The shear modulus $G=C_{44}=44.9$ GPa
is obtained directly.

With these elastic parameters, we further calculate the bulk modulus [22]

$K=\frac{C_{11}+2C_{12}}{3}=132.7~{}GPa,$ (1)

Young’s modulus

$E=\frac{9KG}{3K+G}=121.0~{}GPa,$ (2)

and Poisson’s ratio

$\nu=\frac{3K-2G}{2(3K+G)}=0.348.$ (3)

According to Pugh[25], materials with the ratio $K/G>$1.75 are classified as
plastic. For (ScZrNbTa)0.685[RhPd]0.315, this ratio reaches 2.95, indicative
of good plasticity. This is further supported by the Poisson’s ratio. $\nu$ of
covalent systems are known to be small ($\nu\sim 0.1$), while those of ionic
crystals are $\sim 0.25$[26]. The value $\nu=0.348$ in
(ScZrNbTa)0.685[RhPd]0.315 resides in a plastic regime ($\nu>0.31$)[27].

Table 1:  Elastic constants of representative conductors and superconductors
at room temperature. $K$ \- Bulk modulus, $E$ \- Young’s modulus, $G$ \- Shear
modulus, $\nu$ \- Poisson’s ratio. Acoustic Debye temperature $\Theta_{D}$ is
calculated according to the elastic constants.

Materials $T_{c}$ (K) $K$ (GPa) $E$ (GPa) $G$ (GPa) $\nu$ $\Theta_{D}$ (K)
(ScZrNbTa)0.685[RhPd]0.315 7 132.7 121.0 44.9 0.348 277 Cu[28] $-$ 140.2 123.5
45.4 0.350 332 Nb[29] 9.2 174.3 108.9 39.0 0.396 274 NbTi[30] 9.7 131.8 111.7
41.1 0.359 325 Nb3Sn[31] 18 155.6 139.4 51.6 0.351 305 V3Ga[32] 15 115.0
YNi2B2C[33] 15.3 184.7 218.1 83.7 0.303 553 MgB2[34] 39 128.0 245.0 104.0
0.180 971 YBCO[35] 90 115.9 167.9 66.7 0.259 462 LaFeAsO†[36] $-$ 47.3 73.9
29.8 0.240 300

† LaFeAsO becomes a superconductor when electron-doped by partially
substituting O with F, and the $T_{c}$ of LaFeAsO0.89F0.11 reaches 26 K[37].
The elastic moduli of LaFeAsO derived from powder polycrystal are probably
underestimated. The values from first-principles calculation are[26]: $K=97.9$
GPa, $E=141.5$ GPa, $G=56.2$ GPa, $\nu=0.259$, $\Theta_{D}=416$ K.

The optimized elastic constants from RUS measurements also enable us to
estimate some thermodynamic parameters. The Debye temperature ($\Theta_{D}$)
is related to the average sound velocity ($v_{m}$) via [38]

$\Theta_{D}=\frac{h}{k_{B}}[\frac{3n\rho N_{A}}{4\pi M}]^{1/3}v_{m},$ (4)

where $h$ and $k_{B}$ are as conventionally defined in quantum mechanics,
$N_{A}$ is Avogadro’s number, $\rho$=9.610 g/cm3 is the density, $M$ is the
molecular weight of the solid, and $n$=2 is the number of atoms in the CsCl-
type molecule[38]. Here we adopted the average molecular weight $M$=206.376
g/mol for (ScZrNbTa)0.685[RhPd]0.315. The average sound velocity is taken as
[39]

$\frac{1}{v_{m}^{3}}=\frac{1}{3}(\frac{1}{v_{11}^{3}}+\frac{2}{v_{44}^{3}}),$
(5)

where $v_{11}=4477$ m/s and $v_{44}$=2162 m/s are the longitudinal and shear
sound velocities, respectively, and they can be retrieved from
$v_{ii}=\sqrt{C_{ii}/\rho}$ ($i$=1,4). The calculations result in $v_{m}=2430$
m/s and $\Theta_{D}=277$ K.

In addition, the lattice thermal conductivity $\kappa_{L}$ (for
$T>\Theta_{D}$) can be evaluated by[40, 41]

$\kappa_{L}=\frac{A\bar{M}\delta n^{1/3}\Theta_{D}^{3}}{\gamma^{2}T},$ (6)

where $\bar{M}=103.188$ g/mol is the average mass of the atoms in the crystal,
$\gamma\equiv\frac{3}{2}(\frac{3v_{11}^{2}-4v_{44}^{2}}{v_{11}^{2}+2v_{44}^{2}})=2.1$
is the acoustic Grüneisen parameter that characterizes the anharmonicity of a
sample [42, 43], $A=\frac{2.43\times 10^{4}}{1-0.514/\gamma+0.228/\gamma^{2}}$
W/m${}^{2}\cdot$K3, and $\delta^{3}$ signifies the average volume occupied by
one atom in the crystal that can be known from the structural parameters[14].
This gives rise to $\kappa_{L}=16.0$ W/m$\cdot$K at 300 K.

The elastic constants and the estimated thermodynamic parameters for
(ScZrNbTa)0.685[RhPd]0.315 are summarized in Table 1, and for comparison, we
also list other representative conductors and superconductors. It is
interesting to note that the Young’s and shear moduli of
(ScZrNbTa)0.685[RhPd]0.315 are about 10% larger than in Nb and NbTi that have
comparable $T_{c}$. Meanwhile, the Poisson’s ratios of
(ScZrNbTa)0.685[RhPd]0.315, Cu and NbTi are essentially the same, implying
that they have similar ductility. Therefore, filamentary superconducting wires
made of (ScZrNbTa)0.685[RhPd]0.315 / Cu composite are possible.

Figure 3: Low-temperature elastic constants of (ScZrNbTa)0.685[RhPd]0.315.
(a), $C_{11}$ (left) and $C_{12}$ (right); (b), $K$ (left) and $G$ (right);
(c) $E$ (left) and $\nu$ (right). The dashed lines mark $T_{c}$.

In order to study the mechanical performance under cryogenic conditions, we
also performed the RUS measurements at low temperature, and the results are
shown in Fig. 3. Upon cooling, $C_{44}$ ($=G$) increases monotonically below
20 K, and a weak inflection is visible at $T_{c}$, manifesting the
superconducting transition. The same trend is also seen in Young’s modulus.
Other elastic moduli including $C_{11}$, $C_{12}$ and $K$ also increase below
20 K, and tend to level off, but soften slightly near $T_{c}$. The tiny
feature of elastic constants at $T_{c}$ probably implies relatively weak
electron-phonon coupling, which is conceivable here. In particular, the
absence of a step-like discontinuity in $C_{ij}$ around $T_{c}$ evidences that
the coupling between strain and superconducting order parameter is very
weak[44]. At 5.4 K, the base temperature of our measurements, $K=138.6$ GPa,
$E=126.2$ GPa, and $G=46.8$ GPa. It is important to note that for all the
temperatures tested, Poisson’s ratio remains essentially constant about 0.348.
All these suggest that at cryogenic condition this HEA exhibits even better
mechanical performance than at room temperature. We should also point out that
in Nb3Sn, due to the formation of martensitic phase ($\sim 43$ K), Young’s and
shear moduli are softened dramatically, and the values reduce to $E=49$ GPa
and $G=16.8$ GPa at 4.2 K[45]. This makes Nb3Sn rather brittle and in turn
causes the large strain dependence in the critical current density[3, 4]. Such
a shortcoming is absent in (ScZrNbTa)0.685[RhPd]0.315.

Finally, it is important to note that among the SCs listed in Table 1,
(ScZrNbTa)0.685[RhPd]0.315 has the elastic constants most close to Cu; in
other words, the elastic properties of (ScZrNbTa)0.685[RhPd]0.315 and Cu are
in good compatibility. This suggests that the filamentary superconducting wire
made of (ScZrNbTa)0.685[RhPd]0.315 / Cu subjects little strain effect, and
thus will maintain good superconductivity. Also, because all the elastic
moduli of Cu are relatively larger than in (ScZrNbTa)0.685[RhPd]0.315, the
(ScZrNbTa)0.685[RhPd]0.315 / Cu composite wire is expected to exhibit even
better mechanical performance than pure (ScZrNbTa)0.685[RhPd]0.315.

##  4 Conclusion

In conclusion, the elastic properties of the high-entropy alloy superconductor
(ScZrNbTa)0.685[RhPd]0.315 have been studied by Resonant Ultrasound
Spectroscopy measurements. The room-temperature bulk modulus is $K=132.7$ GPa,
Young’ modulus is $E=121.0$ GPa, shear modulus is $G=44.9$ GPa, and Poisson’s
ratio is $\nu=0.348$. The Young’s and shear moduli are $\sim$10% larger than
those in NbTi superconductor with similar $T_{c}$. Most crucially, the
mechanical performance is further improved at cryogenic temperature. These
results illustrate the advantageous elastic properties of high-entropy alloy
superconductors, and suggest the feasibility for industrial applications.

##  5 CRediT authorship contribution statement

Yupeng Pan: Elastic property measurements, Data analysis, Writing. Xiaobo He:
Resistivity measurements. Binjie Zhou: Elastic property measurements. Denver
Strong: Sample synthesis. Jian Zhang: Elastic property measurements. Hai-Bin
Yu: Validation, Guidance. Yunfei Tan: Validation, Guidance. Robert J. Cava:
Sample synthesis, Validation, Guidance. Yongkang Luo: Supervision,
Methodology, Writing.

##  6 Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  7 Acknowledgments

We thank Albert Migliori and Brad Ramshaw for technical aid. This work was
supported by National Natural Science Foundation of China (No. 52077086), and
the sample preparation at Princeton University was supported by the US
Department of Energy Division of Basic Energy Sciences grant number DE-
FG02-98ER45706.

##  8 Author contributions

The manuscript was written with contributions from all the authors. All the
authors have approved the final version of the manuscript.

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