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\begin{document}

\bibliographystyle{acm}

\title{Generator Sets for the Alternating Group}

\author[Aviv Rotbart]{Aviv Rotbart$^ \dagger$}

\thanks{$^\dagger$ Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel.\newline
Email:  {\tt aviv.rotbart@gmail.com}} 
\thanks{This paper is part of the author's MSc thesis, which was written at Bar-Ilan University under the supervision of Prof. R.\ M.\ Adin and Prof. Y.\ Roichman.} 

\begin{abstract}
Although the alternating group is an index 2 subgroup of the
symmetric group, there is no generating set that gives a Coxeter structure
on it.  Various generating sets were suggested and studied by Bourbaki,
Mitsuhashi, Regev and Roichman, Vershik and Vserminov, and others.  In a recent
work of Brenti, Reiner and Roichman, it is explained that palindromes in
Mitsuhashi's generating set play a role similar to that of reflections in
a Coxeter system.

We study in detail the length function with respect to the set of
palindromes. Results include an explicit combinatorial description, a
generating function, and an interesting connection to Broder's restricted
Stirling numbers.
\end{abstract}

\maketitle

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\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 65 \rms (2011), Article~B65b\hfill}
\def\thepage{}

\section{Introduction}
The study of parameters (statistics) of the symmetric group and other related groups is a very active branch of combinatorics in recent years. A major step was made about one hundred years ago, when MacMahon~\cite{MM} showed that the parameters \emph{major index} and \emph{inversion number} are equi-distributed on the symmetric group, $S_n$. This important result is the foundation of the field, and stimulated many subsequent generalizations and refinements. 

It is well known that several statistics on $S_n$ may be defined via its Coxeter generators (simple reflections) $\{ s_i=(i,i+1)\mid 1\le i \le n-1\}$, or via the transpositions (reflections) $\{ t_{ij}=(i,j)\mid 1\le i<j \le n\}$. Unfortunately, the alternating group $A_n \subseteq S_n$ is not a Coxeter group. Our goal is to study generating sets for the alternating group that play a role similar to that of reflections in the symmetric group, and to explore the combinatorial properties of $A_n$ based on these sets.

A good candidate is the set $\{s_1s_{i+1}=(1,2)(i+1,i+2)\mid 1< i < n-1\}$. Mitsuhashi~\cite{MI} pointed out that these generators for the alternating group play a role similar to that of the above Coxeter generators of $S_n$. Regev and Roichman~\cite{RR} describe a canonical presentation of the elements in $A_n$ based on this set. They also calculate the generating functions of length and other statistics, with respect to this set of generators.

Our work deals with $A_n$-statistics calculated with respect to a new set of generators, $\{s_1t_{ij}=(1,2)(i,j)\mid 1\le i<j \le n\}$. This set consists of palindromes in Mitsuhashi's generators discussed above. As Brenti, Reiner and Roichman [3] explain, these palindromes play a role similar to that of reflections in the symmetric group. The following diagram describes the relations between the four generating sets mentioned above. 

$$
\begin{matrix}
  \text{$S_n$-Coxeter} & \underrightarrow{\text{conjugate by $S_n$}} & \text{$S_n$-Transpositions} \\
  C=\{ s_i\mid 1\le i \le n-1\}&          & T=\{ t_{ij}\mid 1\le i<j \le n\}&\\ \\
 \Biggm\downarrow{\text {multiply by $s_1$}}&                   &\Biggm\downarrow{\text {multiply by $s_1$}} & \\ \\
  \text{$A_n$-Coxeter (Mitsuhashi)}&  \underrightarrow{\text{take palindromes}} & \text{$A_n$-Transpositions} \\
    C(A_n)=\{ s_1s_{i+1}\mid 1< i \le n-1\}&         & T(A_n)=\{s_1t_{ij}\mid 1\le i<j \le n\}&\\
\end{matrix}
$$ 

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\markboth{\SMALL AVIV ROTBART}{\SMALL GENERATOR SETS FOR THE
ALTERNATING GROUP}
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\vskip15pt

Various aspects of the generating set $T(A_n)$ are studied in this work, including: canonical forms of elements in $A_n$ with respect to $T(A_n)$; length of elements and the relation between length and number of cycles; a generating function for length expectation and variance for length; and finally a connection with Broder's restricted Stirling numbers~\cite{BRO}.

The methods used in this work include: manipulations on generating functions of Stirling numbers; theorems on the number of cycles of a permutation and on the number of permutations of a given length in $S_n$; bijections between certain subsets of permutations in $A_n$, $S_n$ and permutations with Broder's property (see Definition \ref{dfrstr1}). 

The paper is organized as follows. Detailed background and notations for the symmetric and alternating groups, as well as for Stirling numbers, is given in Section~2. In Section~3 we present the main results achieved in this work. The \emph{A} canonical presentation is analyzed in Section~4. In Section~5 we discuss refined counts of permutations in $A_n$, while the relation between length and the number-of-cycles statistic is analyzed in Section~6. In Section~7 we calculate the generating function of length with respect to the generating set $T(A_n)$. The expectation and variance of the length function are studied in Section~8. The relation between our results and restricted Stirling numbers is analyzed in Section~9.

\section{Background}

\subsection{The Symmetric Group}
In this subsection, we present the main notations, definitions and theorems on the symmetric group, denoted by $S_n$.

\begin{nota}
Let $n$ be a nonnegative integer, then $[n]:=\{1,2,3,\dots ,n\}$ (where $[0]:=\emptyset$). 
\end{nota}


\begin{df}
Denote by $\mathbb{N}$ the set of natural numbers.
The \emph{symmetric group} on $n\in \mathbb{N}$ letters (denoted by $S_n$)  is the group consisting of all permutations on $n$ letters, with composition as the group operation.
\end{df}

\begin{df}
Given a permutation $v\in S_n$, we say that a pair $(i,j)\in [n]\times [n]$ is an \emph{inversion} of $v$ if $i<j$ and $v(i)>v(j)$. If $(i,i+1)$ is a transposition of $v$, then it is called an \emph{adjacent transposition}.
\end{df}

\begin{df}
The \emph{Coxeter generators} of $S_n$ are 
$$
\{ s_i=(i,i+1)\mid 1\le i \le n-1\},
$$ 
i.e., all the adjacent transpositions.
\end{df}

It is a well-known fact that the \emph{symmetric group} is a \emph{Coxeter
group} with respect to the above generating set. The following natural statistic describes the length of permutations in the symmetric group, with respect to the Coxeter generating set:

\begin{df}\label{lengthSnC} The \emph{length} of a permutation $v\in S_n$ with respect to the Coxeter generators is defined to be 
$$
\ell_{C}(v):=\min\{\ r \geq 0 \ | \  v=s_{i_1} \ldots s_{i_r} \mbox{ for some } i_1,\ldots,i_r \in [n-1] \ \}.
$$
\end{df}

\begin{df}
The \emph{inversion number} of $v\in S_n$ is 
$$
inv(v):=|\{(i,j) \ | \ 1 \leq i<j\leq n, \ v(i)>v(j)\}|.
$$
\end{df} 

\begin{fac}
For each $v\in S_n$, we have
$$
inv(v)=\ell_{C}(v).
$$
\end{fac}



Another important set of generators for $S_n$ is the set of all transpositions.

\begin{nota}
Denote by $T$ the set of all transpositions in $S_n$, i.e., 
$$
T=\{(i,j)\mid 1\le i<j \le n\}.
$$
\end{nota}

The definition of length with respect to $T$ is similar.

\begin{df}\label{lengthSnT}
Let $v\in S_n$, then  
$$
\ell_{T}(v):=\min\{\ r \geq 0 \ | \  v=t_1
\ldots t_r,\quad t_i \in T \ \}.
$$
\end{df}

A well known result describes the connection between the number of cycles and this length statistics in $S_n$.
\begin{thm}
If $cyc(v)$ is the number of cycles in $v\in S_n$, then
$$
\ell_{T}(v) + cyc(v)=n
$$
\end{thm} 

This result will be useful in some of the proofs in this work.

\subsection{The Alternating Group}\label{altgrp_sec}
In this section we define the alternating group, which is a subgroup of the symmetric group. We also describe a known generating set for this group and the corresponding generating function of length.

\begin{df}
The \emph{Alternating Group} on $n$ letters, denoted by $A_n$, is the group consisting of all even permutations in the symmetric group $S_n$; i.e., $A_n:=\{v\in S_n\mid sign(v)=1\}$. 
\end{df}

Following Mitsuhashi~\cite{MI} we let
$$
a_i:=s_1s_{i}=(1,2)(i,i+1)\qquad (2\le i\le n-1).
$$

The set $C(A_n):=\{a_i\ | \ 2\le i\le n-1\}$ generates the alternating group on $n$ letters, $A_{n}$.

\medskip

Regev and Roichman~\cite{RR} used Mitsuhashi's generators to describe a covering map  $f:A_{n+1} \to S_n$, which allows us to translate $S_n$-identities into corresponding $A_{n+1}$-identities. They gave a formula for the generating function of length with respect to these generators.

\begin{pro}[\sc {\cite[\sc Thm.~6.1]{RR}}]
Denoting by $\ell_{C(A_n)}(\cdot)$ is the length with respect to Mitsuhashi's generators, we have
$$
\sum_{w\in A_{n+1}}q^{\ell_{C(A_n)}(w)} = (1+2q)(1+q+2q^2)\cdots (1+q+\ldots + q^{n-2}+2q^{n-1}).
$$
\end{pro}

\subsection{Stirling Numbers}
For basic properties of Stirling numbers the reader is referred to ~\cite{St}. In this subsection we describe one important generalization of them, Broder's ~\cite{BRO} restricted Stirling numbers.

\begin{df}\label{dfrstr1}
The unsigned $r$-restricted Stirling number of the first kind, denoted by $\ribua{n}{k}_r$, is the number of permutations of the set $\{1,2,\dots,n\}$ with $k$ disjoint cycles, with the restriction that the numbers $1, 2, \dots, r$ belong to distinct cycles. The case $r=1$ gives the usual unsigned Stirling numbers of the first kind.
\end{df}


\begin{df}
The Kronecker delta function is defined as follows:
$$
\delta_{i,j}=
\begin{cases}
1,& \text{if } i=j; \\
0,& \text{otherwise.}
\end{cases}
$$
\end{df}

\begin{cla}
$r$-Stirling numbers of the first kind satisfy the same recurrence relation as unsigned Stirling numbers of the first kind, except for the initial conditions:
$$
\ribua{n}{k}_r=(n-1){\ribua{n-1}{k}}_r+{\ribua{n-1}{k-1}}_r, \qquad (r<k<n),
$$
with the following initial conditions:
\begin{align*}
&\ribua{n}{k}_r=0, & (k<r \text{ or } n<k); \\
&\ribua{n}{r}_r=\frac{(n-1)!}{(r-1)!},  &(r \leq n); \\
&\ribua{n}{n}_r=1, &(r \leq n). 
\end{align*}

\end{cla}

\begin{thm}[\sc{\cite[\S 6.9]{BRO}}]
The generating function of unsigned $r$-restricted Stirling numbers of the first kind is
$$
\sum_{k=0}^{n} {\ribua{n}{k}}_r\cdot x^{k}=
\begin{cases}
x^r(x+r)(x+r+1)\cdots (x+n-1)& \text{if } 1\leq r\leq n; \\
0& \text{otherwise}.
\end{cases}
$$
\end{thm}

\begin{df}\label{dfrstr2}
The $r$-restricted Stirling number of the second kind, denoted by ${\silsul{n}{k}}_r$, is the number of ways to partition the set $\{1,2,\dots,n\}$ into $k$ nonempty disjoint subsets with the restriction that the numbers $1, 2, \dots, r$ belong to distinct subsets. The case $r=1$ gives the usual Stirling numbers of the second kind.
\end{df}

\begin{cla}
$r$-restricted Stirling numbers of the second kind satisfy the same recurrence relation as Stirling numbers of the second kind, except for the initial conditions.
$$
\silsul{n}{k}_r={k\cdot \silsul{n-1}{k}}_r+{\silsul{n-1}{k-1}}_r, \qquad (r<k<n),
$$
with the following initial conditions:
\begin{align*}
&\silsul{n}{k}_r=0, & (k<r \text{ or } n<k); \\
&\silsul{n}{k}_r=r^{n-r},  &(r\leq n); \\
&\silsul{n}{n}_r=1, &(r\leq n). 
\end{align*}

\end{cla}

\begin{thm}[\sc{\cite[\S 6.10]{BRO}}]
The generating function of $r$-restricted Stirling numbers of the second kind is
$$
\sum_{n=0}^{\infty} {\silsul{n}{k}}_r\cdot x^{n}=
\begin{cases}
\frac{x^k}{(1-rx)(1-(r+1)x)\cdots (1-kx)},& \text{if } 1\leq r\leq k; \\
0,& \text{otherwise}.
\end{cases}
$$
\end{thm}

Restricted Stirling numbers of the first and second kind satisfy the same orthogonality relation as the usual (unsigned) Stirling numbers, as described in the following theorem.

\begin{thm}[\sc{\cite[\S 4.5]{BRO}}]
\label{thmres}
We have
$$
\sum_{k=0}^{n} {\ribua{n}{k}}_r\cdot{\silsul{k}{m}}_r\cdot (-1)^k=
\begin{cases}
(-1)^n\cdot\delta_{m,n}, & \text{if } r\leq m\leq n; \\
0,& \text{otherwise}.
\end{cases}
$$
\end{thm}
\subsection{Harmonic Numbers}
\begin{df}\label{def_harmc}
The \emph{$n$-th harmonic number}, denoted by $H_n$, is the sum of the reciprocals of the first $n$ positive integers:
$$
H_n= 1+\frac{1}{2}+\frac{1}{3}+\cdots + \frac{1}{n}.
$$
\end{df}
\begin{df}\label{def_genharmc}
The \emph{generalized $n$-th harmonic number of order m}, denoted by $H_{n,m}$, is 
$$
H_{n,m}= 1+\frac{1}{2^m}+\frac{1}{3^m}+\cdots + \frac{1}{n^m}.
$$
\end{df}


\section{Main Results}
In this section we present the main results of this paper. Details and proofs will be given in Sections~4--8.

Let
$$
a_{ij} := s_1t_{ij}=(12)(ij), \qquad (1 \le i < j \le n).
$$
The set of \emph{$A$-transpositions} 
$$
T(A_n):=\{a_{ij} \mid 1 \le i < j \le n \}
$$ 
generates the alternating group on $n$ letters. The length of an element $v\in A_n$ can be naturally defined with respect to the above generators:
$$
 \ell_{T(A_n)}(v)= \min \{k\geq 0 \mid v= v_1 \cdots v_k,\quad v_i\in T(A_n)\}.
$$


\begin{nota}
Denote by $a(n,m)$ the number of elements of length $m$ in $A_n$.
\end{nota}

Our first result is a Stirling-type recursion for $a(n,m)$.
\begin{pro}[{\it Corollary \ref{conc_len}\rm}]
$$
a(n,m)=(n-1)\cdot a(n-1,m-1) + a(n-1,m), \qquad (0<m<n),
$$
with initial conditions $a(n,0)=1$ for $n\geq 0$, and $a(n,n)=0$ for $n>0$. 
\end{pro}


The following result relates the length function to the cycle number.

\begin{pro}[{\it Corollary \ref{conc_cyc}\rm}]
Let $v\in A_n$, $n\geq 2$, Then
\begin{equation*}
\ell_{T(A_n)}(v)= \\
\begin{cases}
n-cyc(v)&  \text{if 1,2 are in different cycles of v;} \\
n-cyc(v)-1&  \text{if 1,2 in the same cycle of v.}
\end{cases}
\end{equation*}
\end{pro}
(For $n\leq 2$, $A_n$ contains only the identity permutation.)

Note that the length function $\ell_{T(A_n)}(v)$ is odd if and only if $1,2$ are in the same cycle in $v$ (see Corollary~\ref{pro_nice}).

\begin{pro}[{\it Theorem \ref{thm_genfunc}\rm}] 
For $n\geq 2$, we have
\begin{align*}
\sum_{v\in A_n} x^{\ell_{T(A_n)}(v)}
&=\sum_{k=0}^n a(n,k)\cdot x^{k} \\
&=(1+2x)(1+3x)\cdots (1+(n-1)x) \\
&= \prod_{t=2}^{n-1}(1+tx). \nonumber
\end{align*}
\end{pro}

\begin{thm}[{\it Theorem \ref{thm_exp_var}\rm}]
The expected value of $\ell_{T(A_n)}$ is
$$
E[\ell_{T(A_n)}]=n-H_n-\frac{1}{2},
$$
and its variance is
$$
Var[\ell_{T(A_n)}]=H_n-H_{n,2}-\frac{1}{4}.
$$
\end{thm}


Finally, we discuss a certain generalization of Stirling numbers and relate it to our statistic $a(n,k)$. The generalization discussed is Broder's restricted Stirling numbers~\cite{BRO}, see Definitions \ref{dfrstr1} and \ref{dfrstr2}. The connection was initially established using the On-Line Encyclopedia of Integer Sequences~\cite{EIS}.

\begin{pro}[{\it Theorem \ref{thm_str}\rm}] 
For $0\leq k\leq n-2$,
we have
$$
a(n,k)={\ribua{n}{n-k}}_2
$$
\end{pro}


\medskip

\section{The $A$ Canonical Presentation}
%\begin{rem} 
%In this section we present the main results obtained in this paper according to their order of appearance. 
%\end{rem}
In this section we consider a canonical presentation of elements in $A_n$ by the corresponding $s_1t_{ij}$ generators.

\subsection{A Generating Set for $A_n$}\label{subsdefan}
We let
$$
a_{ij} := s_1t_{ij}=(12)(ij), \qquad (1 \le i < j \le n).
$$

Denote by $T(A_n):=\{a_{ij} \mid 1 \le i < j \le n \}$ the set of $A$-transpositions.

\begin{df}\label{def_newgen}
For $n \geq 3$ define the following subset of permutations in $A_n$:
\begin{equation*}
 R_n= \{ (12)(in) \mid 1 \leq i<n \} \cup \{ e \}. 
\end{equation*}
\end{df}

\begin{nt}
$R_n$ is a subset of $T(A_n)$,  $R_n=(T(A_n)\setminus T(A_{n-1}))\cup \{e\} .$
\end{nt} 


\begin{thm}\label{thm_canon}
Let $v \in A_n$, $n \geq 3 $. Then there exist unique elements $v_i \in R_i$, $3 \leq i \leq n$, such that $v=v_3\cdots v_n$. Call it the canonical presentation of $v$.
\end{thm}

\begin{lem}\label{lem_pres}
Let $k\in \mathbb{N}$, $m_1,\cdots,m_{2k}\in \{1,\cdots,n\}$ be distinct and let
$$
v=(m_1m_2)\cdots(m_{2k-1}m_{2k})\in S_n,\quad m_1\neq m_2,\cdots, m_{2k-1}\neq m_{2k}
$$ 
be a product of transpositions. Then, for every $1\leq i\leq 2k$, there exists a presentation of $v$ as a product of transpositions in which $m_i$ appears in the rightmost factor only. 
\end{lem}

\begin{proof}[Proof of Lemma \ref{lem_pres}]
We provide a proof for $i=1$. 
First we prove the assertion for the case $k=2$, i.e.,
for the case where $v$ is a product of two cycles.
If $\{m_1,m_2 \} \cap \{m_3,m_4 \}= \varnothing$, then $v=(m_1m_2)(m_3m_4)=(m_3m_4)(m_1m_2)$, as required. Else, if $m_2=m_3$, then $v=(m_1m_2)(m_2m_4)=(m_1m_2m_4)=(m_2m_4m_1)=(m_2m_4)(m_4m_1)$, as required. The case $m_2=m_4$ is similar. Else, if $m_1=m_3$, then $v=(m_1m_2)(m_1m_4)=(m_2m_1m_4)=(m_4m_2m_1)=(m_4m_2)(m_2m_1)$, as required. The case $m_1=m_4$ is similar. If $m_1=m_3$ and $m_2=m_4$, then $v$ is the identity, thus its cycles are disjoint and commute with each other. All possible cases were checked and thus we are finished.   

Now we turn to the general case, where $v$ is a product of $k$ cycles. By induction the lemma applies also for this case as we can perform the same steps described in the simple case repeatedly until the desired form of $v$ is achieved. 
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm_canon}] 
We proceed by induction on $n$. For $n=3$, we have
$$A_3=\{(12)(13),(12)(23),e \}=R_3,$$ 
and thus the claim holds. Now assume that each $w \in A_{n-1}$, $n\geq 4$, has a unique canonical presentation $w=w_3\cdots w_{n-1}$, $w_i\in R_i$. We will show that, if $v \in A_n$, then $v$ has a unique canonical presentation as well. This follows actually from Lemma~\ref{lem_pres}. We assume that $v \in A_n \setminus A_{n-1}$, otherwise the proof follows immediately from the induction hypothesis. First we apply Lemma~\ref{lem_pres} to $v$ to get $n$ in the rightmost factor only. We have $v=g_1\cdots g_k=g_1\cdots g_{k-1}(12)(12)g_k$ with $n$ in $g_k$ only. Now, since $g_1\cdots g_{k-1}(12) \in A_{n-1}$, according to the induction hypothesis it has a unique canonical presentation, say $w_1\cdots w_{t}$. Thus we have $v=w_1\cdots w_{t}(12)g_k$ and this is unique canonical presentation for $v$, because $(12)g_k$ is unique. 
\end{proof}

By Theorem~\ref{thm_canon}, we obtain the following corollary.

\begin{cor}
The set of $A$-transpositions, $T(A_n):=\{a_{ij} \mid 1 \le i < j \le n \}$, generates the alternating group on $n$ letters.
\end{cor}


\begin{df}\label{deflenm}
For $v\in A_n$ with the canonical presentation $v=v_3 \cdots v_n$, let 
$$
\hat{\ell}(v)=\#\{i\mid v_i\neq e\}.
$$
\end{df}

\begin{thm}\label{thm3} For all $v\in A_n$,
$$
\hat{\ell}(v) = \ell_{T(A_n)}(v).
$$
\end{thm} 
In other words the length of the canonical presentation coincides with the natural length with respect to the generating set $T(A_n)$.

\begin{proof}[Proof of Theorem~\ref{thm3}] 
It suffices to show that, if $\hat{\ell}(v)=r$, then $v$ can not be presented as a product of less than $r$ generators. For $n=3$ it was shown that $A_3=R_3$, thus all the elements in $A_3$ are of length $1$, except for the identity $e$ whose length is $0$. For $n > 3$ denote the length of the canonical presentation of $v$ by 
$\hat{\ell}(v)=r$. Denote the shortest presentation of $v$  by $v_2=b_1\cdots b_k$, $b_i\in T(A_n)$. Then $\ell_{T(A_n)}(v)=k$. Now we can apply the corollary of Lemma~\ref{lem_pres} described in the proof of Theorem~\ref{thm_canon} to turn $v_2$ into a canonical presentation of $v$, say $v_2'$, with $\ell_{T(A_n)}(v_2')=k$. Since $v_1$ and $v_2'$ are two canonical presentations of the same permutation, according to Theorem~4.2 they are actually the same presentation, i.e., $r=k$. 
\end{proof}   

In the rest of this paper, we explore the natural length function with respect to $T(A_n)$. For this purpose, we use the equivalence to the length of the canonical expression proved above, as needed. 


\section{Counting Permutations in $A_n$}
In this section we study the number of permutations in $A_n$ of a given length with respect to $T(A_n)$. A Stirling-type recurrence relation for this statistic is described. 

\begin{df}
Let 
$$
A(n,m) = \{v\in A_n \mid \ell_{T(A_n)}(v)=m \}
$$
and 
$$
a(n,m)=|A(n,m)|.
$$
\end{df}

\begin{pro}\label{prop1} For $n\geq 3$,
$$
a(n,1)=a(n-1,1)+n-1.
$$
\end{pro}


\begin{proof}[Proof of Proposition~\ref{prop1}]
By Definition \ref{def_newgen}, $R_n \setminus \{e\}$ is the subset of generators of $A_n$ that do not belong to $A_{n-1}$; namely $R_n \setminus \{e\}=(T(A_n)\setminus T(A_{n-1}))=\{(12)(nj)\mid 1\leq j <n\}$. These are the generators that involve the new letter $n$. Thus $|R_n|=n$. 

For every $n$, $A(n,1)=T(A_n)$, and since $T(A_n)=T(A_{n-1})\cup (R_n \setminus \{e\})$ (this is a disjoint union), we conclude $a(n,1)=a(n-1,1)+n-1$.  
\end{proof}


\begin{thm}\label{thm_groups}
We have
$$
A(n,m) = A(n-1,m-1)\cdot R_n~ \cup ~A(n-1,m),
$$
where the union on the right-hand side is a disjoint union.
\end{thm}

\begin{proof}[Proof of Proposition~\ref{thm_groups}] 
We prove the claim by two-sided set inclusion. 
First we prove that $A(n,m) \supseteq A(n-1,m-1)\cdot R_n \cup A(n-1,m)$. Note that the right-hand side of the equation is a disjoint union, according to the properties of the $A_n$ canonical presentation.

On the other hand, we also have $A(n-1,m) \subseteq A(n,m)$ because a permutation $v$ of length $m$  in $A_{n-1}$ is also of length $m$ in $A_n$. The new generators in $A_n$ can not shorten the length of $v$ because they involve the new letter $n$ which is a fixed point in $v$. 
 
Let $v\in A(n-1,m-1)$, and consider its canonical presentation. Multiply $v$ by $w \in R_n$ from the right side to have the canonical presentation of a permutation $v\cdot w\in A_n$ of length $m-1+1=m$, i.e., $v\cdot w\in A(n,m)$.

We showed that each part of the union on the right-hand side of the equation is contained in the left-hand side, therefore the union itself is also contained in the left-hand side. This proves the first inclusion.
Now we show that $A(n,m) \subseteq A(n-1,m-1)\cdot R_n \cup A(n-1,m)$. Let $v\in A(n,m)$.
\begin{enumerate}
\item
If $n$ is a fixed point in $v$, then $v\in A(n-1,m)$ with the same canonical presentation. 
\item 
Otherwise, $n$ is not a fixed point, and therefore the canonical presentation of $v$ reads as follows:
$$
v= \underbrace{r_{1}\cdots r_{k-1}}_\text{$\in A(n-1,m-1)$} \cdot \underbrace{r_{n}}_\text{$\in (R_n \setminus \{e\})$}, \quad r_{i}\in R_{i}, \quad 1\leq i \leq n.
$$
We have $r_{n}\in R_n \setminus \{e\}$ because $n$ is not a fixed point and must appear in the presentation. The above canonical presentation of $v$ establishes the required inclusion.
\end{enumerate} 
\end{proof}

From Proposition~\ref{prop1} and  Theorem~\ref{thm_groups} we can conclude the following relation.
\begin{cor}\label{conc_len} For $1\leq m\leq n-2$,
we have
$$
a(n,m)=a(n-1,m-1)\cdot (n-1) + a(n-1,m).
$$
\end{cor}


\section{Relation between Length and Cycle Number}
In this section we show that the length $\ell_{T(A_n)}(\cdot)$ and the number of cycles $cyc(\cdot)$ are strongly related statistics on $A_n$.

\begin{obs}\label{obs_cyc}
For $n\geq 3$, and $v\in A(n,1)$, $cyc(v)=n-2$.
\end{obs}
In other words, the number of cycles in a generator of $A_n$ is $n-2$.

\begin{proof}[Proof of Observation \ref{obs_cyc}] Every generator $v\in A_n$ is of the form $(12)(in)$, where $1\leq i<n$. If $i=1$, or $i=2$, then $v$ has one cycle of length $3$ and $n-3$ cycles of length $1$ (fixed points). This sums to $n-2$ cycles. Otherwise $i>2$, and then $v$ has two cycles of length $2$ and $n-4$ more cycles of length $1$. This also sums to $n-2$ cycles in $v$. 
\end{proof} 




\begin{cor}\label{pro_nice}
For every $n\geq 2$ and $v\in A_n$, the letters $1,2$ are in the same cycle in $v$ if and only if $\ell_{T(A_n)}(v)$ is odd.
\end{cor}

\begin{proof}[Proof of Corollary~\ref{pro_nice}] If $v\in A_n$ is of length one, $1,2$ share the same cycle since the structure of a generator is $(12)(ij)$. In length two, $1,2$ appear in different cycles because of the multiplication process described in the proof of Theorem~\ref{thm_groups}. For length three, $1,2$ are in the same cycle according to the same process, and so on and so forth. For odd length, the letters $1,2$ are in the same cycle, and for even length they are in different cycles. This proves both implications of the equivalence. 
\end{proof}

\begin{thm}\label{thm_cyc}
For $n\geq 3$ and $v\in A_n$,
we have
$$
cyc(v)=
\begin{cases}
n-\ell_{T(A_n)}(v)& \text{if $\ell_{T(A_n)}(v)$ is even}, \\
n-\ell_{T(A_n)}(v)-1&  \text{if $\ell_{T(A_n)}(v)$ is odd}.
\end{cases} 
$$
\end{thm}

\begin{proof}[Proof of Theorem~\ref{thm_cyc}]
We proceed by induction on $n$. For $n=3$, we have 
$$A_3=\{(12)(13)=(213),(12)(23)=(123),e=(1)(2)(3) \},$$ 
and the claim follows. Assume that, for each $v\in A_n$, we have
$$
cyc(v)=
\begin{cases}
n-\ell_{T(A_n)}(v)& \text{if $\ell_{T(A_n)}(v)$ is even}, \\
n-\ell_{T(A_n)}(v)-1&  \text{if $\ell_{T(A_n)}(v)$ is odd}.
\end{cases} 
$$
Now, $w\in A_{n+1}$ can be obtained in two ways by Theorem~\ref{thm_groups}. First, by multiplying $v\in A_n$ by $r\in R_{n+1}$, and secondly by adding the letter $n+1$ as fixed point to some $v\in A_n$. Both cases will be analyzed.
\begin{enumerate}
\item 
In this case $w=vr$ for $v\in A_n$, $r\in R_{n+1}$. If $\ell_{T(A_n)}(v)$ is even, then, by Corollary~\ref{pro_nice}, the letters $1,2$ are in different cycles in $v$, and therefore they will be in the same cycle in $w$, thus $cyc(w)=cyc(v)-1$ (see Theorem~\ref{thm_groups} for details). The length of $w$ is $\ell_{T(A_{n+1})}(w)=\ell_{T(A_n)}(v)+1$. By the induction hypothesis,
$$
cyc(w)=cyc(v)-1=n-\ell_{T(A_n)}(v)-1=n-\ell_{T(A_{n+1})}(w)=(n+1)-\ell_{T(A_{n+1})}(w)-1,
$$ 
as required. If $\ell_{T(A_n)}(v)$ is odd, then, by Corollary~\ref{pro_nice} and Theorem~\ref{thm_groups}, $cyc(w)=cyc(v)+1$ and $\ell_{T(A_{n+1})}(w)=\ell_{T(A_n)}(v)+1$.  By the induction hypothesis,
$$
cyc(w)=cyc(v)+1=n-\ell_{T(A_n)}(v)-1+1=n-\ell_{T(A_{n+1})}(w)+1=(n+1)-\ell_{T(A_{n+1})}(w),
$$
as required.
\item 
In this case $w=v$ for some $v\in A_n$, where the letter $n+1$ is a fixed point in $w$. Here, $cyc(w)=cyc(v)+1$ and $\ell_{T(A_{n+1})}(w)=\ell_{T(A_n)}(v)$. If $\ell_{T(A_n)}(v)$ is even,
$$
cyc(w)=cyc(v)+1=n-\ell_{T(A_n)}(v)+1=n-\ell_{T(A_{n+1})}(w)+1=(n+1)-\ell_{T(A_{n+1})}(w),
$$
as required, and, if $\ell_{T(A_n)}(v)$ is odd,
$$
cyc(w)=cyc(v)+1=n-\ell_{T(A_n)}(v)-1+1=n-\ell_{T(A_{n+1})}(w)=(n+1)-\ell_{T(A_{n+1})}(w)-1,
$$
as required.
\end{enumerate}
In both cases the relation between cycle number and length holds, therefore the theorem is proved.
\end{proof}


Corollary~\ref{pro_nice} and Theorem~\ref{thm_cyc} imply the
following relation.

\begin{cor}\label{conc_cyc}
Let $v\in A_n$.
\begin{equation}\label{len_cyc}
\ell_{T(A_n)}(v)= \\
\begin{cases}
n-cyc(v)&  \text{if\/ $1,2$ are in different cycles of v}, \\
n-cyc(v)-1&  \text{if\/ $1,2$ are in the same cycle of v}.
\end{cases}
\end{equation}
\end{cor}

Equation~(\ref{len_cyc}) provides a simple way to find the length of a permutation $v$ given as a product of disjoint cycles. 

Theorem~\ref{thm_cyc} implies that all the permutations of the same length in $A_n$ have the same number of cycles.
\begin{df}
$$
m(n,k) = \emph{number of cycles in a permutation $v\in A_n$ of length $\ell_{T(A_n)}(v)=k$}.
$$
\end{df}

\section{Generating Function of Length in $A_n$}\label{secgenfun}
An explicit formula for the generating function of the length in $A_n$, with respect to the generating set $T(A_n)$, is given in this section.

%\begin{obs}\label{obs1}
According to Theorem~\ref{thm_cyc}, the number $m(n,k)$, of cycles in a permutation $v\in A_n$ of length $k$, can be calculated by the following formula:
\begin{equation}\label{eq_gen1}
m(n,k)= \\
\begin{cases}
n-k,& \text{if $k$ is even}, \\
n-k-1,&  \text{if $k$ is odd.}
\end{cases} 
\end{equation}
A well known result in $S_n$ is
\begin{equation}\label{eq_gen2}
m(n,k)=n-k
\end{equation}
where the length $k$ is taken with respect to the generating set $T=\{(ij)\mid 1\leq i < j\leq n\}$, that is, all the transpositions in $S_n$. Since the number of cycles in a permutation is independent of the generating set, from Equations~(\ref{eq_gen1}) and (\ref{eq_gen2})
we can conclude that, for $v\in A_n$, we have
\begin{equation}\label{eq_gen3}
\ell_{T}(v)=
\begin{cases}
\ell_{T(A_n)}(v),& \text{if $\ell_{T(A_n)}(v)$ is even}, \\ 
\ell_{T(A_n)}(v)+1,& \text{if $\ell_{T(A_n)}(v)$ is odd,}
\end{cases} 
\end{equation}
where $\ell_{T}(v)$ is the length with respect to $T$.
Note that, in each of the cases, $\ell_{T}(v)$ is even, which complies with the fact that we deal with even permutations in $S_n$.
From Equation~(\ref{eq_gen3}) we can conclude that the number of permutations  of even length $k$ in $S_n$ equals the sum of the number of permutations of lengths $k$ and $k-1$ in $A_n$. Since the number of permutations of length $k$ in $S_n$ with respect to $T$ is the unsigned Stirling number of the first kind $c(n,n-k)$, we can deduce the following equation for even $k$:
\begin{equation}\label{eq_gen4}
c(n,n-k)=a(n,k)+a(n,k-1).
\end{equation}
%\end{obs}
A further consequence is the following.

\begin{cla}\label{cla1}
Equation~\eqref{eq_gen4} holds also for odd $k\in \mathbb{N}$.
\end{cla}

\begin{proof}[Proof of Claim \ref{cla1}] 
Let $k+1$ be even. Using the recursive relation of Stirling numbers,
we can expand the left-hand of Equation~\eqref{eq_gen4} to obtain
\begin{align*}
c(n&,n-(k+1))=c(n-1,n-(k+1)-1)+c(n-1,n-(k+1))\cdot(n-1)\\
&= 
c(n-1,n-1-(k+1))+c(n-1,n-1-((k+1)-1))\cdot (n-1)\\
&= 
c(n-1,n-k-2)+c(n-1,n-1-k)\cdot (n-1).
\end{align*}
Rearrangement of terms gives the following result:
$$
(n-1)\cdot c(n-1,n-1-k)=c(n,n-k-1)-c(n-1,n-k-2).
$$
The expressions at the right-hand side represent even length, so we can use Equation~\eqref{eq_gen4} and Conclusion~\ref{conc_len} to obtain the desired result:
\begin{align*}
(n-1)\cdot &c(n-1,n-1-(k-1))= a(n,k)+a(n,k-1)-a(n-1,k)-a(n-1,k-1)\\
&= 
a(n-1,k-1)\cdot (n-1) + a(n-1,k)+a(n-1,k-2)\cdot (n-1)\\
&\kern2cm + a(n-1,k-1)-a(n-1,k) 
-a(n-1,k-1)\\
&=a(n-1,k-1)\cdot (n-1)+a(n-1,k-2)\cdot (n-1).
\end{align*}
Dividing both sides by $(n-1)$, we deduce, for odd $k$, 
the equation
$$
c(n-1,n-1-k)=a(n-1,k)+ a(n-1,k-1).
$$
\end{proof}


The following generating function for unsigned Stirling numbers of the first kind is well known~\cite [pp. 213]{CO}:
\begin{equation}\label{str_book}
\sum_{k=1}^n c(n,k)\cdot x^{n-k}=(1+x)(1+2x)\cdots (1+(n-1)x).
\end{equation}
By Equation (\ref{eq_gen4}), we have
$$
\sum_{k=0}^n c(n,n-k)\cdot x^{k}=\sum_{k=0}^n a(n,k)\cdot x^{k} + \sum_{k=0}^n a(n,k-1)\cdot x^{k}.
$$
Using Equation \eqref{str_book} for the left-hand side, we obtain
\begin{multline*}
(1+x)(1+2x)\cdots (1+(n-1)x)\\= 
 \sum_{k=0}^n a(n,k)\cdot x^{k} + x\cdot \sum_{k=0}^n a(n,k-1)\cdot x^{k-1}= (1+x)\cdot \sum_{k=0}^n a(n,k)\cdot x^{k}.
\end{multline*}
Dividing both sides by $(x+1)$, we get the generating function 
of length in $A_n$ with respect to the generating set $T(A_n)$.

\begin{thm}\label{thm_genfunc}
We have
\begin{align}\label{eq_genfunc}
\sum_{k=0}^n a(n,k)\cdot x^{k}
&=(1+2x)(1+3x)\cdots (1+(n-1)x) \\
&= \prod_{t=2}^{n-1}(1+tx). \nonumber
\end{align}
\end{thm}


\section{Expectation and Variance}
In this section, the expectation and variance of the length function in $A_n$ are studied.

\begin{df}
Let $A$ be a finite set, and $s:A\rightarrow \mathbb{R}$ a real function. The expectation of $s$ is defined by
$$
E[s]:=\frac{1}{|A|} \sum_{a\in A} s(a),
$$
and the variance of $s$ is defined by
$$
Var[s]:=E[s^2]-E^2[s].
$$
\end{df}

Given a generating function of $s$, we can use it to calculate these statistics. The following formulas are well-known.

\begin{pro} \label{pro_proba}
Let 
$$
F_s(x) := \sum_{a\in A} x^{s(a)}
$$ 
be the generating function of $s$. Then 
$$
E[s]=\frac{1}{|A|} F'_s(x)\Bigg\vert_{x=1},
$$ 
and
$$
Var[s]=\frac{1}{|A|}\Big[F''_s(x) + F'_s(x) - \frac{1}{|A|}(F'_s(x))^2\Big]\Bigg\vert_{x=1}.
$$
\end{pro}

\begin{df}
We recall the definitions of harmonic numbers (see Definitions \ref{def_harmc} and \ref{def_genharmc}),
$$
H_n= 1+\frac{1}{2}+\frac{1}{3}+\cdots + \frac{1}{n}
$$
and generalized harmonic numbers,
$$
H_{n,m}= 1+\frac{1}{2^m}+\frac{1}{3^m}+\cdots + \frac{1}{n^m}
$$
\end{df}


\begin{thm}\label{thm_exp_var}

The expected value of $\ell_{T(A_n)}$ is
$$
E[\ell_{T(A_n)}]=n-H_n-\frac{1}{2},
$$
and its variance is
$$
Var[\ell_{T(A_n)}]=H_n-H_{n,2}-\frac{1}{4}.
$$
\end{thm}

\begin{proof}[Proof of Theorem~\ref{thm_exp_var}] 
Compute the derivative of the generating function with respect
to length (see Theorem~\ref{thm_genfunc}) as a product of functions:
$$
 \bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)'=  \bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)\sum_{t=2}^{n-1}{\frac{t}{1+tx}}.
$$
Thus, by Proposition~\ref{pro_proba},
$$
E[\ell_{T(A_n)}]=\frac{1}{|A_n|}\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)\sum_{t=2}^{n-1}{\frac{t}{1+tx}}\Bigg\vert_{x=1}=\frac{2}{n!}\cdot \frac{n!}{2}\bigg(n-2-\sum_{t=2}^{n-1}\frac{1}{1+t}\bigg)=n-H_n-\frac{1}{2}.
$$
For the variance, we have
\begin{align*}
\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)''
&=\bigg[\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)\sum_{t=2}^{n-1}{\frac{t}{1+tx}}\bigg]' \\
&\kern-2pt
=\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)\sum_{t=2}^{n-1}{\frac{t}{1+tx}}\sum_{t=2}^{n-1}\frac{t}{1+tx} +\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)\sum_{t=2}^{n-1}\frac{-t^2}{(1+tx)^2} ,
\end{align*}
and thus
\begin{align*}
Var[\ell_{T(A_n)}&]=\frac{1}{|A_n|}\bigg[\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)''+\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)'-\frac{1}{|A_n|}\bigg(\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)'\bigg)^2\bigg]_{x=1} \\
&=\frac{2}{n!}\bigg[\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)\sum_{t=2}^{n-1}{\frac{t}{1+tx}}\sum_{t=2}^{n-1}\frac{t}{1+tx} 
+\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)\sum_{t=2}^{n-1}\frac{-t^2}{(1+tx)^2} \\
&\kern20pt
+\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)\sum_{t=2}^{n-1}\frac{t}{1+tx} -\frac{2}{n!}\bigg(\bigg(\prod_{t=2}^{n-1}(1+tx)\bigg)\sum_{t=2}^{n-1}\frac{t}{1+tx}\bigg)^2\bigg]_{x=1} \\
&=\frac{2}{n!}\bigg[\frac{n!}{2}\bigg(n-H_n-\frac{1}{2}\bigg)^2 + \frac{n!}{2}\bigg(2H_n+\frac{1}{4}-n-H_{n,2}\bigg)+\frac{n!}{2}
\left(n-H_n-\frac{1}{2}\right) \\
&\kern20pt
-\frac{2}{n!}\bigg(\frac{n!}{2}(n-H_n-\frac{1}{2}\bigg)^2\bigg] \\
&= H_n-H_{n,2}-\frac{1}{4}.
\end{align*}
\end{proof}


\section{Connection with Restricted Stirling Numbers}
This section discusses the relation between our statistic $a(n,m)$ and $2$-restricted Stirling numbers of the first kind (see Broder~\cite[\S 1]{BRO}). This relation was initially observed using the On-Line Encyclopedia of Integer Sequences~\cite{EIS}.


\medskip
Recall Definition \ref{dfrstr1} of the $r$-restricted Stirling numbers of the first kind, $\ribua{n}{k}_r$. We shall use it with $r=2$. The number $\ribua{n}{k}_1=c(n,k)$ is the usual (unrestricted) Stirling number of the first kind.

\begin{cla}[{See Broder~\cite[\S~3, Thm.~3]{BRO} for a generalized version}]\label{cla_str}
We have 
$$
c(n,k)={\ribua{n}{k}}_2+{\ribua{n}{k+1}}_2.
$$
\end{cla}



\begin{thm}\label{thm_str}
The number of permutations in $A_n$ of length $\ell_{T(A_n)}(\cdot)=k$ is equal to a corresponding 2-restricted Stirling number. Namely,
$$
a(n,k)={\ribua{n}{n-k}}_2, \quad (0\leq k\leq n-2).
$$
\end{thm}

We give two proofs to Theorem~\ref{thm_str}. The first proof is algebraic, and the second consists of a direct bijection between two sets.

\begin{proof}[Proof of Theorem~\ref{thm_str}] 
From Claims \ref{cla_str} and \ref{cla1} we can deduce the following equation:
$$
a(n,k)+a(n,k-1)={\ribua{n}{n-k}}_2+{\ribua{n}{n-k+1}}_2, \quad (0\leq k\leq n-1).
$$ 
Now the theorem can be proved by induction on $k$. By assumption, $n\geq 2$. For $k=0$, we have $a(n,0)=1$ and ${\ribua{n}{n}}_2=1$. The claim $a(n,k)={\ribua{n}{n-k}}_2$ now follows by induction on $k$.
\end{proof}

\begin{df}
Let
$$
P(n,k)=\{v\in S_n \mid cyc(v)=k \text{ and } 1,2 \text{ are in different cycles in } v\}.
$$
\end{df}

We now present an explicit bijection between the sets 
$A(n,k)$ and $P(n,n-k)$.
 
\begin{proof}[A Bijective Proof of Theorem~\ref{thm_str}]
Define a map $f:A(n,k)\rightarrow P(n,n-k)$ by
$$
f(v)=
\begin{cases}
v& \text{if $\ell_{T(A_n)}(v)$ is even}, \\
(1,2)v&  \text{if $\ell_{T(A_n)}(v)$ is odd}.
\end{cases} 
$$
We will show that $f$ is one-to-one and onto $P(n,n-k)$.
\begin{enumerate}
\item
Consider $v_1,v_2\in A(n,k)$ with $f(v_1)=f(v_2)$. If $v_1,v_2$ are both of even length, or both of odd length, then, by the definition of $f$, $v_1=v_2$.
If $v_1$ is of even length and $v_2$ is of odd length or vice versa, then, by the definition of $f$ and the fact that $f(v_1)=f(v_2)$, $v_1=(1,2)v_2$. This contradicts the assumption that $v_1,v_2\in A(n,k)$, and is therefore impossible. Since only the first case is feasible, $v_1=v_2$ and $f$ is one-to-one.
\item 
Consider $w\in P(n,n-k)$. The length of $w$ in $S_n$, $\ell_{T}(w)$, is $k$. If $k$ is even, then $w\in A_n$. By Corollaries~\ref{pro_nice} and \ref{conc_cyc}, we have $\ell_{T(A_n)}(w)=k$, therefore $w\in A(n,k)$ and $f(w)=w$. If $k$ is odd, then $(1,2)w\in A_n$. By Corollaries~\ref{pro_nice} and \ref{conc_cyc}, we have $\ell_{T(A_n)}((1,2)w)=k$, therefore $(1,2)w\in A(n,k)$ and $f((1,2)w)=w$. This proves that $f$ is onto $P(n,n-k)$. 
\end{enumerate}
\end{proof}

\section*{Acknowledgements}
I wish to thank my supervisors Prof.\ Ron M.\ Adin and Prof.\ Yuval Roichman for their professional guidance, patience and for introducing me to the fascinating world of combinatorial research. 

I want to thank my parents for their love and support.

I would like to thank all those who helped me and supported me during this work.

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\end{document}

\end{document}
