simplifying the proof.

######  Definition 5.2.3.

A permutation $\pi$ in $\mathfrak{S}_{n}$ is called an involution if it is a
root of unity of order $2$.

######  Remark 5.2.4.

In our notation, the set of involutions is just
$\mathcal{R}_{2}{\left(\mathfrak{S}_{n}\right)}$.

######  Theorem 5.2.5.

The set $\mathcal{R}_{2}{\left(\mathfrak{S}_{n}\right)}$ has a cyclic descent
extension if and only if $n$ is even.

###### Proof.

Recall that $RSK$ is a bijective function between permutations in
$\mathfrak{S}_{n}$ and pairs of Standard Young Tableaux $(P,Q)$ of the same
shape, such that if $RSK(\pi)=(P,Q)$ then $RSK(\pi^{-1})=(Q,P)$ and
$\operatorname{Des}(\pi)=\operatorname{Des}(Q)$. Since, $\pi$ is an involution
if and only if $\pi^{-1}=\pi$, we have from the properties of $RSK$ that $\pi$
is an involution if and only if $RSK(\pi)=(P,P)$ for some $P$. Hence, we have
$Des(\mathcal{R}_{2}{\left(\mathfrak{S}_{n}\right)})=Des(SYT(n))$ as
multisets. Therefore,

$\mathcal{Q}(\mathcal{R}_{2}{\left(\mathfrak{S}_{n}\right)})=\sum_{\lambda\vdash
n}s_{\lambda},$

which obviously implies that $s_{(n-k,1^{k})}$ appears exactly once in
$\mathcal{Q}(\mathcal{R}_{2}{\left(\mathfrak{S}_{d}\right)})$. Hence we have,

$H_{\mathcal{R}_{2}{\left(\mathfrak{S}_{n}\right)}}(x)=\sum_{i=0}^{n-1}x^{i}$

which is divisible by $(1+x)$ if and only if $n$ is even, as claimed. ∎

###  5.3 Reduction to unimodal permutations

In this section, we show that for a Schur-positive subset $A$ of
$\mathfrak{S}_{n}$, the study of hook multiplicities in $\mathcal{Q}(A)$ can
be reduced to the study of multiplicities of unimodal permutations in $A$.
First, we prove a slightly different form of lemma 4.1.12.

######  Lemma 5.3.1.

Let $A$ be a Schur-positive set with
$\mathcal{Q}(A)={\operatorname{ch}}(\phi)$ and $h_{k}$ defined as in lemma
4.1.12. Then

$h_{k}=\big{|}\bigl{\\{}a\in
A:\operatorname{Des}(a)=\left\\{n-k,\dots,n-1\right\\}\bigr{\\}}\big{|}.$

###### Proof.

Let $T\in{\rm SYT}(n-k,1^{k})$ be the hook with $1,\dots,n-k$ in the first
row, and ${1,n+k+1,\dots,n}$ in the first column.

$T=\ytableausetup{mathmode,boxframe=normal,boxsize=2em}\ytableau
1&\none[\dots]{\scriptstyle n-k}\\\ {\scriptscriptstyle n-k+1}\\\
\none[\vdots]\\\ n.$

Clearly, we have $\operatorname{Des}(T)=\\{n-k,\dots,n-1\\}.$. Hence,

$h_{k}\leq\big{|}\bigl{\\{}a\in
A:\operatorname{Des}(a)=\left\\{n-k,\dots,n-1\right\\}\bigr{\\}}\big{|}.$

Now, if $\operatorname{Des}(S)=\left\\{n-k,\dots,n-1\right\\}$ for some
standard Young tableau $S$, then the numbers $1,\dots,n-k$ must appear in the
first row, since $i+1$ appearing south of $i$ would imply that
$i\in\operatorname{Des}(S)$. Now, $n-k\in\operatorname{Des}(S)$ implies that
$n-k+1$ appears south of $n-k$ and can not appear in the first row. On the
other hand, it must appear in position $(2,1)$; otherwise, there must be a
number larger than $n-k+1$, which appears west or north of $n-k$,
contradicting $S$ being a standard Young tableau. By induction, $n-k+j$
appears in position $(j+1,1)$ for every $1\leq j\leq k$. Hence, we must have
$S=T$. Hence, there exists a unique standard Young tableau $T$ with
$Des(T)=\left\\{n-k,\dots,n-1\right\\}$ . Since $A$ is Schur-positive we have
by Theorem 2.5.12

$\sum_{a\in A}x^{\operatorname{Des}(A)}=\sum_{\lambda\vdash
n}c_{\lambda}\sum_{T\in{\rm SYT}(\lambda)}x^{\operatorname{Des}(T)},$

where $c_{\lambda}$ is the multiplicity of $s_{\lambda}$ in $\mathcal{Q}(A)$.
The desired result follows by comparing coefficients of
$x^{\left\\{n-k,\dots,n-1\right\\}}$ on both sides of the equation. ∎

For a permutation $\pi\in\mathfrak{S}_{n}$, the condition

$\operatorname{Des}(\pi)=\left\\{n-k,\dots,n-1\right\\}$

implies that $\pi_{i}<\pi_{i+1}$ for $1<i<n-k$ and $\pi_{i}>\pi_{i+1}$ for
$i\geq n-k$. Namely, if we view $\pi$ as a finite sequence, it is an ascending
sequence up to the position $n-k$ and a descending sequence after. Recall that
such sequences are called unimodal. In particular, the last lemma can be
restated for permutations in the following form.

######  Lemma 5.3.2.

Let $A$ be a Schur-positive subset of $\mathfrak{S}_{n}$. Then, $h_{k}$ equals
the number of unimodal permutations in $A$ that have the maximum at $n-k$.

Therefore, the calculation of
$h_{k}(\mathcal{R}_{d}{\left(\mathfrak{S}_{n}\right)})$ reduces to counting
the number of unimodal permutations that have the maximum at $n-k$.

######  Definition 5.3.3.

For a subset $A$ of $\mathfrak{S}_{n}$, let $u_{k}(A)$ denote the number of
unimodal permutations in $A$ that have the maximum at $k$. Define the unimodal
maxima polynomial of $A$ by

$U_{A}(x)=\sum_{k=1}^{n}u_{k}(A)x^{k}.$

The following lemma allows us to work with $U_{A}(x)$ instead of $H_{A}(x)$ to
check an existence of a cyclic descent extension for a Schur-positive set $A$,
as in Lemma 4.1.14

######  Lemma 5.3.4.

Let $A$ be a Schur-positive subset of $\mathfrak{S}_{n}$ with unimodal maxima
and hook polynomials $U_{A}(x)$ and $H_{A}(x)$ respectively. Then $U_{A}(x)$
is divisible by $1+x$ with the quotient having non-negative coefficients if
and only if $H_{A}(x)$ is.

###### Proof.

By Lemma 5.3.2, we have $h_{k}(A)=u_{n-k}(A)$, which implies the equality
$U_{A}(x)=x^{n}H_{A}(\frac{1}{x})$. If

$H_{A}(x)=(1+x)(a_{0}+\dots a_{n-1}x^{n-1}),$

then

$\begin{split}U_{A}(x)&=x^{n}(1+\frac{1}{x})(a_{0}+\dots
a_{n-1}\frac{1}{x^{n-1}})\\\ &=(1+x)(a_{0}x^{n-1}+\dots a_{n-1})\end{split}$

Hence, divisibility by $1+x$ of $H_{A}(x)$ with quotient coefficients being
non-negative implies the same for $U_{A}(x)$. The converse follows similarly.
∎

######  Corollary 5.3.5.

A Schur-positive subset $A$ of $\mathfrak{S}_{n}$ has a cyclic descent
extension if and only if $U_{A}(x)$ is divisible by $1+x$ with coefficients of
the quotient being non-negative integers.

###  5.4 The case of prime $n$

In this section, we prove the existence of a cyclic descent extension for the
set $\mathcal{R}_{p}{\left(\mathfrak{S}_{p}\right)}$ when $p$ is a prime
number.

Before we proceed to calculations, we cite a result by Gannon from [20]. Let
$\Delta_{n}^{m}$ denote the set of all unimodal $n$-cycles with the maximum at
$m$.

######  Lemma 5.4.1.

The number of unimodal cycles of length $n$, with maximum at $m$ is

$\displaystyle|\Delta_{n}^{m}|$
$\displaystyle=\delta_{n,1}+\frac{(-1)^{n-m}}{n}\sum_{\begin{subarray}{c}d|n\\\
d\leq n-m\end{subarray}}\mu(d)\sum_{j=1}^{(n-m)/d}(-1)^{j}\binom{n/d}{j}$
$\displaystyle=\delta_{n,1}+(-1)^{m+1}\delta_{n,2}+\frac{(-1)^{n-m+1}}{n}\sum_{\begin{subarray}{c}d|n\\\
d<m\end{subarray}}\mu(d)\sum_{j=1}^{\left\lceil
m/d\right\rceil-1}(-1)^{j+n/d}\binom{n/d}{j}.$

where $\mu$ is the Möbius function on ${\mathbb{N}}$.

######  Observation 5.4.2.

Binomial coefficients satisfy the following identity

$\sum_{i=0}^{k}(-1)^{i}\binom{n}{i}=(-1)^{k}\binom{n-1}{k}.$

or equivalently

$\sum_{i=1}^{k}\binom{n}{i}(-1)^{i}=(-1)^{k}\binom{n-1}{k}-1.$

######  Lemma 5.4.3.

When $p$ is prime and $n=p$, the polynomial
$U_{\mathcal{R}_{n}{\left(\mathfrak{S}_{n}\right)}}\left(x\right)$ is
divisible by $1+x$ and the quotient has non-negative coefficients.

###### Proof.

Observe that roots of unity of order $p$ in $\mathfrak{S}_{p}$ are either
cycles of length $p$ or the identity permutation. Replace

$\sum\limits_{j=1}^{(n-m)/d}(-1)^{j}\binom{n/d}{j}$

by

$(-1)^{(n-m)/d}\binom{n/d-1}{(n-m)/d}-1$

in

$|\Delta_{n}^{m}|=\delta_{n,1}+\frac{(-1)^{n-m}}{n}\sum_{\begin{subarray}{c}d|n\\\
d\leq n-m\end{subarray}}\mu(d)\sum_{j=1}^{(n-m)/d}(-1)^{j}\binom{n/d}{j}$

to obtain

$|\Delta_{n}^{m}|=\delta_{n,1}+\frac{(-1)^{n-m}}{n}\sum_{\begin{subarray}{c}d|n\\\
d\leq
n-m\end{subarray}}\mu(d)\left((-1)^{(n-m)/d}\binom{n/d-1}{(n-m)/d}-1\right).$

Note that no unimodal cycles of length $p$ have their maximum at $p$; the only
unimodal permutation with the maximum at $p$ is the identity permutation. The
case $p=2$ has already been handled in Theorem 5.2.5. Therefore, we assume
that $p\geq 3$ is an odd prime. Since $p$ is prime, the only divisor $d$ of
$p$ with $d<p-m$ for $m\geq 1$ is $d=1$ with $\mu(d)=1$ and obviously
$\delta_{p,1}=0$. Thus

$\begin{split}\Delta_{p}^{m}&=\frac{1}{p}\left(\binom{p-1}{p-m}-(-1)^{p-m}\right)\\\
&=\frac{1}{p}\left(\binom{p-1}{m-1}+(-1)^{m}\right).\end{split}$

Substituting
$u_{m}(\mathcal{R}_{p}{\left(\mathfrak{S}_{p}\right)})=\Delta_{p}^{m}$ for
$1\leq m\leq p-1$ and
$u_{p}(\mathcal{R}_{p}{\left(\mathfrak{S}_{p}\right)})=1$ we obtain

$\begin{split}U_{\mathcal{R}_{p}{\left(\mathfrak{S}_{p}\right)}}(x)&=x^{p}+\frac{1}{p}\sum_{m=1}^{p-1}\left(\binom{p-1}{m-1}+(-1)^{m}\right)x^{m}\\\
&=x^{p}+\frac{1}{p}\left(\sum_{m=1}^{p}\binom{p-1}{m-1}x^{m}+\sum_{m=1}^{p}(-1)^{m}x^{m}\right)\\\
&=\frac{1}{p}\left(\sum_{m=1}^{p}\binom{p-1}{m-1}x^{m}+px^{p}+\sum_{m=1}^{p}(-1)^{m}x^{m}\right)\\\
&=\frac{1}{p}\left(\sum_{m=1}^{p}\binom{p-1}{m-1}x^{m}+\sum_{m=1}^{p-1}\left(x^{p}-(-1)^{m}x^{m}\right)\right)\\\
&=\frac{1}{p}\left(\left(1+x\right)^{p-1}+\left(1+x\right)\sum_{m=1}^{p-1}\sum_{j=m}^{p-1}(-1)^{j}x^{j}\right)\\\
&=\frac{1+x}{p}\left(\sum_{m=1}^{p-1}\binom{p-2}{m-1}x^{m}+\sum_{m=1}^{p-1}(-1)^{m}mx^{m}\right)\\\
&=(1+x)\sum_{m=1}^{p-1}\frac{1}{p}\left(\binom{p-2}{m-1}+(-1)^{m}m\right)x^{m}.\end{split}$

We have to show that the coefficients in the summation are non-negative
integers. For $2\leq m\leq p-2$, $\binom{p-2}{m-1}\geq p-2$ and also
$\binom{p-2}{0}-1=0$ (for $m=1$) and $\binom{p-2}{p-2}+(p-1)=p$ (for $m=p-1$).
Therefore the coefficient is non-negative, and it remains to prove that they
are integers. It is sufficient to show that

$\binom{p-2}{m-1}+(-1)^{m}{m}\equiv 0\ (\mathrm{mod}\ p)$

for every $1\leq m\leq p-1$ or equivalently,

$\binom{p-2}{m}+(-1)^{m+1}(m+1)\equiv 0\ (\mathrm{mod}\ p)$

for every $0\leq m\leq p-2$.

We proceed by reverse induction. For $m=p-2$

$\binom{p-2}{p-2}+(-1)^{p-1}(p-1)=p\equiv 0\ (\mathrm{mod}\ p).$

Assume that for some $m<p-2$,

$\binom{p-2}{m+1}\equiv(-1)^{m+1}(m+2)\ (\mathrm{mod}\ p)$

holds. Recall that

$\binom{p-2}{m}=\binom{p-2}{m+1}\frac{m+1}{p-2-m}.$

But

$\binom{p-2}{m+1}\equiv(-1)^{m+1}(m+2)\ (\mathrm{mod}\ p),$

therefore

$\begin{split}\binom{p-2}{m}&\equiv(-1)^{m+1}(m+2)\frac{m+1}{p-2-m}\
(\mathrm{mod}\ p)\\\ &\equiv(-1)^{m+1}(m+2)\frac{m+1}{-(m+2)}\ (\mathrm{mod}\
p)\\\ &\equiv(-1)^{m}(m+1)\ (\mathrm{mod}\ p)\end{split}$

as desired. We have proved that the coefficients of
$\frac{U_{\mathcal{R}_{p}{\left(\mathfrak{S}_{p}\right)}}(x)}{1+x}$ are non-
negative integers. ∎

Applying Corollary 5.3.5, we immediately obtain the following.

######  Corollary 5.4.4.

For every prime $p$, the set $\mathcal{R}_{p}{\left(\mathfrak{S}_{p}\right)}$
has a cyclic descent extension.

######  Corollary 5.4.5.

There exists an involution

$\phi:\mathcal{R}_{p}{\left(\mathfrak{S}_{p}\right)}\cap\Delta(p)\to\mathcal{R}_{p}{\left(\mathfrak{S}_{p}\right)}\cap\Delta(p)$

such that $m_{\phi(\pi)}=m_{\pi}\pm 1$ where $m_{\phi(\pi)}=m_{\pi}\pm 1$ and
$m_{\phi(\pi)}$ denote the positions of maxima of $\phi(\pi)$ and $\pi$
respectively.

###  5.5 Gluing unimodal permutations

In this section, we study the cycle structure of unimodal permutations.
Intuitively, we show the existence of a total ordering $<_{\Delta}$ on
_unimodal full cycles_ and define a partial action $\oplus$ on unimodal full
cycles, such that all unimodal permutations can be described as
$\sigma_{1}\oplus\dots\oplus\sigma_{k}$ where $\sigma_{1},\dots,\sigma_{k}$
are unimodal full cycles. We then proceed to show that
$\sigma_{1},\dots,\sigma_{k}$ is defined if any only in
$\sigma_{1},\dots,\sigma_{k}$ is a unimodal sequence relative to $<_{\Delta}$.

Most of the definitions, results and basic notation in this section are based
on the work of Gannon [20] on the structure of unimodal permutations. Details
and further references can be found there. There are other approaches to the
enumeration of unimodal permutations with given cycle structure and maximum,
for example, in the work of Thibon [47], who employed methods similar to those
of Gessel and Reutenauer [22].

We use the following notation. $\Delta(n)$ denotes the set of unimodal
permutations in $\mathfrak{S}_{n}$. The set of unimodal full cycles in
$\mathfrak{S}_{n}$ will be denoted by $\Delta_{n}$. Denote by
$\Delta(*)=\biguplus\limits_{n\in{\mathbb{N}}}\Delta(n)$ the set of all
unimodal permutations, and
$\Delta_{*}=\biguplus\limits_{n\in{\mathbb{N}}}\Delta_{n}$ the set of all
unimodal full cycles. The maximum of a permutation $\pi\in\mathfrak{S}_{n}$ is
$m_{\pi}=\pi^{-1}(n)$.

For a linearly ordered set $J=\left\\{j_{1}<\dots<j_{n}\right\\}$, let
$\Omega_{J}:[n]\to J$ be the unique order preserving bijection, namely
$\Omega_{J}(i)=j_{i}$. For a linearly ordered set $X$ and a function $f:X\to
X$, when $f$ preserves a finite $J\subseteq X$ and $f|_{J}$ is a bijection,
the permutation type or shape of $f|_{J}$ is the permutation in
$\pi\in\mathfrak{S}_{n}$ obtained by relabeling $j_{i}$ with $i$ or
equivalently $\pi=\Omega_{J}^{-1}\circ f|_{J}\circ\Omega_{J}$.

We proceed to describe the key results from [20] that we use in our work.

######  Lemma 5.5.1.

[20] Let $\pi\in\Delta(n)$ and $J\subseteq[n]$ such that $\pi$ preserves $J$.
Then $\pi|_{J}$ is unimodal. In particular, if $\pi=\sigma_{1}\dots\sigma_{k}$
is the cycle decomposition of $\pi$, then $\sigma_{i}$ is unimodal for every
$1\leq i\leq k$.

######  Definition 5.5.2.

Let $\pi_{i}\in\Delta(n_{i})$ for $i=1,\dots,k$, $n=n_{1}+\dots+n_{k}$ and
disjoint $J_{i}\subseteq[n]$ with $|J|_{i}=n_{i}$, and
$\Omega_{J_{i}}:[n_{i}]\to J_{i}$ be the unique order preserving bijections.
Define the permutation $\pi\in S_{n}$ by

$\pi(x)=\left(\pi_{1}\oplus_{J_{1}}\dots\oplus_{J_{k-1}}\pi_{k}\right)(x)=\left(\Omega_{J_{i}}\circ\pi_{i}\circ\Omega_{J_{i}}^{-1}\right)(x)\
\text{for}\ x\in J_{i}.$

Note, that $\pi$ is always well defined, because $\left\\{J_{1},\dots
J_{k},\right\\}$ is a set partition of $[n]$.

######  Remark 5.5.3.

An alternative notation will be used. When $k=2$, we sometimes write $J$ and
$\overline{J}$ instead of $J_{1}$ and $J_{2}$ respectively. When we do that,
we also write $\Omega$ and $\overline{\Omega}$ instead of $\Omega_{J_{1}}$ and
$\Omega_{J_{2}}$. Often, we write $\Omega_{i}$ instead $\Omega_{J_{i}}$ and
$\oplus_{i}$ instead of $\oplus_{J_{i}}$. When the context is clear, the
indices may be omitted.

Using the $\oplus$ notation, we make the following observation. Let
$\pi\in\Delta(*)$ and let ${\pi=\sigma_{1}\circ\dots\circ\sigma_{k}}$ be the
cycle decomposition of $\pi$. Let $J_{i}$ be the subset of $[n]$ preserved by
$\sigma_{i}$, $n_{i}$ be the maximum of $J_{i}$ and
$m_{i}=\sigma_{i}^{-1}(n_{i})$. Let $\pi_{i}$ be the shape of $\sigma_{i}$.

Assume that $m_{i}<m_{i+1}$, since it is always possible to enumerate the
cycles in this way. Then,
$\pi=\pi_{1}\oplus_{J_{1}}\dots\oplus_{J_{k-1}}\pi_{k}$ and the sequence
$n_{1},\dots,n_{k}$ has to be unimodal. We obtain the following observation.

######  Observation 5.5.4.

Every unimodal permutation $\pi\in\Delta(*)$ can be uniquely represented as
$\pi_{1}\oplus_{J_{1}}\dots\oplus_{J_{k-1}}\pi_{k}$ with
$\Omega_{i}(m_{i})<\Omega_{i+1}(m_{i+1})$ and the sequence $\Omega_{i}(n_{i})$
is unimodal.

From now on, unless stated otherwise, when
$\pi=\pi_{1}\oplus_{1}\dots\oplus_{k-1}\pi_{k}$ is unimodal and $m_{i}$ denote
the maxima of $\pi_{i}$, then $i<j$ implies that
$\Omega_{i}(m_{i})<\Omega_{j}(m_{j})$. In particular, when indices are omitted
and the sum $\pi_{1}\oplus\dots\oplus\pi_{k}$ is unimodal, we always mean that
$\Omega_{i}(m_{i})<\Omega_{j}(m_{j})$ for $i<j$.

The lemma below is essential for building unimodal permutations from smaller
ones. A simple observation is that if $\pi=\pi_{1}\oplus_{J}\pi_{2}$ holds,
then the maximum $m$ of $\pi$ is equal to either $\Omega(m_{1})$ or
$\overline{\Omega}(m_{2})$ where $m_{i}$ denotes the maximum of $\pi_{i}$.

######  Lemma 5.5.5.

[20, Lemma 1] Let $\pi_{1}\in\Delta(n_{1}),\pi_{2}\in\Delta(n_{2})$,
$n=n_{1}+n_{2}$ and $J\subseteq[n]$ with $|J|=n_{1}$ such that
$\Omega(m_{1})<\overline{\Omega}(m_{2})$. Let $\pi=\pi_{1}\oplus\pi_{2}$. Use
$m_{1},m_{2},m$ to denote maxima of $\pi_{1},\pi_{2},\pi$ respectively. Assume
that $\pi_{1}\oplus\pi_{2}\in\Delta(n)$. Then for every $i_{1}\in[n_{1}]$ and
$i_{2}\in[n_{2}]$ we have:

  1. 1.

If $i_{1}>m_{1}$ and $i_{2}\geq m_{2}$, then $\Omega(i_{1})>m$ and
$\overline{\Omega}(i_{2})\geq m$.

  2. 2.

If $i_{1}\leq m_{1}$ and $i_{2}<m_{2}$ then $\Omega(i_{1})\leq m$ and
$\overline{\Omega}(i_{2})<m$.

  3. 3.

If $i_{1}>m_{1}$ and $i_{2}<m_{2}$ then
$\Omega(i_{1})>\overline{\Omega}(i_{2})$.

  4. 4.

If $i_{1}\leq m_{1}$ and $i_{2}\geq m_{2}$ then
$\Omega(i_{1})<\overline{\Omega}(i_{2})$.

We have the following relation on the maxima unimodal sums, which we use
extensively in our calculations.

######  Corollary 5.5.6.

Let $\pi_{1},\pi_{2},\pi,m_{1},m_{2},m,\Omega,\overline{\Omega}$ be as in
Lemma 5.5.5. Then

  1. 1.

$m=m_{1}+m_{2}-1$ if $m=\Omega(m_{1})$.

  2. 2.

$m=m_{1}+m_{2}$ if $m=\overline{\Omega}(m_{2})$.

###### Proof.

By parts 1 and 2 of 5.5.5, we have $\big{|}\\{i\in[n_{1}]:\Omega(i)\leq
m\\}\big{|}=m_{1}$ and

$\big{|}\\{j\in[n_{2}]:\overline{\Omega}(j)\leq
m\\}\big{|}=\left\\{\begin{array}[]{rl}m_{2}-1&\text{when}\ \Omega(m_{1})=m\\\
m_{2}&\text{when}\ \Omega(m_{2})=m.\end{array}\right.$

Next, observe that

$m=\big{|}\\{i\in[n_{1}]:\Omega(i)\leq
m\\}\big{|}+\big{|}\\{j\in[n_{2}]:\overline{\Omega}(j)\leq m\\}\big{|}.$

When $m=\Omega(m_{1})$, we have
$\big{|}\\{j\in[n_{2}]:\overline{\Omega}(j)\leq m\\}\big{|}=m_{2}-1$, and
therefore $m=m_{1}+m_{2}-1$. Otherwise, we have $m=m_{1}+m_{2}$, as desrired.
∎

The following theorem is essential for building unimodal permutations with a
given cycle structure.

######  Theorem 5.5.7.

[20, Theorem 3] Let $\pi_{1}\in\Delta_{n_{1}}$ and $\pi_{2}\in\Delta_{n_{2}}$
be unimodal full cycles with maxima $m_{1}$ and $m_{2}$ respectively. There
exists a unique $J$ with $\pi_{1}\oplus_{J}\pi_{2}$ unimodal satisfying
$\Omega(m_{1})<\overline{\Omega}(m_{2})$.

######  Definition 5.5.8.

[20, discussion before Theorem 3] Let $\pi_{1}\in\Delta_{n_{1}}$ and
$\pi_{2}\in\Delta_{n_{2}}$ be two unimodal cycles. Let $J$ and $\Omega$ be as
in Theorem 5.5.7. For every $(i_{1},i_{2})\in[n_{1}]\times[n_{2}]$, write
$i_{1}\prec i_{2}$ if $\Omega(i_{1})<\overline{\Omega}(i_{2})$ and $i_{1}\succ
i_{2}$ otherwise.

######  Remark 5.5.9.

To avoid ambiguity, we sometimes write $\prec_{ij}$ and $\succ_{ij}$ when
working with more than two cycles.

######  Remark 5.5.10.

Theorem 5.5.7 implies that for unimodal full cycles $\pi_{1}\in\Delta_{n_{1}}$
and $\pi_{2}\in\Delta_{n_{2}}$, there exist unique $J_{1},J_{2}$ such that
$\pi_{1}\oplus_{J_{1}}\pi_{2}$ and $\pi_{2}\oplus_{J_{2}}\pi_{1}$ are
unimodal. Observe that when $\pi_{1}=\pi_{2}$ then $J_{1}=J_{2}$, and there
exists exactly one unimodal sum $\pi\oplus\pi$. The uniqueness of the set
together with the convention that $\Omega(m_{1})<\overline{\Omega}(m_{2})$
allow us to omit the index $J$ in the sum $\pi_{1}\oplus_{J}\pi_{2}$ when it
is unimodal.

The following result is a corollary of Theorem 5.5.7 and determines the
structure of unimodal sum $\pi\oplus\pi$ of unimodal cycle $\pi$ with itself.
In particular, it determines the maximum of this sum.

######  Corollary 5.5.11.

[20, Corollary 4] Let $\pi\in\Delta_{n}$ with the maximum at $m$ and let $J$
be the unique set such that $\pi\oplus_{J}\pi$ is unimodal. Then the following
hold.

  1. 1.

The set $J$ contains exactly one element from from $\left\\{2k-1,k\right\\}$
for every $1\leq k\leq n$.

  2. 2.

If $n\equiv m\pmod{2}$ then $(\pi\oplus\pi)^{-1}(2n)=2m$ and
$(\pi\oplus\pi)^{-1}(2n)=2m-1$ otherwise.

######  Definition 5.5.12.

[20] We say that $\pi\in\Delta_{n}$ is acute if $m\equiv n\ (\mathrm{mod}\ 2)$
or equivalently $(\pi\oplus\pi)^{-1}(2n-1)<(\pi\oplus\pi)^{-1}(2n)$ (the
maximal of subcycles of $\pi\oplus\pi$ run diagonally SW-NE like), and we say
that $\pi\in\Delta_{n}$ is grave otherwise, namely, the maxima of the
subcycles of $\pi\oplus\pi$ run diagonally NW-SE (which implies that $2n$
occurs before $2n-1$.)

For a unimodal permutation $\pi\in\Delta(n)$ (not necessarily a cycle), let
$\mathcal{A}(\pi)$, denote the set of all acute unimodal cycles
$\sigma\in\Delta_{*}$ for which exist $J\subseteq[n]$ such the shape of
$\pi|_{J}$ that have shape $\sigma$. Similarly, $\mathcal{G}(\pi)$, will
denote the grave shapes in $\pi$. Define $\mathcal{A}_{>}(\pi)$ as the
multiset $\sigma\in\mathcal{A}(\pi)$, for which exists $J\subseteq[n]$, that
have shape $\sigma$ and the maximum of $\pi|_{J}$ is larger than the maximum
of $\pi$. $\mathcal{A}_{<}(\pi),\mathcal{G}_{>}(\pi),\mathcal{G}_{<}(\pi)$ are
defined equivalently.

######  Proposition 5.5.13.

[20, Proposition 5] Let $\pi_{i}\in\Delta_{n_{i}}$ be unimodal cycles for
$i=1,2,3$ with maxima $m_{i}$ respectively. For $i<j$ let $\pi_{ij}$ be the
unimodal sum $\pi_{i}\oplus\pi_{j}$ and $\prec_{ij},\succ_{ij}$ the order
corresponding to $\prec$ Definition 5.5.8 for $\pi_{i}$ and $\pi_{j}$.

  1. 1.

Assume that either $n_{1}\prec_{12}n_{2}$ or $n_{2}\succ_{23}n_{3}$. Then for
every $a\in[n_{1}],b\in[n_{2}],c\in[n_{3}]$ we have both

    1. (a)

$a\prec_{12}b$ and $b\prec_{23}c$ implies $a\prec_{13}c$.

    2. (b)

$a\succ_{12}b$ and $b\succ_{23}c$ implies $a\succ_{13}c$.

  2. 2.

There exist $J_{i}$ such that
$\Omega_{1}(m_{1})<\Omega_{2}(m_{2})<\Omega_{3}(m_{3})$ and
$\pi_{1}\oplus_{J_{1}}\pi_{J_{2}}\oplus_{2}\pi_{3}$ is unimodal if and only if
either $n_{1}\prec_{12}n_{2}$ or $n_{2}\succ_{23}n_{3}$. Moreover, when such
$J_{i}$ exist, they will be unique.

  3. 3.

Suppose that $\pi\in\Delta_{k}$ and $\sigma\in\Delta_{l}$. and let $I$ and $J$
be the unique $I$ and $J$ such that $\pi\oplus_{I}\sigma$ and
$\sigma\oplus_{J}\pi$ are unimodal. Then $k\prec_{I}l$ if and only if
$l\succ_{J}k$.

######  Remark 5.5.14.

Since $J_{1},J_{2},J_{3}$ such that
$\Omega_{1}(m_{1})<\Omega_{2}(m_{2})<\Omega_{3}(m_{3})$ and the sum
$\pi=\pi_{1}\oplus_{J_{1}}\pi_{2}\oplus_{J_{2}}\pi_{3}$ is unimodal, we will
write $\pi_{1}\oplus\pi_{2}\oplus\pi_{3}$ to denote this sum.

Part 3 of the preceding theorem says that if $\pi\in\Delta_{k}$ and
$\sigma\in\Delta_{l}$ are unimodal full cycles, then either the maxima of
$\pi\oplus_{I}\sigma$ and $\sigma\oplus_{j}\pi$ occur both either in
$\Omega(\pi)$ or in $\Omega(\sigma)$. Therefore, we can define a total
ordering on $\Delta_{*}$.

######  Definition 5.5.15.

[20] Part 3 of the previous proposition defines total order $<_{\Delta}$ on
$\Delta_{*}$ as follows: For $\pi_{1}\in\Delta_{n_{1}}$ and
$\pi_{2}\in\Delta_{n_{2}}$, we say that $\pi_{1}<_{\Delta}\pi_{2}$ if
$n_{1}\prec n_{2}$ in the unimodal sum $\pi_{1}\oplus\pi_{2}$.

######  Observation 5.5.16.

Let $\pi_{i}\in\Delta_{n_{i}}$ for $i=1,\dots,k$, then the terms of the
sequence $n_{1},\dots n_{k}$ are totally ordered by the relation $\prec$,
defined as follows: $n_{i}\prec n_{j}$ when $i<j$ and $n_{i}\prec_{ij}n_{j}$
or $j<i$ and $n_{j}\succ_{ji}n_{i}$. Stated explicitly, $n_{i}\prec n_{j}$ if
and only if,

  1. 1.

$\pi_{i}\neq\pi_{j}$ and $\pi_{i}<_{\Delta}\pi_{j}$ for $\pi_{i}\neq\pi_{j}$.

  2. 2.

$\pi_{i}=\pi_{j}$, $i<j$ and $\pi$ is acute.

  3. 3.

$\pi_{i}=\pi_{j}$, $j<i$ and $\pi$ is grave.

Note that every two terms with different entities are treated as different
objects. In addition, $\prec$ depends on $\pi_{1},\dots,\pi_{k}$.

The following theorem provides the conditions under which the unimodal sum of
two unimodal permutations exists.

######  Theorem 5.5.17.

[20, Theorem 6] Let $\pi_{i}\in\Delta(n_{i})$ for $i=1,2$. Let
$m_{i}=\pi_{i}^{-1}(n_{i})$ and $\hat{\pi}_{i}$ be the shape of the cycle
containing $m_{i}$ in $\pi_{i}$. Then

  1. 1.

if either $\mathcal{A}_{>}(\pi_{1})\cap\mathcal{A}_{>}(\pi_{2})$ or
$\mathcal{G}_{<}(\pi_{1})\cap\mathcal{G}_{<}(\pi_{2})$ are nonempty, then
there no unimodal sums of form $\pi_{1}\oplus\pi_{2}$ or
$\pi_{2}\oplus\pi_{1}$.

  2. 2.

if instead $\hat{\pi}_{1}\in\mathcal{G}_{<}(\pi_{2})$ or
$\hat{\pi}_{2}\in\mathcal{A}_{>}(\pi_{1})$, then no unimodal sum of form
$\pi_{1}\oplus\pi_{2}$ with $m_{1}\prec m_{2}$ exists.

  3. 3.

otherwise, there is exactly one unimodal sum of form $\pi_{1}\oplus\pi_{2}$
with $m_{1}\prec m_{2}$.

######  Definition 5.5.18.

[20] When exists, the unique unimodal sum $\pi_{1}\oplus_{J}\pi_{2}$ with
$\Omega(m_{1})\prec\overline{\Omega}(m_{2})$ will be denoted by
$\pi_{1}\boxplus\pi_{2}$.

######  Definition 5.5.19.

We say that $\pi_{1}$ and $\pi_{2}$ in $\Delta(*)$ are cycle disjoint, if
there is no $\sigma\in\Delta_{*}$ such that both $\pi_{1}$ and $\pi_{2}$ have
a subcycle of shape $\sigma$.

######  Corollary 5.5.20.

[20] If both $\pi_{1},\pi_{2}\in\Delta(*)$ are shape disjoint, then both
$\pi_{1}\boxplus\pi_{2}$ and $\pi_{2}\boxplus\pi_{1}$ exist.

######  Observation 5.5.21.

We may not omit braces when using $\oplus$ since $\oplus$ is not necessarily
associative. For example, for $\pi_{1},\pi_{2},\pi_{3}\in\Delta_{*}$
$\pi_{1}<_{\Delta}\pi_{2}<_{\Delta}\pi_{3}$,
$\pi_{2}\oplus(\pi_{1}\oplus\pi_{3})$ is defined, while
$\pi_{2}\oplus\pi_{1}\oplus\pi_{3}$ is not. For example, let
$\pi_{1}<_{\Delta}\pi_{2}$ be two unimodal cycles, such that $\pi_{1}$ is
grave. Then $(\pi_{1}\oplus\pi_{1})\oplus\pi_{2}$ =
$\pi_{1}\oplus(\pi_{2}\oplus\pi_{1})$, while
$\pi_{1}\oplus(\pi_{1}\oplus\pi_{2})$ does not exist by Part 2 of Theorem
5.5.17.

One case where braces can be omitted without risking ambiguity or having the
sum not defined is the case where all $\pi_{i}$ are cycles of the same shape
since the sum $\underset{k\ times}{\underbrace{\pi\oplus\dots\oplus\pi}}$
exists and must be unique by Theorem 5.5.17 and the relations $\prec_{ij}$ and
$\succ_{ij}$.

######  Definition 5.5.22.

For $k>0$ and $\pi\in\Delta_{*}$ write

$k\pi=\underset{k\,times}{\underbrace{\pi\oplus\dots\oplus\pi}}$

to be the unimodal sum $\pi\oplus\dots\oplus\pi$ where $\pi$ appears exactly
$a$ times. We say that a unimodal permutation $\pi\in\Delta(*)$ is pure if all
the subcycles of $\pi$ have the same shape.

######  Remark 5.5.23.

From now on, we use the notation $\pi=\pi_{1}\oplus\dots\oplus\pi_{k}$ where
for every $i$, $\pi_{i}$ is pure and $\pi$ is defined.

The following lemma gives the sufficient and necessary conditions for the
existence of the unimodal sum $\pi=\pi_{1}\oplus\dots\oplus\pi_{k}$ when
$\pi_{i}$ is a full cycle for every $1\leq i\leq k$.

######  Lemma 5.5.24.

Let $\pi_{1}\in\Delta_{n_{1}},\dots,\pi_{k}\in\Delta_{n_{k}}$. Then there
exists a unique unimodal $\pi$ such

$\pi=\pi_{1}\oplus\dots\oplus\pi_{k}$

if and only if the sequence $n_{1},\dots,n_{k}$ is unimodal when compared by
$\prec$ as in Observation 5.5.16.

###### Proof.

If a unimodal permutation $\pi=\pi_{1}\oplus\dots\oplus\pi_{k}$ is unimodal
permutation, then we have $\Omega_{i}(m_{i})<\Omega_{i+1}(m_{i+1})$ and
$\pi(\Omega_{i}(m_{i}))=\Omega_{i}(n_{i})$. Therefore,
$\Omega_{1}(n_{1}),\dots,\Omega_{k}(n_{k})$ is a unimodal sequence, which is
equivalent to $n_{1},\dots,n_{k}$ being unimodal when compared by $\prec$.

Uniqueness follows from Theorem 5.5.7 and Proposition 5.5.13. To see that,
observe that if $\pi=\pi_{1}\oplus\dots\oplus\pi_{k}$ is unimodal, for every
$1\leq i<j\leq k$ and $a\in[n_{i}],b\in[n_{j}]$, $\Omega_{i}(a)<\Omega_{j}(b)$
if and only if $a\prec_{ij}b$. But $\prec_{ij}$ does not depend on
$\Omega_{i}$ and $\Omega_{j}$. Therefore, there is at most one way to order
the elements of $\biguplus_{i=1}^{k}[n_{i}]$. (Note again, that for every
$i\neq j$ the elements of $[n_{i}]$ and $[n_{j}]$ are treated as different
entities). Therefore, if $\Omega_{1},\dots,\Omega_{k}$, must be unique.

We prove the existence of the unimodal sum $\pi=\pi_{1}\oplus\dots\pi_{k}$ by
induction. The case of $k=1$ is trivial, and when $k=2$, there is a unique
unimodal $\pi$ such that $\pi=\pi_{1}\oplus\pi_{2}$ by Theorem 5.5.7. Now
assume that the statement holds for $k$ and show that it holds for $k+1$. Let
$n_{1},\dots,\pi_{k},n_{k+1}$ ordered by $\prec$. By definition,
$\mathcal{A}_{>}(\pi_{k+1})=\mathcal{G}_{<}(\pi_{k+1})=\emptyset$. Therefore,
by Theorem 5.5.17, $\pi\oplus\pi_{k+1}$ exists unless
$\pi_{k+1}\in\mathcal{A}_{>}(\pi)$ which is possible if and only if
$\pi_{k+1}$ is acute and there exist $1\leq j<k$ such that $n_{j}\succ n_{i}$
for every $i\neq j$ with $1\leq i\leq k$ and $k\geq l>j$ such that
$\pi_{l}=\pi_{k+1}$. But then we have $n_{l}\prec n_{k+1}$ and $n_{l}\prec
n_{j}$, and $j<l<k+1$, which contradicting the unimodality of
$n_{1},\dots,n_{k+1}$. ∎

###  5.6 Proof of Theorem 5.1.6

To complete the proof of the theorem, explore the connection between the
maxima of unimodal full cycles and unimodal permutations that can be built
from them by using the unimodal sum operation described in the previous
section. As noted in Observation 5.5.4, every unimodal permutation can be
represented as the unimodal sum of its cycle shapes.

We begin with the simplest case.

######  Lemma 5.6.1.

Let $\pi\in\Delta_{*}$ with the maximum $m$. Then, for every $k$, the maximum
of $k\pi$ is equal to

  1. 1.

$k(m-1)+1$ when $\pi$ is grave.

  2. 2.

$km$ when $\pi$ is acute.

###### Proof.

The case for $k=1$ is trivial, and the case of $k=2$ is part 2 of Corollary
5.5.11. Denote $m_{k}$ the maximum of $k\pi$. Assume that the statement is
true for $k-1$. Then $k\pi=\pi\oplus((k-1)\pi)$ with $J=J_{1}$ where
$J_{1}\dots J_{k}$ are the unique subsets of $[kn]$ such that $k\pi$ is
unimodal and $J_{i}(m)<J_{i+1}(m)$ for every $1\leq i<k$.

If $\pi$ is grave, then $n\succ_{1}i$ holds for every $1<i\leq k$, which
implies that maximum of $\pi\oplus((k-1)\pi)$ of is $\Omega(m)$. By induction
assumption $\pi_{k-1}=(k-1)(m-1)+1$ by Corollary 5.5.6, the maximum of $k\pi$
is $m_{k}=m+m_{k-1}+1=m+(k-1)(m-1)-1=k(m-1)+1$ as desired.

If $\pi$ is acute, then we have $n\prec_{1}i$ for every $1<i\leq k$, which
implies that the maximum of $\pi\oplus((k-1)\pi)$ is
$\overline{\Omega}(m_{k-1})$, and $m_{k}=m+m_{k-1}$ by Corollary 5.5.6. By
induction assumption, $m_{k-1}=(k-1)m$, therefore $m_{k}=m+(k-1)m=km$ as
desired. ∎

We are interested in determining the maxima of permutations with a given
number of subcycle shapes. We start with a simple observation.

######  Lemma 5.6.2.

Let $\pi_{i}\in\Delta_{n_{i}}$ for $i=1,\dots,k$ such that
$\pi=\pi_{1}\oplus\dots\oplus\pi_{k}$ is defined. Let
$\sigma,\tau\in\Delta_{*}$ such that $\sigma<_{\Delta}\tau$. Then for any
$1\leq i\leq k$ with $\pi_{i}=\tau$,

  1. 1.

If $\sigma$ is acute, there is at most one $j$ with $i<j$ such that
$\pi_{j}=\sigma$.

  2. 2.

If $\sigma$ is grave, there is at most one $j$ with $j<i$ such that
$\pi_{j}=\sigma$.

###### Proof.

By Lemma 5.5.24 $\pi=\pi_{1}\oplus\dots\oplus\pi_{k}$ is defined if and only
if $n_{1},\dots,n_{k}$ ordered by $\prec$ is unimodal. If the sequence
$\sigma$ is acute and there exist $j_{2}>j_{1}$ such that
$\pi_{j_{1}}=\pi_{j_{2}}=\sigma$, then we have $n_{i}\succ n_{j_{1}}\prec
n_{j_{2}}$, which contradicts unimodality. Similarly, if $\sigma$ is grave and
there exist $j_{1}<j_{2}<i$ such that $\pi_{j_{1}}=\pi_{j_{2}}$ then, we have
$n_{j_{1}}\succ n_{j_{2}}\prec n_{i}$, which again contradicts the unimodality
of $n_{1},\dots,n_{k}$. ∎

######  Definition 5.6.3.

Let $a_{1},\dots,a_{l}$ positive integers and $l$ distinct unimodal
$\sigma_{1},<_{\Delta}\sigma_{k}\in\Delta_{*}$ unimodal full cycles. Define

${\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}=\left\\{\pi\in\Delta(*):\text{$\pi$
has exactly $a_{i}$ subcycles of shape $\sigma_{i}$ and no other}\right\\}.$

######  Corollary 5.6.4.

Let $\sigma_{1}<_{\Delta}\dots<_{\Delta}\sigma_{l}$ be $l$ different unimodal
cycles, and $a_{1},\dots,a_{l}$ postive integers. Then the number of unimodal
permutations in $\Delta(*)$ with exactly $a_{i}$ subcycles of shape
$\sigma_{i}$ is $2^{l-1}$.

###### Proof.

We prove the statement by induction. If $l=1$, there is one such permutation,
namely $a_{l}\sigma_{l}$. Next, assume that the statement is true for $l$ and
prove it for $l+1$. Let $\sigma_{1},\dots,\sigma_{l+1}$ be $l+1$ unimodal full
cycles, $a_{1},\dots,a_{l+1}$ positive integers. By the induction assumption,
there are $2^{l-1}$ unimodal permutations with $a_{i}$ subcycles of shape
$\sigma_{i}$ for $2\leq i\leq l+1$. Next, observe that if
$\pi_{1},\dots,\pi_{k}\in\Delta_{*}$ and $\pi=\pi_{1}\oplus\dots\oplus\pi_{k}$
is defined, if $\pi_{i}<\pi_{j}$ and $\pi_{i}<\pi_{k}$, then we can’t not have
$j<i<k$, as it would imply $n_{j}\succ n_{i}\prec n_{k}$ contradicting Lemma
5.5.24. Intuitively, ”largest shapes appear in the middle.” Therefore, every
$\sigma\in{\left\langle
a_{1}\sigma_{1},\dots,a_{l+1}\sigma_{l+1}\right\rangle_{\boxplus}}$ is of form
$i\sigma_{1}\oplus\pi\oplus j\sigma_{2}$ where $\pi\in{\left\langle
a_{2}\sigma_{2},\dots,a_{l+1}\sigma_{l+1}\right\rangle_{\boxplus}}$, and $i,j$
are nonnegative integers and $i+j=a_{1}$.

By Lemma 5.6.2, we have $j\leq 1$ when $\sigma$ is acute, $i\leq 1$ when
$\sigma$ is grave. Therefore, for every
$\pi=\pi_{1}\dots\oplus\dots\pi_{m}\in{\left\langle
a_{2}\sigma_{2},\dots,a_{l+1}\sigma_{l+1}\right\rangle_{\boxplus}}$ there are
at most 2 unimodal permutations in ${\left\langle
a_{1}\sigma_{1},\dots,a_{l+1}\sigma_{l+1}\right\rangle_{\boxplus}}$ such that
all the cycles of shape $\sigma_{2},\dots,\sigma_{l+1}$ appear in exactly the
same order they appear in $\pi$. On the other hand, both unimodal sums are
defined. To see that, suppose that $\pi\in{\left\langle
a_{2}\sigma_{2},\dots,a_{l+1}\sigma_{l+1}\right\rangle_{\boxplus}}$ is a
unimodal permutation, expressed as a unimodal sum of full cycles
$\pi=\pi_{1}\oplus\dots\oplus\pi_{m}$. Let $n_{i}$ denote the maximum of
$\pi_{i}$ and $n_{\sigma}$ the maximum of $\sigma_{1}$. Then, by Lemma 5.5.24
the sequence $n_{1},\dots,n_{m}$ ordered by $\prec$ is unimodal. But then, by
definition of $\prec$ the sequences

$\underset{a_{1}\
times}{\underbrace{n_{\sigma},\dots,n_{\sigma}}},n_{1},\dots,n_{m}$

and

$\underset{a_{1}-1\
times}{\underbrace{n_{\sigma},\dots,n_{\sigma}}},n_{1},\dots,n_{m},n_{\sigma}$

are unimodal and therefore both unimodal sums

$\underset{a_{1}\
times}{\underbrace{\sigma_{1}\oplus\dots\oplus\sigma_{1}}}\oplus\pi_{1}\oplus\dots\oplus\pi_{m}$

and

$\underset{a_{1}-1\
times}{\underbrace{\sigma_{1}\oplus\dots\oplus\sigma_{1}}}\oplus\pi_{1}\oplus\dots\oplus\pi_{m}$

are defined. When $\sigma_{1}$ is grave, the argument is symmetric. ∎

######  Corollary 5.6.5.

When $\sigma_{1}<_{\Delta}\dots<_{\Delta}\sigma_{l}$, Every element of
${\left\langle a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}$
has unique representation, of $a_{1}\sigma_{1}\boxplus\pi$ or $\pi\boxplus
a_{l}\sigma_{1}$ where $\pi\in{\left\langle
a_{2}\sigma_{2},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}$.

###### Proof.

By the proof of Corollary 5.6.4, for every
$\sigma\in\boxplus\left\\{a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\\}$,
there exist
$\pi\in\boxplus\left\\{a_{2}\sigma_{2},\dots,a_{l}\sigma_{l}\right\\}$, $i,j$
such that $i+j=a_{1}$ and $i\sigma_{1}\oplus\pi\oplus j\sigma_{1}$. If
$\sigma$ is acute, then $j=0$ or $j=1$. By definition of $\boxplus$, if $j=0$,
then

$\sigma=i\sigma_{1}\oplus\pi=a_{1}\sigma_{1}\oplus\pi=a_{1}\sigma_{1}\boxplus\pi,$

and if $j=1$ then

$i\sigma=\sigma_{1}\oplus\pi\oplus
j\sigma=(a_{1}-1)\sigma_{1}\oplus\pi\oplus\sigma_{1}=\pi\boxplus
a_{1}\sigma_{1}.$

Similarily, when $\sigma$ is grave, we have either $i=1$ or $i=0$,

$\sigma=\pi\oplus a_{1}\sigma_{1}=\pi\boxplus a_{1}\sigma_{1}$

when $i=0$ and

$\sigma=\sigma_{1}\oplus\pi\oplus(a_{1}-1)\sigma_{1}$

when $i=1$. ∎

In the following proofs, we often rely on the observation stated below.

######  Observation 5.6.6.

Let $A\subseteq\mathfrak{S}_{n}$ be a set of unimodal permutations in
$\mathfrak{S}_{n}$. Then unimodal maxima polynomial $U_{A}(x))$, just counts
the number of elements in $A$ with given maximum and can be expressed as

$\sum_{\pi\in A}x^{m_{\pi}},$

where $m_{\pi}$ is the maximum of $\pi$.

######  Lemma 5.6.7.

Let $a_{1},\dots,a_{l}>0$ and $\sigma_{1}<_{\Delta}\dots<_{\Delta}\sigma_{l}$
different full cycles, and let $m_{i}$ denote the maximum of $a_{i}\sigma_{i}$
for $1\leq i\leq l$ and let $m=m_{1}+\dots+m_{l}$. Let $U_{{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}(x)$ be the
unimodal maxima polynomial of ${\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}$. Then,

$U_{{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}(x)=x^{m-l+1}(1+x)^{l-1}.$

In particular, for every $2<l$

$1+x|U_{{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}(x)$

and the quotient has non-negative coefficients.

###### Proof.

Let $m_{1}$,…$m_{l}$ be the maxima of $a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}$
respectively. We proceed by induction. When $l=1$, $U_{\left\langle
a_{1}\sigma_{1}\right\rangle_{\boxplus}}(x)=x^{m}_{1}$ and result is trivial.
When $l=2$, by Corollary 5.6.5, the elements of ${\left\langle
a_{1}\sigma_{1},a_{2}\sigma_{2}\right\rangle_{\boxplus}}$ are
$a_{1}\pi_{1}\boxplus a_{2}\pi_{2}$ and $a_{2}\pi_{2}\boxplus a_{1}\pi_{1}$.
By Observation 5.5.6, and by definition of $<_{\Delta}$, the maximum of
$a_{1}\pi_{1}\boxplus a_{2}\pi_{2}$ is $m_{1}+m_{2}$ and the maximum of
$a_{2}\pi_{2}\boxplus a_{1}\pi_{1}$ is $m_{1}+m_{2}-1$, and therefore
$U_{{\left\langle
a_{1}\sigma_{1},a_{2}\sigma_{2}\right\rangle_{\boxplus}}}(x)=x^{m_{1}+m_{2}-1}(1+x)$
and the statement is true for $l=2$.

Now, assume that the statement is true for $l-1$. By Corollary 5.6.5, for
every element of $\sigma\in{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}$ exists a
unique $\pi\in{\left\langle
a_{2}\sigma_{2},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}$ such that
either $\sigma=a_{1}\sigma_{1}\boxplus\pi$ or $\sigma=\pi\boxplus
a_{1}\sigma_{1}$.

Denote $m_{\pi},m_{\sigma}$ denote the maxima of $\pi,\sigma$ respectively. By
applying Observation 5.5.6 and the definition of $<_{\Delta}$ again, we have
$m_{\sigma}=m_{1}+m_{\pi}$ for $\sigma=a_{1}\sigma_{1}\boxplus\pi$ and
$m_{\sigma}=m_{1}+m_{\pi}-1$ for $\sigma=\pi\boxplus a_{1}\sigma_{1}$ is
$m_{1}+m_{\pi}-1$. By Observation 5.6.6,

$\begin{split}U_{{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}(x)&=\sum_{\sigma\in{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}x^{m_{\sigma}}\\\
&=\sum_{\pi\in{\left\langle
a_{2}\sigma_{2},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}x^{m_{1}+m_{\pi}-1}+x^{m_{1}+m_{\pi}}\\\
&=x^{m_{1}-1}(1+x)\sum_{\pi\in{\left\langle
a_{2}\sigma_{2},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}x^{m_{\sigma}}\\\
&=x^{m_{1}-1}(1+x)U_{{\left\langle
a_{2}\sigma_{2},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}(x)\end{split}$

By the induction hypothesis,

$U_{{\left\langle
a_{2}\sigma_{2},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}(x)=x^{m^{\prime}-l+2}(1+x)^{l-2}$

where $m^{\prime}=m_{2}+\dots+m_{l}$. Therefore,

$\begin{split}U_{{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}(x)&=x^{m_{1}-1}(1+x)x^{m^{\prime}-l+2}(1+x)^{l-2}\\\
&=x^{m_{1}+m^{\prime}-l-1}(1+x)^{l-1}=x^{m-l+1}(1+x)^{l-1}\end{split}$

as desired. ∎

######  Observation 5.6.8.

When $\pi\in\Delta(n)$ is a pure permutation, with maximum $m$, the maximum of
$\pi\oplus\pi$ behaves precisely in the same way as the unique shape $\sigma$
of its subcycles. Namely, if $\sigma$ is acute, the maximum of $\pi+\pi$ is
$2m$ when $\sigma$ is acute, and $2m-1$ when $\sigma$ is grave, parity of
$n-m$. Hence, in the following discussion, we will refer to the pure
permutations as acute or grave, even when they are not cycles.

The involution $\phi$ from Corollary 5.4.5 plays a crucial role in proving our
theorem. For a permutation $\pi$, let $m_{\pi}$ denote the minimum of $\pi$.
Recall that $m_{\phi}(\pi)=m_{\pi}\pm 1$.

In the proof of Theorem 5.1.6, we construct unimodal permutations from $\pi$
and $\phi(\pi)$. The following lemma is crucial for understanding the behavior
of maxima of unimodal permutations built this way.

######  Lemma 5.6.9.

Let $\pi$ and $\sigma$ be two pure permutations, such that
$m_{\sigma}=m_{\pi}+1$ holds. Then for every $k$ and $l$ the set of maxima of
$k\pi\boxplus l\sigma$ and $l\pi\boxplus k\sigma$ is

  1. 1.

$\left\\{(k+l)m_{\pi}+1,(k+l)m_{\pi}\right\\}$ when $\pi$ is acute.

  2. 2.

$\left\\{(k+l)m_{\pi}+l-k+1,(k+l)m_{\pi}+l-k\right\\}$ when $\pi$ is grave.

###### Proof.

By Observation 5.5.6 the minima of $k_{\pi}\boxplus l\sigma$ and
$l\sigma\boxplus k\pi$ are $m_{k\pi}+m_{l\sigma}$ and
$m_{k\pi}+m_{l\sigma}-1$. Also, both numbers appear as the minimum of one of
the sums, regardless of the order between $\sigma$ and $\pi$.

If $\pi$ is acute, then $\sigma$ is grave, the maximum of $k\pi$ is
$km_{\pi}$, and the maximum of $l\sigma$ is $l(m_{\sigma}-1)+1$. By
substituting $(m_{\sigma}-1)$ with $m_{\pi}$ and applying Corollary 5.5.6,
shows conclude the set of the maxima of $k\pi\boxplus l\sigma$ and
$l\sigma\boxplus k\pi$ is $\left\\{(k+l)m_{\pi}+1(k+l)m_{\pi}\right\\}$.

Similarily, if $\pi$ is grave then $\sigma$ is acute, the the maximum of
$k\pi$ is $k(m_{\pi}-1)+1$ and the maximum of $l\sigma$ is $lm_{\sigma}$. By
substituing $m_{\sigma}$ with $m_{\pi}+1$ and applying Corollary 5.5.6, we
show that the set of maxima of $k\pi\boxplus l\sigma$ and $l\sigma\boxplus
k\pi$ is $\left\\{m_{k\pi}+m_{l\sigma},m_{k\pi}+m_{l\sigma}-1\right\\}$, as
desired. ∎

We are ready to complete the proof of Theorem 5.1.6.

######  Theorem 5.6.10.

For a prime $p$, a cyclic descent extension for
$\mathcal{R}_{p}{\left(\mathfrak{S}_{n}\right)}$ exists if and only if $p\mid
n$.

###### Proof.

Let $p$ be a prime number. We prove that if $p|n$, then a cyclic descent
extension exists for $\mathcal{R}_{p}{\left(\mathfrak{S}_{n}\right)}$. The
proof of the converse, namely that if a cyclic descent extension exists for
$\mathcal{R}_{p}{\left(\mathfrak{S}_{n}\right)}$, then $p|n$ was given in
Lemma 5.2.1.

By the given assumption, $p$ divides $n$, and hence $q=\frac{n}{p}$ is an
integer. For any unimodal element
$\pi\in\mathcal{R}_{p}{\left(\mathfrak{S}_{n}\right)}$, there exists a unique
set of unimodal cycles $\sigma_{1},\dots,\sigma_{l}$ such that
$\sigma_{i}\in\Delta_{p}\cup\Delta_{1}$ for $1\leq i\leq l$, and positive
numbers $a_{1},\dots a_{l}$, such that $\pi$ has exactly $a_{i}$ subcycles of
shape $\sigma_{i}$ and no other subcycle shapes. In other words,
$\pi\in{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}$.

Let $S$ be the set of all possible combinations of
${{a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}}}$ such that

${{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}\cap\mathcal{R}_{n}{\left(\mathfrak{S}_{p}\right)}\neq\emptyset}.$

Then

$\mathcal{R}_{p}{\left(\mathfrak{S}_{n}\right)}\cap\Delta(n)=\biguplus_{{a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}}\in
S}{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}.$

Obviously,

$U_{\mathcal{R}_{p}{\left(\mathfrak{S}_{n}\right)}}(x)=U_{\mathcal{R}_{p}{\left(\mathfrak{S}_{n}\right)}\cap\Delta(n)}(x).$

Therefore,

$U_{\mathcal{R}_{p}{\left(\mathfrak{S}_{n}\right)}}(x)=\sum_{\\{a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\\}\in
S}U_{{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}(x).$

By Lemma 5.6.7, when $l\geq 2$. $U_{{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}(x)$ is
divisible by $1+x$, with the coefficients of the quotient being non-negative.

o finish the proof, we need to handle the case when $l=1$, as
$U_{{\left\langle q\sigma\right\rangle_{\boxplus}}}=x^{m_{q\sigma}}$ is not
divisible by $1+x$. To address this issue, we use the involution $\phi$ from
Corollary 5.4.5.

By using $\phi$, we can partition
$\mathcal{R}_{p}{\left(\mathfrak{S}_{p}\right)}\cap\Delta(p)$ into pairs
${\pi,\phi(\pi)}$. Without loss of generality, we assume that
$m_{\phi(\pi)}=m_{\pi}+1$. If this is not the case, we can replace $\pi$ with
$\phi(\pi)$.

Let $S_{\pi}$ denote all possible pairs ${a_{\pi}\pi,a_{\phi(\pi)}\phi(\pi)}$
with $a_{\pi}+a_{\phi(\pi)}=q$, where $q=\frac{n}{p}$. If $\pi=e$ or
$\phi(\pi)=e$, we replace $e$ with $p(1)$ when definitions require it. Observe
that $e=p(1)$ can be treated as a regular acute full cycle for calculation
purposes by Observation 5.6.8.

Note that the parities of $m_{\pi}$ and $m_{\phi(\pi)}$ are different, meaning
that $\pi$ is acute if and only if $\phi(\pi)$ is grave.

We proceed to calculate

$\sum_{\\{a_{\pi}\pi,a_{\phi(\pi)}\phi(\pi)\\}\in S_{\pi}}U_{{\left\langle
a_{\pi}\pi,a_{\phi(\pi)}\phi(\pi)\right\rangle_{\boxplus}}}(x).$

If $\pi$ is acute and $\phi(\pi)$ is grave, then the maxima of ${\left\langle
a_{\pi}\pi,a_{\phi(\pi)}\phi(\pi)\right\rangle_{\boxplus}}$ for every $1\leq
a_{\pi}\leq q-1$ are $qm_{\pi}+1$ and $qm_{\pi}$, with maximum of $q\pi$ being
$qm_{\pi}$ and the maximum of $q\phi(\pi)$ being $qm_{\pi}+1$, by Lemma 5.6.9.
Hence,

$\sum_{\\{a_{\pi}\pi,a_{\phi(\pi)}\phi(\pi)\\}\in S_{\pi}}U_{{\left\langle
a_{\pi}\pi,a_{\phi(\pi)}\phi(\pi)\right\rangle_{\boxplus}}}(x)=q(x^{qm_{\pi}}+x^{qm_{\pi}+1})$

which is divisible by $1+x$ with quotient $qx^{qm_{\pi}}$. If $\pi$ is grave
then the maxima of $q\pi$ is $q(m_{\pi}-1)+1$ and the maximum of $q\phi(\pi)$
is $q(m_{\pi}+1)$. For every $1<a_{\pi}<q-1$, by Lemma 5.6.9 the maxima of
${\left\langle a_{\pi}\pi,a_{\phi(\pi)}\phi(\pi)\right\rangle_{\boxplus}}$ are
$(a_{\pi}+a_{\phi(\pi)})m_{\pi}+q-2a_{\pi}+1$ and
$(a_{\pi}+a_{\phi(\pi)})m_{\pi}+q-2a_{\pi}$. Thus, we have

  1. 1.

$U_{{\left\langle q\pi\right\rangle_{\boxplus}}}=x^{qm_{\pi}-q+1}$,

  2. 2.

$U_{{\left\langle q\phi(\pi)\right\rangle_{\boxplus}}}(x)=x^{q(m_{\pi}+1)}$

  3. 3.

$U_{{\left\langle
a_{\pi}\pi,a_{\phi(\pi)}\phi(\pi)\right\rangle_{\boxplus}}}=x^{qm_{\pi}+q-2a_{\pi}+1}+x^{qm_{\pi}+q-2a_{\pi}}$
for $1\leq a_{\pi}\leq q-1$.

Therefore,

$\begin{split}&\sum_{{\left\langle
a_{\pi}\pi,a_{\phi(\pi)}\phi(\pi)\right\rangle_{\boxplus}}\in
S_{\pi}}U_{{\left\langle
a_{\pi}\pi,a_{\phi(\pi)}\phi(\pi)\right\rangle_{\boxplus}}}(x)=\\\
&x^{qm_{\pi}-q+1}+\sum_{a_{\pi}=1}^{q-1}(x^{qm_{\pi}+q-2a_{\pi}+1}+x^{qm_{\pi}+q-2a_{\pi}})+x^{q(m_{\pi}+1)}=\\\
&x^{qm_{\pi}-q+1}\sum_{i=1}^{2q-1}x^{i}\end{split}$

which is also divisible by $1+x$, with the quotient having negative
coefficients. Hence, the terms of

$U_{\mathcal{R}_{p}{\left(\mathfrak{S}_{n}\right)}}(x)$

coming from $S_{\pi}$ sum up to polynomial divisible by $1+x$ with non-
negative coefficients in the quotient, and other terms are form
$U_{{\left\langle
a_{1}\sigma_{1},\dots,a_{l}\sigma_{l}\right\rangle_{\boxplus}}}(x)$ with
$l\geq 2$ and are also divisible by $1+x$ with non-negative coefficients in
the quotient. Thus $U_{\mathcal{R}_{p}{\left(\mathfrak{S}_{n}\right)}}(x)$ is
divisible by $1+x$, and the quotient has non-negative coefficients. The result
follows. ∎

## Bibliography

  * [1] R. M. Adin, S. Elizalde and Y. Roichman, Cyclic descents for near-hook and two-row shapes, Europ. J. Combin. 79 (2019), 152-178.
  * [2] R. M. Adin, I. M. Gessel, V. Reiner, Y. Roichman Cyclic quasi-symmetric functions, Isr. J. Math. 243, 437-500 (2021).
  * [3] R. M. Adin, P. Hegedüs and Y. Roichman, Higher Lie characters and cyclic descent extension on conjugacy classes, arXiv:1909.04460.
  * [4] R. M. Adin, A. Postnikov and Y. Roichman, On characters of Weyl groups, Discrete Math. 226 (2001), 355-358.
  * [5] R. M. Adin, A. Postnikov and Y. Roichman, Combinatorial Gelfand models, J. Algebra 320 (2008), 1311-1325.
  * [6] R. M. Adin, V. Reiner and Y. Roichman, On cyclic descents of standard Young tableaux, Int. Math. Res. Not. IMRN, (2020), 10231-10276.
  * [7] R. M. Adin and Y. Roichman, Cyclic descents for involutions and matchings, in preparation.
  * [8] R. M. Adin and Y. Roichman, Matrices, characters and descents. Linear Algebra Appl. 469 (2015), 381-418.
  * [9] R. M. Adin and Y. Roichman, On maximal chains in the non-crossing partition lattice, J. Combin. Theory Ser. A 125 (2014), 18-46.
  * [10] D. Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups. Mem. Amer. Math. Soc. 202 (2009).
  * [11] H. Barcelo and A. Ram, Combinatorial representation theory, New Perspectives in Algebraic Combinatorics, MSRI Book Series 38, Cambridge University Press, New York, 1999, pp. 23-90.
  * [12] A. Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), 159–183.
  * [13] A. Björner, F. Brenti Combinatorics of Coxeter groups, Graduate Texts in Math., vol. 231, Springer, New York, 2005,
  * [14] A. Björner and M. Wachs, On lexicographically shellable posets, Trans. Amer. Math. Soc. 277 (1983), 323-341.
  * [15] P. Cellini, Cyclic Eulerian elements, Europ. J. Combinatorics 19 (1998), 545–552.
  * [16] K. Dilks, K. Petersen and J. Stembridge, Affinne descents and the Steinberg torus, Adv. in Applied Math, 42 (2009), 423-444.
  * [17] S. Elizalde and Y. Roichman, Arc permutations, J. Algebr. Comb. 39 (2014), 301-334.
  * [18] S. Elizalde and Y. Roichman, On rotated Schur-positive sets, J. Combin. Theory Ser. A 152 (2017), 121-137.
  * [19] D. Foata and M. P. Schützenberger, Major index and inversion number of permutations, Math. Nachr., (1978), 143-159.
  * [20] T. Gannon, The cyclic structure of unimodal permutations, Discrete Mathematics 237(1-3), 149-161.
  * [21] I. M. Gessel, Multipartite $P$-partitions and inner products of Schur functions, Contemp. Math. 34 (1984), 289–302.
  * [22] I. M. Gessel and C. Reutenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory Ser. A 64 (1993), 189–215.
  * [23] I. Goulden and A. Yong, Tree-like Properties of Cycle Factorizations, J. Combin. Theory Ser. A 98 (2002), 106–117.
  * [24] B. Han, Cyclic descents for Motzkin paths, arXiv:2104.03774.
  * [25] B. Huang, Cyclic descents for general skew tableaux, J. Combin. Theory Ser. A 169 (2020), 45 pages.
  * [26] A. Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), 1–61.
  * [27] Y. H. Khachtryan-Raziel, On the center of the Hurwitz graph, arXiv:1508.02620.
  * [28] A. A. Klyachko, Lie elements in a tensor algebra, (Russian) Sibirsk. Mat. Z . 15 (1974), 1296-1304.
  * [29] G. Kreweras, Sur les partitions non-croisées d’un cycle, Discrete Math. 1 (1972), 333-350.
  * [30] J. McCammond, Noncrossing partitions in surprising locations, Amer. Math. Monthly 113 (2006), 598-610.
  * [31] P. McNamara, EL-labelings, supersolvability and 0-Hecke algebra actions on posets, J. Combin. Theory Ser. A 101 (2003), 69-89.
  * [32] O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory Ser. A 125 (2014), 357-378.
  * [33] T. K. Petersen, P. Pylyavskyy and B. Rhoades, Promotion and cyclic sieving via webs, J. Algebraic Combin. 30 (2009), 19-41.
  * [34] Y. Poupard, Etude et dénombrement paralleles des partitions non croisees d’un cycle et des coupage d’un polygone convexe, Discrete Math. 2 (1972), 279–288.
  * [35] B. Rhoades, Cyclic sieving, promotion, and representation theory J. Combin. Theory Ser. A 117, (2010), 38-76.
  * [36] Y. Roichman, A recursive rule for Kazhdan-Lusztig characters, Adv. Math. 129 (1997), 24 - 45.
  * [37] Y. Roichman, A note on the number of k-roots in $\mathcal{S}_{n}$, Sém. Lothar. Combin. 70 (2014), Article B70i, 5pp (electronic).
  * [38] B. E. Sagan, The symmetric group. Representations, combinatorial algorithms, and symmetric functions, 2nd edition , Grad. Text in Mathematics 203. Springer-Verlag, 2001.
  * [39] R. Simion, Noncrossing partitions, Discrete Math. 217 (2000), 367–409.
  * [40] R. P. Stanley, Ordered structures and partitions (revision of 1971 Harvard University thesis). Memoirs Amer. Math. Soc. 119 (1972).
  * [41] R. P. Stanley, Parking functions and noncrossing partitions, Electronic J. Combin. 4 (1997), R20.
  * [42] R. P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (new series) 1 (1979), 475-511.
  * [43] R. P. Stanley. Enumerative Combinatorics: Volume 1 (2nd. ed.), Cambridge University Press, USA. (2011).
  * [44] R. P. Stanley, Enumerative combinatorics, Volume 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge Univ. Press, Cambridge, 1999.
  * [45] R. P. Stanley, Positivity problems and conjectures in algebraic combinatorics, in: Mathematics: Frontiers and Perspectives (V. Arnold, M. Atiyah, P. Lax, and B. Mazur, eds.), Amer. Math. Soc., Providence, RI, 2000, pp. 295–319.
  * [46] V. Strehl, Minimal transitive products of transpositions - The reconstruction of a proof by A. Hurwitz, Sém. Lothar. Combin. 37 (1996).
  * [47] J.-Y. Thibon, The Cycle Enumerator of Unimodal Permutations. Annals of Combinatorics 5, 493-500 (2001).
  * [48] https://en.wikipedia.org/wiki/Ascending_and_Descending


# Can extended Chaplygin gas source a Hubble tension resolved emergent
universe ?

Rikpratik Sengupta1, Prasenjit Paul2, B C Paul3 and Mehedi Kalam1 1 Department
of Physics, Aliah University, Kolkata 700160, West Bengal, India  2 Department
of Physics, Indian Institute of Engineering Science and Technology, Shibpur,
Howrah 711103, West Bengal, India  3 Department of Physics, Department of
Physics, North Bengal University, Siliguri 734014, West Bengal, India.  $^1$
<EMAIL_ADDRESS>$^2$<EMAIL_ADDRESS>$^3$
<EMAIL_ADDRESS>$^1$<EMAIL_ADDRESS>
###### Abstract

In this paper, we attempt to explore the possibility of a obtaining a viable
emergent universe scenario supported by a type of fluid known as the extended
Chaplygin gas, which extends a modification to the equation of state of the
well known modified Chaplygin gas by considering additional higher order
barotropic fluid terms. We consider quadratic modification only. Such a fluid
is capable of explaining the present cosmic acceleration and is a possible
dark energy candidate. We construct a theoretical model of the emergent
universe assuming it is constituted from such a fluid. It interestingly turns
out that the theoretical constraints we obtain on the extended Chaplygin gas
parameters from our emergent universe model are well in agreement with the
observational constraint on these parameters from BICEP2 data. Our model is
found to replicate the late time behaviour really well and reproduces
$\Lambda$-CDM like behaviour, as evident from the analysis of the statefinder
parameters. Moreover, the Hubble parameter analysis shows that for
theoretically constrained values of the ECG parameters, the Hubble tension can
be resolved yielding higher values of the present Hubble parameter $H_{0}$ in
all possible cases. Also, the value of $H(z)$ at a redshift $z=2.34$ fits
better than $\Lambda-CDM$ with recent observations in some cases. This leads
us to the realization that such a fluid is not only a probable candidate for
dark energy, but also sources an emergent universe unlike modified Chaplygin
gas and the initial singularity problem can be resolved in a flat universe
within the standard relativistic context.

##  1 Introduction

Over the years, the Big bang model has come to be accepted as the standard
model of cosmology. The predictions of the Big bang model are compatible to
quite a large extent with the presently available plethora of observational
data. However, there are some problems with the standard Big bang model.
Firstly, there is a ‘beginning’ of time, the point at which the field
equations describing the spacetime are no longer capable of describing the
physical situation due to presence of the “initial singularity[1].” As a
consequence of this, the standard Big bang model can not answer the question
about how the universe came into existence. There have been a number of
efforts to resolve this initial singularity problem and to find an answer to
the question regarding the coming into existense of the universe. Most
physicists believe that at such an early phase of the universe, the energy
densities were considerably high enough and the length scales involving the
universe were comparable to or lower than the Planck length, making quantum
gravity effects increasingly significant.

The main concern in dealing with the singularity problem is that there is no
single consistent theory of Quantum Gravity (QG) at present. The two main
theoretical setups in which a lot of efforts are being invested in this
direction are the higher dimensional Superstring/ M- theories[3, 2] and the
Loop Quantum Gravity (LQG)[4, 5]. Neither of the two scenarios have been fully
developed till date, but there has been a lot of progress in both the fields
in the last few decades. However, it is interesting to note that in a number
of investigations involving both the setups, there have been a number of
cosmological solutions in which the initial singularity is absent. In the LQG
context, cosmological models involving a “regular bounce" have been
proposed[6]. Such a bounce leads to a “cyclic” universe, where there are
repeated non singular big bangs and big crunches. In the braneworld gravity
context, which is a higher dimensional scenario inspired from the Superstring/
M- theories, similar cosmological solutions with non-singular bounce have been
obtained[7], which may also be cyclic in nature by introducing a cosmological
turnaround mechanism involving a homogenous scalar field[8]. In the context of
$11$-dimensional M- theories, a “cyclic” picture of the universe has been
obtained, where the Big bang is believed to be a collision between two dynamic
braneworlds[9].

The second problem with the standard Big bang model is that it requires
additional ingredients to explain the “dark sector” of the universe. The “dark
sector” refers to the dark energy which is believed to be responsible for the
presently observed accelerating phase of the universe[10, 11] and the dark
matter whose effect can be realized via its gravitational interaction[12, 13].
In order to explain these effects, the standard relativistic Big bang scenario
requires introduction of additional scalar fields capable of violating the
strong energy condition like the quintessence[14, 15], tachyon[16] or
phantom[17] fields, or exotic fluids like the Skyrme fluid[18]. Such a field
is required to exist in the universe in addition to the “inflaton” scalar
field responsible for the rapid accelerating expansion phase in the early
universe realized through expoonential or power law type of scale factor.
Inflation can also be realized via a tachyon field[19, 20]. For dark matter
candidates, particles which have not yet been detected experimentally and are
not predicted by the standard model of particle physics have to be taken into
account. In an attempt to resolve these problems, there have been a number of
cosmological models which make use of a modified Einstein-Hilbert action by
considering additional terms in the Lagrangian of the geometry or the matter
sector or both, contributing to non-conventional effects[21, 22, 23, 24]. The
dark energy problem can also be resolved in the higher dimensional braneworld
gravity setup[25, 26, 27] and the cyclic universe scenario obtained from the
M- theory setup[9]. The simplest resolution is by considering a generalized
Randall-Sundrum single brane model[28] which is characterized by perfect fluid
bulk matter, resulting in an ‘effective’ fluid leading to accelerated
expansion on the brane. In the cyclic picture[9], the inflationary phase is
absent and a single scalar field governs all the phases of evolution of the
universe from triggering the bounce to causing accelerated expansion. A
cosmological bounce can also be obatained in the context of modified
gravity[29, 30]. The dark matter problem can also be resolved in the context
in the context of braneworld gravity, where the gravitational effect of any
such hypothetical matter can be replaced by the higher dimensional braneworld
effects.[31]

In the pure geometrical context, using conformal spacetime geometry, another
cyclic cosmological model known as the Conformal cyclic cosmology has been
proposed by Penrose[32]. The Emergent Universe (EU) scenario has been proposed
by Ellis and Maartens[33] in 2004 and is a non-singular alternative to the Big
bang resolving the initial singularity problem but it is different from the
other scenarios in the context that there is no bounce or QG regime and also
it is not a cyclic model. The scenario assumes an initial Einstein Static
Universe (ESU), which corresponds to length scales greater than the Planck
scale to avoid the QG regime and is subsequently followed by the standard
inflationary and reheating phases. The EU scenario was originally proposed in
a relativistic context, considering the presence of a positive curvature term
in the Friedmann equations. Such an EU scenario[34] was also obtained in the
relativistic context by considering a minimally coupled scalar field with a
physically interesting potential to be the dominant source term in the early
universe. An identical EU scenario[35] can also be obtained by adding a term
quadratic in scalar curvature with a negative coupling parameter to the
gravitaional Lagrangian. It was shown that the EU scenario can be realized
with a spatial curvature term absent in the Friedmann equations[36], in the
context of semi-classical Starobinsky gravity[37]. A generalized Equation of
State (EoS) describing an EU[38] was found in the relativistic context,
accomodating normal matter, exotic matter as well as dark energy. The EU
scenario has been successfully studied in braneworld context[39] as well as
the LQG context[40]. The EU scenario can also be realised in the framewrok of
Einstein-Gauss-Bonnet gravity in four dimensions coupled with a dilaton
field[41] as well as in the modified Gauss-Bonnet gravity[42].

A type of fluid known as “Chaplygin gas”, with its origin in string
theories[43] had been used as a dark energy candidate to explain the late time
acceleration of the universe[44, 45]. Such a fluid is characterized by the EoS
$p=-\frac{B_{1}}{\rho}$, with $p$ and $\rho$ denoting the pressure and energy
density, respectively in a comoving frame, while $B_{1}$ denotes the Chaplygin
gas parameter. It is assumed that the energy density is positive and the
constant $B_{1}>0$. However, for constructing a physically consistent
cosmological model[46], the concerned EoS had to be modified to
$p=-\frac{B_{1}}{\rho^{\alpha}}$, introducing an additional free parameter
$\alpha$, such that $0\leq\alpha\leq 1$. This type of fluid is called the
Generalized Chaplygin gas (GCG)[47, 48, 49, 50]. Initially it exhibits a dust
like behaviour, but at late times it behaves asymptotically as a cosmological
constant term, thus explaining the present acceleration of the universe. The
EoS for GCG was further modified[51, 52, 53] for better correspondance with
observational data, known as modified GCG[54, 55, 56, 57, 58]. The EoS for
modified GCG has the form

$p=A_{1}\rho-\frac{B_{1}}{\rho^{\alpha}},$ (1)

where another additional constant parameter $A_{1}$ is introduced.

The Extended Chaplygin gas (ECG) EoS was proposed[59, 60, 61, 62, 63, 64, 65]
to recover barotropic fluid having a quadratic or higher order EoS, which is
basically a higher order generalization of the modified GCG at least upto the
second order. The Van der Waals fluid is an alternative to the idea of a
perfect fluid, that acts as the single source term in describing the evolution
of the universe in both the matter dominated and present accelerating phases,
identical to the Chaplygin gas family of fluids. It is also described a non-
linear Equation of state[66] just like the Chaplygin gas family. The EoS for
ECG may be written in general as

$P=\sum_{n=1}^{\infty}A_{n}\rho^{n}-\frac{B_{1}}{\rho^{\alpha}}$ (2)

Just like the other Chaplygin gas models, ECG is also used as a dark energy
candidate to explain the present cosmic acceleration[59, 60, 61, 62]. Such
models usually violate the strong energy condition of standard General
Relativity (GR), which raises a possibility of violation of the null energy
condition in the relativistic context. For obatining an EU scenario, it is
likely that the null energy condition of GR must be violated[67]. So, it is
worth investigating the possibility whether an EU scenario in the relativistic
context is supported by such a dark energy candidate. Such an investigation
has been performed for modified GCG[68], but it turns out that in order to
make the EU scenario viable, the choice of the modified GCG parameters that
are to be made are physically unrealistic. So, modified GCG does not support a
viable EU scenario but since ECG incorporates higher order barotropic fluid
extension of the modified GCG, it may be worth exploring whether this type of
a fluid, suitable as a possible dark energy candidate can support an EU
scenario or not. Moreover, ECG is known to admit bouncing and cyclic types of
universes in the relativistic context[69].

In the following section, we shall consider the mathematical details of the
possible EU scenario sourced by the ECG. In the final section we shall discuss
the physical significance of the conclusions that can be inferred from our
obtained results.

##  2 Mathematical model

Putting $n=2$ in Eq. (2), the EoS for Extended Chaplygin Gas (ECG) shall be
upto the second order, which physically represents the quadratic barotropic
EoS given by

$P=A_{1}\rho+A_{2}\rho^{2}-\frac{B_{1}}{\rho^{\alpha}},$ (3)

where $\rho$ represents the energy density, pressure is denoted by $P$ and
$A_{1}$, $A_{2}$, $B_{1}$ and $\alpha$ denote the ECG parameters.

Here, for simplicity of the model, such that the number of free parameters be
reduced and the energy density may be obtained analytically, the following
realistic assumptions are made following the previous works[62, 70, 71]

$\displaystyle\alpha=1$ $\displaystyle A_{1}=A_{2}-1$ (4) $\displaystyle
B_{1}=2A_{2}.$

Conservation of energy-momentum tensor can be written as

$\dot{\rho}+3H(\rho+P)=0,$ (5)

where $H=\frac{\dot{a}}{a}$ denotes the Hubble parameter and $a$ is the scale
factor.

Plugging in the EoS given by Eq. (3) into the conservation Eq. (5), we solve
for the energy density in terms of the scale factor $a$, which turns out to be

$\rho=1+\frac{2+\sqrt{5a^{30A_{2}}e^{\frac{3\pi}{2}}-1}}{a^{30A_{2}}e^{\frac{3\pi}{2}}-1}.$
(6)

Our analysis is valid for all epochs of cosmic evolution such that we consider
the early universe has considerably high energy densities ($\rho>>1$) and then
the density gradually decreases before reaching an asymptotic value. For the
late time $\rho<<1$. In order to obtain Eq. (6) we have used the
approximations $tan^{-1}(\rho+1)\approx\frac{\pi}{2}$ for $\rho>>1$ and
$tan^{-1}(\rho+1)\approx\frac{\pi}{4}$ for $\rho<<1$.

The first Friedmann equation is given as,

$\rho=3H^{2}=3\bigg{(}\frac{\dot{a}}{a}\bigg{)}^{2}.$ (7)

We choose $\kappa^{2}=8\pi G=1$.

From Eqs. (6) and (7), we obtain a solution for the scale factor $a$ as

$a=\frac{\frac{2\sqrt{3}e^{\frac{15t}{\sqrt{3}A_{2}}}}{e^{\frac{15t}{A_{2}}-1}}-\frac{\sqrt{5}}{e^{\frac{15t}{A_{2}}-1}e^{\frac{3\pi}{4}}}}{\frac{2\sqrt{3}e^{\frac{15t}{\sqrt{3}A_{2}}}}{e^{\frac{15t}{A_{2}}-1}}-\frac{\sqrt{5}}{e^{\frac{15t}{A_{2}}-1}e^{\frac{3\pi}{4}}}-2}.$
(8)

In obtaining the above solution, we have used the approximation
$lna=\frac{1}{a-1}$. This approximation is justified as solution (6)
approximately represents the late time behaviour particularly well. We shall
compare this scale factor to the one used to describe an Emergent Universe
(EU), which is also valid for all time scales, to establish a correspondance
between the independent ECG parameter and the free parameter of the EU scale
factor. That will allow us to obtain an estimate for the ECG parameters if ECG
is to be considered as the constituent of an EU.

The expression of scale factor for the emergent Universe can be written of the
form[36]

$a(t)=a_{0}(10^{4}+e^{\frac{\sqrt{3}Bt}{2}})^{\frac{2}{3(A+1)}}.$ (9)

As we can see, the above scale factor does not vanish as we go backwards in
time to $t=0$ or beyond as in the standard Big bang picture. As a result there
is no appearance of singularity in the Friedmann equations governing the
dynamics of the universe. This physically means that the universe has no
beginning at a particular point of time and the big bang is replaced by an
ESU, whose length scales are large enough to avoid a quantum gravity regime.

If the universe with ECG as constituent is capable of behaving as an EU, then
the scale factors obtained in Eqs. (8) and (9) must be identical. Comparing
the two, expanding both the scale factors binomially and equating the
coefficients for the first terms after expansion, we can obtain a
correspondance between the free EU parameter $a_{0}$ and the independent ECG
parameter $A_{2}$ as

$a_{0}=\frac{2\sqrt{2}\times
10^{5}}{2e^{\frac{15}{A_{2}}-1}e^{\frac{3\pi}{4}}+2\sqrt{2}\times 10^{5}}$
(10)

We know that the deceleration parameter $q(z)$ is defined as,

$q(z)=\frac{\ddot{z}(1+z)}{\dot{z}^{2}}-2,$ (11)

where $z$ denotes the redshift parameter.

The scale factor in Eq. (9) can be written in terms of the redshift $z$ as

$z=\frac{1}{a}-1=\frac{1}{a_{0}\bigg{(}10^{4}+e^{\frac{\sqrt{3}Bt}{2}}\bigg{)}}-1$
(12)

Taking first order derivative with respect to time, we get

$\dot{z}=-\frac{B}{a_{0}\sqrt{3}\bigg{(}A+1\bigg{)}}\frac{e^{\frac{\sqrt{3}Bt}{2}}}{\bigg{(}10^{4}+e^{\frac{\sqrt{3}Bt}{2}}\bigg{)}^{\frac{3A+5}{3A+3}}}.$
(13)

Writing down the above expression for $\dot{z}$ in terms of the redshift
parameter $z$, we obtain

$\dot{z}=-\frac{B}{\sqrt{3}\bigg{(}A+1\bigg{)}}\bigg{[}\frac{\big{(}z+1\big{)}^{\frac{2-3A-3A^{2}}{6(A+1)}}}{a_{0}^{\frac{4+9A+3A^{2}}{6(A+1)}}}-10^{4}\frac{\big{(}z+1\big{)}^{\frac{3A+5}{6(A+1)}}}{a_{0}^{\frac{3A+1}{6(A+1)}}}\bigg{]}$
(14)

Figure 1: variation of deceleration parameter $q(z)$ with redshift $z$

Taking second order derivative of $z$ with respect to time, we get

$\ddot{z}=\frac{\frac{B^{2}}{2(A+1)}e^{\frac{\sqrt{3}Bt}{2}}}{a_{0}\bigg{(}10^{4}+e^{\frac{\sqrt{3}Bt}{2}}\bigg{)}^{\frac{3A+5}{3A+3}}}-\frac{\frac{3B^{2}}{4}e^{\frac{\sqrt{3}Bt}{2}}\frac{(3A+5)^{2}}{3A+3}}{a_{0}\bigg{(}10^{4}+e^{\frac{\sqrt{3}Bt}{2}}\bigg{)}^{\frac{2(3A+4)}{3(A+1)}}}.$
(15)

This can be expressed in terms of the redshift parameter $z$ as

$\displaystyle\ddot{z}=\frac{3B^{2}}{6(A+1)}\bigg{[}\frac{(z+1)^{\frac{2-3A-3A^{2}}{6(A+1)}}}{a_{0}^{\frac{3A^{2}+9A+1}{6(A+1)}}}-10^{4}\frac{(z+1)^{\frac{3A+5}{6(A+1)}}}{a_{0}^{\frac{3A+1}{6(A+1)}}}\bigg{]}$
$\displaystyle-\frac{3B^{2}(3A+5)}{2(3A+3)^{2}}\bigg{[}\frac{(z+1)^{\frac{1}{3}}}{a_{0}^{\frac{2}{3}}}-\frac{2X10^{4}(z+1)^{\frac{3A+5}{6}}}{a_{0}^{\frac{1-3A}{6}}}+10^{8}a_{0}^{\frac{3A+1}{3}}(z+1)^{\frac{3A+4}{3}}\bigg{]}.$
(16)

Figure 2: variation of jerk parameter $j(z)$ with redshift $z$

The observational constraints on the EU parameters $A$ and $B$ respectively
are $-0.034\leq A\leq 0.0014$[72] and $0.003<B<0.5996$[73]. Using Eqs. (12),
(14) and (16) in Eq.(11), it turns out that the deceleration parameter is
independent of the second EU parameter $B$. So, we obtain the deceleration
parameters for the lower and upper limits of $A$, respectively.

For the lower limit of $A$, we have

$q(z)=\frac{1.45}{1-10^{4}a_{0}^{1.45}(z+1)^{1.45}}-0.55~{}[forA=-0.034]$ (17)

and for the upper limit, we obtain

$q(z)=\frac{1.5}{1-10^{4}a_{0}^{1.5}(z+1)^{1.5}}-0.47~{}[forA=0.0014]$ (18)

The variation of the two deceleration parameters obtained above, along the
redshift parameter $z$ have been plotted in Figure 1. The curve in red color
represents the variation of the deceleration parameter with $z$ for the
observationally constrained lower limit of EU parameter $A$, while the curve
denoted in blue color represents the variation of deceleration parameter with
$z$ for the observationally constrained upper limit of $A$. We shall discuss
the physical consequence of the obtained plots in the concluding section.

The statefinder parameters[74] related to the late time behaviour are defined
as

$\displaystyle j=\frac{\ddot{H}}{H^{3}}-3q-2$ $\displaystyle
s=\frac{j-1}{3(q-\frac{1}{2})},$ (19)

where $j$ is known as the jerk parameter and $s$ is known as the snap
parameter.

For the lower obsernational bound of $A$, we obtain the jerk and the snap
parameters as follows

$j(z)=\frac{4.205+2.1045\times
10^{4}a_{0}^{1.45}(1+z)^{1.45}}{\bigg{[}1-10^{4}a_{0}^{1.45}(z+1)^{1.45}\bigg{]}^{2}}-\frac{1.74}{1-10^{4}a_{0}^{1.45}(1+z)^{1.45}}+0.055,$
(20)

and

$\displaystyle s(z)$ $\displaystyle=$ $\displaystyle\frac{4.205+2.1045\times
10^{4}a_{0}^{1.45}(1+z)^{1.45}}{3\bigg{[}0.4+1.05\times
10^{4}a_{0}^{1.45}(1+z)^{1.45}\bigg{]}\bigg{[}1-10^{4}a_{0}^{1.45}(z+1)^{1.45}\bigg{]}}-\frac{1.74}{3\bigg{[}0.4+1.05\times
10^{4}a_{0}^{1.45}(1+z)^{1.45}\bigg{]}}$ (21)
$\displaystyle-\frac{0.945\bigg{[}1-10^{4}a_{0}^{1.45}(1+z)^{1.45}\bigg{]}}{\bigg{[}0.4+1.05\times
10^{4}a_{0}^{1.45}(1+z)^{1.45}\bigg{]}}.$

Figure 3: variation of snap parameter $s(z)$ with redshift $z$

For the higher observational bound on $A$, the jerk and snap parameters are
evaluated as

$j(z)=\frac{4.5+2.25\times
10^{4}a_{0}^{1.5}(1+z)^{1.5}}{\bigg{[}1-10^{4}a_{0}^{1.5}(z+1)^{1.5}\bigg{]}^{2}}-\frac{1.32}{1-10^{4}a_{0}^{1.5}(1+z)^{1.5}}-0.0282,$
(22)

and

$\displaystyle s(z)$ $\displaystyle=$ $\displaystyle\frac{4.5+2.25\times
10^{4}a_{0}^{1.5}(1+z)^{1.5}}{3\bigg{[}0.53+0.97\times
10^{4}a_{0}^{1.5}(1+z)^{1.5}\bigg{]}\bigg{[}1-10^{4}a_{0}^{1.5}(z+1)^{1.5}\bigg{]}}-\frac{1.32}{3\bigg{[}0.53+0.97\times
10^{4}a_{0}^{1.5}(1+z)^{1.5}\bigg{]}}$ (23)
$\displaystyle-\frac{1.0282\bigg{[}1-10^{4}a_{0}^{1.5}(1+z)^{1.5}\bigg{]}}{\bigg{[}0.53+0.97\times
10^{4}a_{0}^{1.5}(1+z)^{1.5}\bigg{]}}.$

The variation of the jerk and snap parameters along the redshift for both
values of $A$ have been plotted in Figures 2 and 3, respectively.

Finally, we obtain the Hubble parameter $H=\frac{\dot{a}}{a}$ in terms of the
redshift $z$ as

$H(z)=\frac{B}{\sqrt{3}\bigg{(}A+1\bigg{)}}\bigg{[}\frac{\big{(}z+1\big{)}^{-\frac{4+9A+3A^{2}}{6(A+1)}}}{a_{0}^{\frac{4+9A+3A^{2}}{6(A+1)}}}-10^{4}\frac{\big{(}z+1\big{)}^{-\frac{3A+1}{6(A+1)}}}{a_{0}^{\frac{3A+1}{6(A+1)}}}\bigg{]}$
(24)

Figure 4: variation of Hubble parameter $H(z)$ (in km/sec/Mpc) with redshift
$z$.

It is to be noted here that unlike the deceleration parameter, the Hubble
parameter depends on both the EU parameters $A$ and $B$. There is also
dependence on the ECG parameters which can in turn be expressed in terms of
the single parameter $a_{0}$. The variation of the Hubble parameter along the
redshift is given in Figure 4.

The EU parameter $B$ is absent in the deceleration parameter as the same
dependence on $B$ appears in the numerator and denominator of the expression
for the deceleration parameter, making it independent of $B$. The Hubble
parameter is however found to depend on both the EU parameters $A$ and $B$.
Also as mentioned in the manuscript, for obtaining analytic solutions, we have
assumed some correspondence between the different ECG parameters following
previous works, as a result of which the independent ECG parameter is $A_{2}$
which can be expressed in terms of the parameter $a_{0}$ from our analysis. We
have used the plots of the variation of different parameters, namely the
deceleration parameter, the jerk and snap parameters to constraint the value
of the ECG parameters such that for observational bounds available on the EU
parameters, we can obtain best fit for these parameters with the available
observational data. The theoretical constraint on the ECG parameters obtained
from our model in the process are confirmed with the Hubble parameter data as
well and also with the previously obtained observational constraint from the
BICEP2 data and they are found to be in very good agreement.

The physical explanation of the plots have been provided in the concluding
section.

##  3 Discussion and Conclusion

In this paper, we have investigated the possibility of a viable EU scenario
sourced by ECG. As discussed earlier, the different Chaplygin gas models
finding their origin in the higher dimensional String theories are probable
dark energy candidates, capable of explaining the late time accelerating
behaviour of the universe. For obtaining such a behaviour in the relativistic
context, it is expected that the strong energy condition as obtained from GR
must be violated, which raises a possibility of violation of the null energy
condition due to negative pressure. The violation of the null energy condition
is essential for obtaining an EU scenario in the standard relativistic context
in order to avert the initial singularity. So, it is worth investigating
whether such a fluid supports an EU or not. Earlier investigation has revealed
that the modified GCG does not support an EU for the realistic choice of the
parameters concerned with the modified GCG[68]. However, the fluid we are
considering here is an extension of the modified GCG EoS, allowing
consideration of higher order barotropic fluid at least up to the quadratic
term. Consideration of the additional term can possibly modify the fluid
making it capable of supporting an EU, unlike the modified GCG, as it is found
to support bouncing and cyclic types of regular cosmological solutions[69].

We have considered upto second order ECG term, physically representing the
quadratic barotropic fluid. In order to obtain analytical mathematical
solutions, we have assumed cetrain realistic correspondances between three of
the free ECG parameters that have been applied in other investigations
also[62, 70, 71]. The free parameter $\alpha$ involved in the EoS is assumed
to be of unit magnitude without any loss of generality, as in the case of all
the Chaplygin gas candidates[53, 62, 70, 71]. Vanishing divergence of the
energy-momentum tensor yield the conservation equation and the energy density
of the ECG is obtained from it. Using the first Friedmann equation, the scale
factor is obtained using the late time approximation, as the solution for the
energy density is valid for all epochs and represents the late time behaviour
particularly well.

At this point we consider that ECG can support a viable EU scenario in order
to impose some constraints on the ECG parameters from our theoretical model.
So we consider the scale factor describing an EU to be identical to the scale
factor that we have obtained by solving the Friedmann equation for a universe
constituted out of ECG. Expanding both the scale factors binomially and
equating the coefficients for the first terms, a relationship is obtained
between the free parameter $a_{0}$ contained in the expression for the EU
scale factor and the independent ECG parameter $A_{2}$, while the other two
ECG parameters are expressed in terms of $A_{2}$ and $\alpha$ is of the order
of unity. This allows us to constraint the ECG parameters $A_{1}$, $A_{2}$ and
$B_{1}$ from our theoretical model of EU supported by ECG. If the
theoretically obtained constraints on the upper and lower limits are in
agreement with the constraints that have been obtained from the observational
data[62], then it will justify our model and we shall argue that ECG supports
a viable EU scenario.

It is known from observational findings that the present value of the
deceleration parameter is $q\approx-0.57$ (SN+BAO datasets)[75] and the
observationally obtained value of the redshift parameter at which the
deceleration parameter flips sign from negative to positive, which means
physically, the universe makes a transition from the accelerating to the
decelerating phase (actually the reverse is hapenning as we are travelling
forward in time), is typically $z\approx 0.8$[76]. While plotting the $q(z)$
versus $z$ curves for the observationally bound lower and upper limits of the
EU parameter $A$ in Fig. 1, we are free to fix the EU parameter $a_{0}$. For
fixing this parameter within a certain range, we find the best fit curve for
$q(z)$ vs $z$, such that the present value of the deceleration parameter and
the value of the redshift parameter at which the deceleration parameter
vanishes are in close approximity to the observational values. As we see from
the plot, for lower $A$, the rate of decrease of q is higher in the early era
and then it flipped with the onset of the deceleration era, such that the time
of flip is same in both the cases. We find from the best fit plot that in case
of $A$=-0.034: for $z$=0, $q$=-0.61 and for $q$=0, $z$=0.82 and in case
$A$=0.0014: for $z$=0, $q$=-0.59 and for $q$=0, $z$=0.83. For obtaining these
best fit values, we obtain the theoretical constraint on $a_{0}$ as
0.0031$\leq a_{0}\leq$ 0.003753.

Now using Eq. (10), we may constrain the ECG parameter $A_{2}$ as 1.327$\leq
A_{2}\leq$ 1.362 from our EU model. Using our earlier cosiderations in Eq.
(4), we may also correspondingly constrain $A_{1}$ and $B_{1}$, respectively
as 0.327$\leq A_{1}\leq$ 0.362 and 2.654$\leq B_{1}\leq$ 2.724. Thus, we
obtain constraints on the three ECG parameters from our EU model by
theoretically constraining the EU parameter $a_{0}$, making use of the
observational constraint on the EU parameter $A$ and assuming that ECG can
support an EU. These constraints on the ECG parameters obtained from our
theoretical model are in very good agreement with the BICEP2 observational
data which gives the observationally constrained limits on the ECG parameters
as $1.2\leq A_{2}\leq 1.6$, $0.2\leq A_{1}\leq 0.6$ and $2.4\leq B_{1}\leq
3.2$[62]. Thus, we can see that the ECG parameters obtained from our
theoretical model are within the observational range obtained from BICEP2 data
and may be estimated more precisely, as the difference between the lower and
upper bounds appear to be considerably smaller.

In order to analyze the late time behaviour of our obtained EU model more
explicitly, we compute the jerk and the snap parameters. We evaluate the jerk
and snap parameters for the obeservational lower and upper bounds of the EU
parameter $A$ using the best fit value of the EU parameter $a_{0}$, which we
had obtained to evaluate the theoretical constraints on the ECG parameters. As
we can see from Fig. 2, the present value of the jerk parameter at redshift
$z=0$ that we obtain from our theoretical model is very close to 1 for both
values of $A$, and as we go back in time, the jerk parameter first decreases
and then increases but the rate of increase is higher for the upper bound
(higher value) of $A$. This behaviour is in agreement with the $\Lambda$-CDM
(cold dark matter) model. For the snap parameter, we see from Fig. 3 that for
the higher $A$, the snap parameter presently has a vanishingly small positive
value and as we go back in time, the parameter reduces and changes sign to a
negative value. For the lower value of $A$, presently it has a very tiny
negative value and first decreases and then increases as we go back in time.
The $\Lambda$-CDM model predicts the present values of the jerk and snap
parameters to be 1 and 0, respectively. As we see, our model reproduces this
value to a close approximation. Thus the late time behaviour is also well
predicted by our EU model supported by an ECG.

We have also obtained an expression for the Hubble parameter in terms of the
redshift and plotted its variation. We have plotted the variation for all
possible combinations of the lower and upper observational bounds of the EU
parameters $A$ and $B$. The parameter $a_{0}$ which encodes the information
regarding the ECG parameters is chosen for the best fit of the curve and the
corresponding ECG parameter $A_{2}$ turns out to have a value 1.35 for
obtaining the best fit of the Hubble variation to observational data[77, 78].
This value of the ECG parameter is within our obtained range from the analysis
of the other parameters and hence in support of our model. We see from Fig. 4
that as the redshift increases, the Hubble parameter also increases as
expected. We obtain the values of $H(z)$ at two particular redshifts, namely
$z=0$ (denoting the value of the Hubble parameter at present time $H_{0}$) and
$z=2.34$ (in km/sec/Mpc), to tally with observational data. For the first
choice of EU parameters $A=-0.034$ and $B=0.003$, we get $H_{0}=73.254$ and
$H(z=2.34)=233.7$. For $A=-0.034$ and $B=0.5996$, we get $H_{0}=72.302$ and
$H(z=2.34)=240.2$. For $A=0.014$ and $B=0.003$, we get $H_{0}=74.206$ and
$H(z=2.34)=231$. For $A=0.014$ and $B=0.5996$, we get $H_{0}=70.029$ and
$H(z=2.34)=244.2.$ The observed data suggests $H_{0}=73.24\pm
1.74km/sec/Mpc$[77] and $H(z=2.34)=222\pm
7km/sec/Mpc\cite[cite]{[\@@bibref{}{Debulac}{}{}]}$. There is a slight
discrepancy with the $\Lambda-CDM$ estimated values of $H_{0}=67km/sec/Mpc$
and $H(z=2.34)=238km/sec/Mpc$. For the present values of the Hubble parameter,
our model provides a better fit for the observational data for all four
possible combinations of the upper and lower EU parameter bounds, thus
resolving the Hubble tension. For the value at higher redshift, our model
predicts a better fit in two cases while the fit is worse compared to
$\Lambda-CDM$ estimation in the other two cases.

Hence, we make the claim that the ECG fluid does source an EU scenario
consistent with the observational data unlike modified GCG, besides being a
probable dark energy candidate as evident from the fact that it replicates the
late time behaviour very well. This characteristic of supporting an EU, unlike
modified GCG, can be interpreted to be a result of the modification arising
from the higher order quadratic barotropic fluid term in the modified EoS, due
to which the null energy condition (NEC) can be violated while in case of
modified GCG only the strong energy condition is violated. We conclude that
ECG is more open to exploring different cosmological scenarios than the
previous Chaplygin gas candidates as it supports a non-singular universe owing
to NEC violation besides reproducing a $\Lambda$-CDM like behaviour and at
late times. The most important perspective is that the initial singularity
problem can be resolved in a standard relativistic context for a flat universe
and the obtained late time cosmology from the EU model is in good agreement
with observational data, at per with or even better than the standard
$\Lambda-CDM$ model in some cases.

##  4 Appendix

We present a few steps of the derivation of the solution (6) from the
conservation equation (5). The solution is obtained under some approximations
namely $tan^{-1}(\rho+1)\approx\frac{\pi}{2}$ for $\rho>>1$ (early universe)
and $tan^{-1}(\rho+1)\approx\frac{\pi}{4}$ for $\rho<<1$ (late times). If
these approximations are invoked, then the solution (6) will satisfy the
differential equation (5). A few steps towards obtaining the solution are
presented below.

On integrating both sides of the differential equation (5) and simplifying, we
are left with

$lna=\frac{1}{30A_{2}}ln{\bigg{[}\frac{\rho^{2}+2\rho+2}{\rho^{2}-2\rho+1}\bigg{]}}-\frac{\pi}{20A_{2}}+C^{\prime},$
(25)

where $C^{\prime}$ is a constant of integration which we assume to be zero for
simplicity.

Upon further simplification, this may be expressed as a quadratic equation of
$\rho$ having the form

$(a^{30A_{2}}-e^{-\frac{3\pi}{2}})\rho^{2}-2(a^{30A_{2}}+e^{-\frac{3\pi}{2}})\rho+(a^{30A_{2}}-2e^{-\frac{3\pi}{2}})=0$
(26)

On taking the positive root solution, we get

$\rho=1+\frac{2+\sqrt{2a^{30A_{2}}e^{\frac{3\pi}{2}}+1+3a^{30A_{2}}e^{\frac{3\pi}{2}}-2}}{a^{30A_{2}}e^{\frac{3\pi}{2}}-1}$
(27)

which on simplification gives Equation (6).

##  5 Acknowlwdgement

MK and BCP is thankful to the Inter-University Centre for Astronomy and
Astrophysics (IUCAA),Pune, India for providing the Visiting Associateship
under which a part of this work was carried out. RS is thankful to the Govt.
of West Bengal for financial support through SVMCM scheme.

## References

  * [1] S. W. Hawking and G. F. R. Ellis, Astrophys. J. 152 (1968) 25.
  * [2] B. Zwiebach, A First Course in String Theory (Cambridge University Press, 2004).
  * [3] J. Polchinski, String Theory, Vol. 2, Superstring Theory and Beyond (Cambridge University Press, 1998).
  * [4] R. Gambini and J. Pullin, A First Course in Loop Quantum Gravity (Oxford University Press, 2011)
  * [5] C. Rovelli, Living Rev. Rel. 11 (2008) 5.
  * [6] M. Bojowald, R. Maartens and P. Singh, Phys. Rev. D 70 (2004) 083517.
  * [7] Y. Shtanov and V. Sahni, Phys. Lett. B 557 (2003) 1.
  * [8] V. Sahni and A. Toporensky, Phys. Rev. D 85 (2012) 123542.
  * [9] P. J. Steinhardt and N. Turok, Phys. Rev. D 65 (2002) 126003.
  * [10] A.G. Riess et al., Astron. J. 116 (1998) 1009.
  * [11] S. Perlmutter et al., Astrophys. J. 517 (1999) 565.
  * [12] F. Zwicky, Helvetica Physica Acta 6 (1933) 110.
  * [13] V. C. Rubin and Jr. W. K. Ford, Astrophys. J. 159 (1970) 379.
  * [14] E. J. Copeland, A. R. Liddle and D. Wands, Phys. Rev. D 57 (1998) 4686.
  * [15] I. Zlatev, L. M. Wang and P. J. Steinhardt, Phys. Rev. Lett. 82 (1999) 896.
  * [16] P. P. Avelino, L. Losano and J.J. Rodrigues, Phys. Lett. B 699 (2011) 10.
  * [17] M. P. Dabrowski, T. Stachowiak, and M. Szydlowski, Phys. Rev. D 68 (2003) 103519.
  * [18] R. Sengupta, B. C. Paul and P. Paul, Pramana – J. Phys. 96 (2022) 114.
  * [19] N. Bilic et al., JCAP 08 (2019) 034.
  * [20] R. Sengupta, P. Paul, B. C. Paul, S. Ray, Int. Jour. of Mod. Phys. D 28 (2019) 1941010.
  * [21] S. Nojiri, S. D. Odintsov and P. V. Tretyakov, Phys. Lett. B 651 (2007) 224.
  * [22] S. Nojiri, S. D. Odintsov and O. G. Gorbunova, J. Phys. A 39 (2006) 6627.
  * [23] M. Szydlowski, A. Kurek and A. Krawiec, Phys. Lett. B 642 (2006) 171.
  * [24] D. Huterer and E. V. Linder, Phys. Rev. D 75 (2007) 023519.
  * [25] G. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B 485 (2000) 208.
  * [26] V. Sahni and Y. Shtanov, JCAP 11 (2003) 014.
  * [27] S. Chakraborty, A. Banerjee and T. Bandyopadhyay, arXiv:0707.0199.
  * [28] L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690.
  * [29] S. D. Odintsov, V. K. Oikonomou and E. N. Saridakis, Annals of Physics 363 (2015) 141.
  * [30] S. K. Tripathy, B. Mishra, S. Ray and R. Sengupta, Chinese Journal of Physics 71 (2021) 610.
  * [31] S. Pal, S. Bharadwaj and S. Kar, Phys.Lett. B 609 (2005) 194.
  * [32] R. Penrose, AIP Conference Proceedings 1446 (2012) 233.
  * [33] G F. R. Ellis and R. Maartens, Class. Quant. Grav. 21 (2004) 223.
  * [34] G F. R. Ellis, J. Murugan and C. G. Tsagas, Class. Quant. Grav. 21 (2004) 233.
  * [35] D. J. Mulryne, R. Tavakol, J. E. Lidsey and G. F. R. Ellis, Phys. Rev. D 71 (2005) 123512.
  * [36] S. Mukherjee, B. C. Paul, S. D. Maharaj and A. Beesham, arXiv:gr-qc/0505103 (2005).
  * [37] A. A. Starobinsky, Phys. Letts. B 91 (1980) 99.
  * [38] S. Mukherjee, B. C. Paul, N. Dadhich, S. D. Maharaj and A. Beesham, Class. Quant. Grav. 23 (2006) 6927.
  * [39] A. Banerjee, T. Bandyopadhyay and S. Chakraborty, Grav. Cosmol. 13 (2007) 290.
  * [40] B. Paik, M. Y. Khlopov, M. Kalam and S. Ray, Physics of the Dark Universe 32 (2021) 100823.
  * [41] B. C. Paul and S. Ghosh, Gen. Rel. and Grav. 42 (2010) 795.
  * [42] B. C. Paul, S. D. Maharaj and A. Beesham, arXiv:2008.00169.
  * [43] M. Bordemann, J. Hoppe, Phys. Lett. B 317 (1993) 315.
  * [44] V. Gorini, A. Kamenshchik and U. Moschella, Phys. Rev. D 67 (2003) 063509.
  * [45] U. Alam, V. Sahni, T. D. Saini and A. A. Starobinsky, Mon. Not. Roy. Astron. Soc. 344 (2003) 1057.
  * [46] M. C. Bento, O. Bertolami and A. A. Sen, Phys. Rev. D 66 (2002) 043507.
  * [47] A. R. Amani and B. Pourhassan, Int. Jour. of Theoretical Phys. 52 (2013) 1309.
  * [48] H. Saadat and B. Pourhassan, Int. Jour. of Theoretical Phys. 52 (2013) 3712.
  * [49] H. Saadat and B. Pourhassan, Int. Jour. of Theoretical Phys. 53 (2014) 1168.
  * [50] A. R. Amani and B. Pourhassan, Int. Jour. of Geom. Methods in Mod. Phys. 11 (2014) 1450065.
  * [51] U. Debnath, A. Banerjee, and S. Chakraborty, Class. Quant. Grav. 21 (2004) 5609.
  * [52] Y-B. Wu et al., Mod. Phys. Lett. A 30 (2015) 1550005.
  * [53] S. Ray et al., Int. Jour. of Mod. Phys. D 30 (2021) 2150093.
  * [54] H. Saadat and B. Pourhassan, Astrophys. Space Sci. 343 (2013) 783.
  * [55] H. Saadat and B. Pourhassan, Astrophys. Space Sci. 344 (2013) 237.
  * [56] B. Pourhassan, Int. Jour. of Mod. Phys. D 22 (2013) 1350061.
  * [57] J. Sadeghi, B. Pourhassan, M. Khurshudyan and H. Farahani, Int. Jour. of Theoretical Phys. 53 (2014) 911.
  * [58] E.O. Kahya, B. Pourhassan and S. Uraz, Phys. Rev. D 92 (2015) 103511.
  * [59] B. Pourhassan and E.O. Kahya, Advances in High Energy Physics 2014 (2014) 231452.
  * [60] E.O. Kahya, M. Khurshudyan, B. Pourhassan and R. Myrzakulov, A. Pasqua, Euro. Phys. Jour. C 75 (2015) 43.
  * [61] B. Pourhassan and E.O. Kahya, Results in Phys. 4 (2014) 101.
  * [62] E. O. Kahya and B. Pourhassan, Astrophys. Space Sci. 353 (2014) 677.
  * [63] J. Sadeghi, H. Farahani, B. Pourhassan, Eur. Phys. J. Plus 130 (2015) 84.
  * [64] B. Pourhassan, Canadian Jour. of Phys. 94 (2016) 659.
  * [65] B. Pourhassan, H. Farahani, S. Upadhyay, New Astronomy 86 (2021) 101569.
  * [66] M. Biswas, S. Maity and U. Debnath, Jour. of Holography Applications in Physics, 1(1) (2021) 71.
  * [67] M. Zhu and Y. Zheng, JHEP 11 (2021) 163.
  * [68] S. Dutta, S. Mukerji and S. Chakraborty, Advances in High Energy Physics, 2016 Article ID 7404218 (2016).
  * [69] A. Salehi, Phys. Rev. D 94 (2016) 123519.
  * [70] B. Pourhassan, Physics of the Dark Universe 13 (2016) 132.
  * [71] E. O. Kahya and B. Pourhassan, Mod. Phys. Lett. A Vol. 30, 13 (2015) 1550070.
  * [72] B.C. Paul, P. Thakur and S. Ghose, Mon. Not. Royal Astron. Society 407 (2010) 15.
  * [73] B.C. Paul, P. Thakur and S. Ghose, Mon. Not. Royal Astron. Society 413 (2011) 686.
  * [74] V. Sahni et. al., J. Exp. Theor. Phys. Lett., 77 (2003) 201.
  * [75] S. K. J. Pacif, S. Arora and P. K. Sahoo, Phys. Dark Universe 32 (2021) 100804.
  * [76] A. Al Mamon and K. Bamba, Eur. Phys. J. C 78 (2018) 862.
  * [77] Riess et al., ApJ 826 (2016) 56.
  * [78] Debulac et al., A and A 574 (2015) A59.