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\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{color}
%\usepackage{graphicx}
\usepackage{ytableau}
\usepackage{tikz-cd}


\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lem}[theorem]{Lemma}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}

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\DeclareMathOperator{\core}{core}
\DeclareMathOperator{\Core}{\mathcal{C}}
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\newcommand{\numcore}{c_t}
\newcommand{\coresum}{C_t}
\newcommand{\ds}{\displaystyle}



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\title
{Random $t$-cores and hook lengths in random partitions}

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%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}

\author{Arvind Ayyer\thanks{\href{mailto:arvind@iisc.ac.in}{arvind@iisc.ac.in}. Partially supported by the UGC Centre for Advanced Studies and by Department of Science and Technology grant EMR/2016/006624.}\addressmark{1} \and Shubham Sinha\thanks{\href{mailto:shs074@ucsd.edu}{shs074@ucsd.edu}. Partially supported by the UGC Centre for Advanced Studies.}\addressmark{2}}



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\address{\addressmark{1}Department of Mathematics, Indian Institute of Science, Bangalore, India \\ \addressmark{2}Department of Mathematics, University of California, San Diego, La Jolla, CA, USA}


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\abstract{
Fix $t \geq 2$. We first give an asymptotic formula for certain sums of the number of $t$-cores. 
We then use this result to compute the distribution of the size of the $t$-core of a uniformly random partition of an integer $n$. 
We show that this converges weakly to a gamma distribution after appropriate rescaling. As a consequence, we find that the size of the $t$-core is of the order of $\sqrt{n}$ in expectation. We then apply this result to show that the probability that $t$ divides the hook length of a uniformly random cell in a uniformly random partition equals $1/t$ in the limit. Finally, we extend this result to all modulo classes of $t$ using abacus representations for cores and quotients.
}

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\keywords{$t$-core, $t$-quotient, uniformly random partition, gamma distribution, hook length, random cell, abacus}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easierHodge Theory and Complex Algebrai
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\begin{document}

\maketitle
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\section{Introduction and statement of results}

The irreducible representation of the symmetric group $S_n$ are indexed by partitions of $n$. While studying modular representations of $S_n$, one naturally encounters special partitions called {\em $t$-cores}~\cite{james-kerber-2009}, which are defined for any integer $t \geq 2$. The $t$-core of a partition $\lambda$, denoted $\core_t(\lambda)$, can be obtained from $\lambda$ by a sequence of operations.
See \cref{sec:core-quo} for the precise definitions.
A partition $\lambda$ is itself called a $t$-core if $\core_t(\lambda) = \lambda$.
Let $\numcore(n)$ be the number of $t$-cores of size $n$. We will be interested in finding the asymptotic behavior of certain sums involving $\numcore(n)$. Anderson~\cite{anderson-2008} has obtained detailed asymptotic results for $\numcore(n)$ using the circle method, but these will not suffice for our purposes. We define the number of partitions obtained by taking $t$-cores of all partitions of $n$ by
\begin{equation}
\label{def-coresum}
\coresum(n):= \# \{\core_t(\lambda) \mid \lambda \quad \text{a partition of $n$} \}. 
\end{equation}
When $t$ is a prime number, $C_t(n)$ can be defined to be the number of $t$-blocks, that is, the number of connected components of the Brauer graph, in the $t$-modular representation theory of the symmetric group $S_n$~\cite{granville-ono-1996}.

Our first main result is the following.

\begin{theorem}
	\label{thm:sum}
	Fix $t \geq 2$. Then
	\begin{equation}
	\coresum(n)= \frac{(2\pi)^{(t-1)/2} }{t^{(t+2)/2} \, \Gamma(\frac{t+1}{2})} \left( n+\frac{t^2-1}{24} \right)^{{(t-1)/2}} + O(n^{(t-2)/2}).
	\end{equation}
\end{theorem}


\begin{cor}
	\label{cor:numcore-size}
	Fix $t \geq 2$. Then $\numcore(n) = O(n^{(t-2)/2})$.
\end{cor}


Let $n$ be a positive integer and $t \geq 2$ be a fixed positive integer as before. Let $\lambda$ be a uniformly random partition of $n$.
Let $Y_n$ be a random variable on $\mathbb{N}_{\geq 0}$ given by
\begin{equation}
Y_n \equiv Y_{n,t} = |\core_t(\lambda)|,
\end{equation}
where $|\cdot |$ denotes the size of the partition.
We will be interested in the convergence of $Y_n$. 
The probability mass function of $Y_n$ is given by
\begin{equation}
\label{def-mu}
\mu_n(k) \equiv \mu_{n,t}(k)=\frac{\#\{\lambda\vdash n:|\core_t(\lambda)|= k \}}{p(n)},
\end{equation}
where $p(n)$ is the number of partitions of $n$.
It can be shown (see \cref{cor:number-fixed-cores}) that $\mu_n$
can be written as 
\begin{equation}
\mu_{n}(k) =\frac{\numcore(k) d_t(n-k)}{p(n)},
\end{equation} 
where $d_t(m)$ is the number of partitions of $m$ with empty $t$-core.
Let $X_n \equiv X_{n,t}$ be continuous random variables defined on $[0, \infty)$ with the probability density function $f_n \equiv f_{n,t}$ given by
\begin{equation}
\label{def-fn}
f_{n}(x) = \frac{\sqrt{n} \, \numcore(\lfloor x \sqrt{n} \rfloor) \, d_t( n- \lfloor x \sqrt{n} \rfloor )}{p(n)}.
\end{equation}
Note that the integral of $f_n$ being $1$ is equivalent to $\sum_k \mu_n(k) = 1$.
Recall that the {\em gamma distribution} with {\em shape parameter} $\alpha > 0$ and {\em rate parameter} $\beta > 0$ is a continuous random variable on $[0,\infty)$ with density given by
\begin{equation}
\label{gamma-density}
\gamma(x)= \begin{cases}
\ds \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1} \exp(-\beta x), & x\ge 0,\\
0, & x<0,
\end{cases}
\end{equation} 
where $\Gamma$ is the standard gamma function. 

\begin{theorem}
	\label{thm:conv}
	The random variable $X_n$ converges weakly to a gamma-distri\-buted random variable $X$ with shape parameter $\alpha=(t-1)/2$ and rate parameter $\beta=\pi/\sqrt{6}$.
\end{theorem}

\begin{center}
	\begin{figure}[htbp!]
		\centering
		\begin{tabular}{c c}
			\includegraphics[scale=0.30]{cdfplots3_t2.png} &
			\includegraphics[scale=0.30]{densityplots_t5.png} \\
			(a) & (b)
		\end{tabular}
		\caption{Comparison of the limiting CDFs and densities for small values of $n$ with $t=5$. A red solid line is used for the limiting distribution, dashed blue for $n = 20$, dash-dotted green for $n = 62$ and dotted magenta for $n=103$.
			In (a) the CDFs, and in (b) the densities, 
			are plotted for $X$ and these $X_n$'s.}
		\label{fig:dist-core5}
	\end{figure}
\end{center}

See \cref{fig:dist-core5} for an illustration of \cref{thm:conv} for $t=5$. Notice that while the distribution seems to converge pointwise in \cref{fig:dist-core5}(a), the density in \cref{fig:dist-core5}(b) does not.
An immediate consequence of \cref{thm:conv} is the following result.

\begin{cor}
	\label{cor:expectation}
	The expectation of the size of the $t$-core for a uniformly random partition of size $n$ is asymptotic to $(t-1)\sqrt{6n}/2\pi$.
\end{cor}

We illustrate \cref{cor:expectation} with the example of $t=3$ in \cref{fig:avg-core3}.
\cref{thm:conv} and \cref{cor:expectation} will be proved in \cref{sec:distrib}.

\begin{center}
	\begin{figure}[htbp!]
		\centering
		\includegraphics[scale=0.5]{avg-core3.pdf}
		\caption{The average size of the $3$-core for partitions of size 1 to 100 in blue circles, along with the result from \cref{cor:expectation}, $\sqrt{6x}/\pi$, as a red line.}
		\label{fig:avg-core3}
	\end{figure}
\end{center}

Let $\lambda$ be a partition of $n$ and $c$ denote a cell in the Young's diagram of $\lambda$. Then we define $h_c$ be the hook length associated to the cell $c$. See \cref{sec:prelim} for the precise definitions.
Our final major result is a statement about the remainder of hook lengths of cells when divided by $t$. 

\begin{theorem}
	\label{thm:random hook length}
	For a uniformly random cell $c$ of a uniformly random partition $\lambda $ of $n$, the probability that the hook length of $c$ in $\lambda$ is congruent to $i$ modulo $t$ is asymptotic to $1/t$ for any $i\in \{0,1,\dots, t-1 \}$.
\end{theorem}

We will devote \cref{sec:prelim} to state some preliminary results. We will sketch the proofs of \cref{thm:sum,thm:conv,thm:random hook length} in \cref{sec:asymp,sec:distrib,sec:hooks} respectively. The complete proofs of these results can be found in the extended version of this paper~\cite{1911.03135}, which will be published elsewhere.



\section{Preliminaries}
\label{sec:prelim}

Recall that an {\em (integer) partition} $\lambda$ of a nonnegative integer $n$ is a nonincreasing tuple of nonnegative integers which sum up to $n$.
If $\lambda$ is a partition of $n$, we write $\lambda \vdash n$ and say that the {\em size} of $\lambda$, denoted $|\lambda|$, is $n$. Let $p(n)$ denote the number of partitions of $n$. We write $\Par$ for the set of all partitions.

Let $c \equiv (i,j)$ be a cell in the Young diagram of $\lambda$.
The {\em hook} of $c$ is a subset of the cells in the Young diagram containing the cells to the right of $c$ in the same row, those below $c$ in the same column, and $c$ itself. 
The {\em hook length} is the cardinality of the cells in the hook of $c$ and is denoted by $h^\lambda_c$.


\subsection{Abacus representation}
\label{sec:abacus}

\begin{defn}
	\label{def:abacus}
	An {\em abacus} or {\em (1-runner)} is a function $w : \mathbb{Z} \to \{0,1\}$ such that there exist $m,n \in \mathbb{Z}$ such that $w_i =1$ (resp. $w_i =0$) for all $i \leq m$ (resp. $i \geq n$).
\end{defn}

Starting from an abacus $w$, consider the up-right path formed by replacing $1$'s by vertical steps and $0$'s by horizontal steps. This path will form the outer boundary of a partition. However, note that this is not a bijective correspondence because any shift of the abacus will lead to the same partition. 

An abacus $w$ is called {\em justified at position $p$} if $w_i = 1$ (resp. $w_i = 0$) for $i < p$ (resp. $i \geq p$). 
%An abacus is called {\em justified} if it is justified at position $p$ for some $p$. 
Any abacus can be transformed to a justified one by moving the $1$'s to the left past the $0$'s start from the leftmost movable $1$. 
An abacus is called {\em balanced} if, after this transformation, it is justified at position $0$. 
Note that balanced abaci are in bijection with partitions.

We now summarize properties of abaci that will be relevant to us. Readers interested in the details can look at~\cite{james-kerber-2009,olsson,loehr-2011}. 

\begin{prop}
	\label{prop:core-facts}
	Let $\lambda$ be a partition with corresponding balanced abacus $w$ and $c$ be a cell in the Young diagram of $\lambda$.
	Then the following properties hold.
	\begin{enumerate}
		
		\item Cells in the Young diagram of $\lambda$ are in bijection with pairs $(i,j)$ such that $i < j$, $w_i = 0$ and $w_j = 1$. The hook length of the cell $c$ corresponding to the pair $(i,j)$ in $w$ is $j-i$. 
		

		
		\item Suppose $c$ has hook length $t$. Then $c$ corresponds to a pair $(i,i+t)$ in $w$. Then, removing a $t$-rim hook for the cell $c$ (see the beginning of \cref{sec:core-quo}) from the Young diagram of $\lambda$ amounts to exchanging the $i$'th and $(i+t)$'th entries in $w$.
		
	\end{enumerate}
	
\end{prop}

\subsection{Cores and quotients}
\label{sec:core-quo}

To describe cores and quotients, we will need some notation. Associated to a partition $\lambda$ and a cell $c$ in its Young diagram, the set of cells joining the two corners in the hook of $c$ along the boundary is known as the {\em rim hook} or {\em ribbon} of $c$. 



The {\em $t$-core of a partition} $\lambda$, denoted $\core_t(\lambda)$, is the partition obtained by removing as many rim hooks of size $t$ from $\lambda$ as possible. Using \cref{prop:core-facts}, we can define $\core_t(\lambda)$ as the partition corresponding to abacus obtained by exchanging as many $(0,1)$ pairs at positions $(i,i+t)$ as possible.


A partition $\lambda$ is called a {\em $t$-core} if $\core_t(\lambda) = \lambda$, or equivalently, if none of the hook numbers in the Young diagram of $\lambda$ is divisible by $t$. 
From \cref{prop:core-facts}, it follows that the partition $\lambda$ is a $t$-core if and only if there is no $i \in \mathbb{Z}$ such that $w_i = 0$ and $w_{i+t} = 1$.
Let $\Core_t$ be set of all $t$-cores.
Let $\numcore(n)$ be the number of $t$-cores of size $n$.
Then it is known~\cite{klyachko-1982} that
\begin{equation}
\label{coret-gf}
\sum_{n=0}^\infty \numcore(n)x^n = \prod_{k=1}^\infty 
\frac{(1-x^{tk})^t}{1-x^{k}}.
\end{equation}



\begin{defn}
	\label{def:t-divisible}
	A partition is said to be {\em $t$-divisible} if it has empty $t$-core.
\end{defn}

Let $\mathcal{D}_t(n)$ be the set of $t$-divisible partitions of $n$ and 
$d_t(n)$ be its cardinality.
Given a partition $\lambda$ with abacus $w$, construct $t$ abaci by letting $\lambda^{i} = (w_{n t + i})_{n \in \mathbb{Z}}$ for $0 \leq i \leq t-1$. The {\em $t$-runner abacus} of $\lambda$ is then $(\lambda^0,\dots,\lambda^{t-1})$ of $1$-runner abaci, where the zeroth position in $\lambda^0$ is underlined. We define the {\em $t$-quotient} of $\lambda$ to be a $t$-tuple of partitions corresponding to the entries of the $t$-runner $(\lambda^0, \dots, \lambda^{t-1})$. We will use the same notation for the $i$'th entry of the $t$-quotient and the $t$-runner.
It turns out that $t$-cores and $t$-divisible partitions have a natural interpretation in terms of $t$-runner abaci. 

\begin{prop}
	\label{prop:t-runner}
	Let $\lambda$ be a partition and $(\lambda^0,\dots,\lambda^{t-1})$ be its $t$-runner abacus. Then
	\begin{enumerate}
		\item $\lambda$ is a $t$-core if and only if $\lambda^i$ is justified for $0 \leq i \leq t-1$, and 
		
		\item $\lambda$ is $t$-divisible if and only if $\lambda^i$ is balanced for $0 \leq i \leq t-1$.
	\end{enumerate}
\end{prop}


A fundamental result on cores and quotients is the {\em partition division theorem}~\cite[Theorem~11.22]{loehr-2011}, which is an analogue of the division algorithm for integers. We restate it in slightly different terminology more suited to our purposes.

\begin{theorem}
	\label{thm:partn-divn}
	Let $t \geq 2$ be an integer. Then there is a natural bijection 
	\[
	\Delta_t: \Par \to \Core_t \times \mathcal{D}_t
	\]
	defined by $\Delta_t(\lambda)=(\rho, \nu )$, where $\rho$ is the $t$-core of $\lambda$ and $\nu$ is a $t$-divisible partition whose $t$-quotient, $(\nu^0,\dots,\nu^{t-1})$, is same as that of $\lambda$. Moreover $|\lambda| = |\rho| + |\nu|$.
\end{theorem}

The following corollary is then immediate.

\begin{cor}
	\label{cor:number-fixed-cores}
	Let $i,n$ be positive integers such that $i < n$. Then 
	\[
	\#\{\lambda\vdash n \mid |\core_t(\lambda)|=i \} = d_t(n-i)\numcore(i).
	\]
\end{cor}



\section{Asymptotics of the number of $t$-cores}
\label{sec:asymp}
The asymptotics of the number of $t$-cores, $\numcore(n)$, was obtained using the circle method by Anderson~\cite[Theorem~2]{anderson-2008} for $t \geq 6$. We demonstrate a new method to obtain these asymptotics for $t \geq 2$.

The following theorem describe the quantity $\numcore(n)$ as the number of integer solutions of a particular quadratic equation. This is a reformulation of \cite[Bijection 2]{garvan-kim-stanton-1990}, and we give a new proof using the abacus representation which can be found in the \cite{1911.03135}.
Let $H_t$ denote the hyperplane in $\mathbb{R}^t$ given by
$
H_t = \{ (x_0,\dots,x_{t-1}) \mid x_0+x_1+\dots x_{t-1}=0 \}.
$


\begin{theorem}
	\label{thm:eq-sols}
	The number of $t$-cores of $n$, $\numcore(n)$, is equal to the number of integer solutions of $F_t(p) = n$ on the hyperplane $H_t$, where $F_t$ is defined by
	\begin{equation}
	\label{eq:sphere}
	F_t(p) = \frac{t}{2} \sum_{i=0}^{t-1}p_i^2
	+ \sum_{i=0}^{t-1}ip_i.
	\end{equation}
\end{theorem}



Observe that subtracting $(t-1) \left( \sum_{i=0}^{t-1}p_i \right)/2$ from  \eqref{eq:sphere} yields the equation of a $t$-dimensional sphere centered at a point in the hyperplane $H_t$ given by
\begin{equation}
\label{eq:c_sphere}
\sum_{i=0}^{t-1} \bigg( p_i- \frac{t-1-2i}{2t} \bigg)^2 = \frac{2}{t}\bigg(n + \frac{t^2-1}{24}\bigg).
\end{equation}
Denote the $(t-1)$-dimensional ball cut out by the hyperplane $H_t$ and the sphere given by \eqref{eq:c_sphere} as $B_{t-1}(n)$. The radius of $B_{t-1}(n)$ is unchanged since the center 
$1/(2t) (t-1, t-3, \dots, -(t-3), -(t-1))$
lies on $H_t$. Then the  volume $V_t(n)$ of $B_{t-1}(n)$, under the induced measure on the hyperplane $H_t$, is 
\begin{equation}\label{volume_sphere}
V_t(n)=\frac{1}{\Gamma(\frac{t+1}{2})} \bigg(\frac{2\pi}{t} \left( n+\frac{t^2-1}{24} \right) \bigg)^{(t-1)/2}.
\end{equation}


Define the lattice $\Lambda_t = H_t\cap \mathbb{Z}^t$. It is clear that $\Lambda_t$ is a full lattice of the codimension one subspace $H_t$ with $\mathbb{Z}$-basis $\{e_0-e_1,e_0-e_2,\dots, e_0-e_{t-1} \}$. We use standard techniques to note that the volume $V(\Lambda_t)$ of the fundamental domain of $\Lambda_t$ in $H_t$ is $\sqrt{t}$.



We obtain the next result by working over the module $(\mathbb{Z}/t\mathbb{Z})^{t}$. To avoid confusion, we will denote points in $(\mathbb{Z}/t\mathbb{Z})^{t}$ with a tilde, e.g. $\widetilde{P},\widetilde{Q}$.

\begin{lem}
	\label{lem:size-Zmodt}
	For any $n \in \mathbb{N}$, the cardinality of the set 
	$ \big\{\widetilde Q\in \widetilde{H}_t \mid F_t(\widetilde Q)\equiv n\mod{t} \big\}$ is $t^{t-2}$,
	where $F_t$ is defined in \eqref{eq:sphere} and $\widetilde{H}_t$ is the hyperplane of $(\mathbb{Z}/t\mathbb{Z})^{t}$ defined by $q_0+q_1+\dots +q_{t-1}=0$.
\end{lem}



We are now in a position to prove the main result of this section. 

\begin{proof}[Proof of \cref{thm:sum}]
	Recall that the number of partitions obtained by taking $t$-cores of all partitions of $n$ is denoted by $\coresum(n)$ and is defined in\eqref{def-coresum}. It is easy to prove the relation 
		\[
		\coresum(n) = \sum_{i=0}^{\lfloor \frac{n}{t}\rfloor} \numcore(n - i \, t).
		\]
	Using \cref{thm:eq-sols}, we conclude that 
	\begin{equation}
	\coresum(n)=\# \big\{Q \in \Lambda_t \cap B_{t-1}(n) \mid F_t(Q)\equiv n \mod t \big\}.
	\end{equation}
		Splitting the set of points in the set on the right hand side according to their remainder modulo $t$, we arrive at 
	\begin{equation}
	\coresum(n)=
	\sum_{\substack{\widetilde{P} \in \Lambda_{t}/t\Lambda_{t} \\ 
			F(\widetilde{P}) \equiv n \mod{t} }}
	\# \left( (\widetilde{P}+t\Lambda_{t}) \cap B_{t-1}(n) \right),
	\end{equation}
	where $\widetilde P+t\Lambda_{t}$ is the lattice $t \Lambda_t$ shifted by $\widetilde{P}$. 
	Clearly, the fundamental domain has volume  $V(\widetilde P+t\Lambda_{t}) =t^{t-1}V(\Lambda_{t})$. We use \cref{lem:size-Zmodt} and standard results on the asymptotic of the number of lattice points inside a $(t-1)$ dimensional sphere to obtain the required approximation in terms of $V_t(n)$ defined in~\eqref{volume_sphere}. 
\end{proof}




\section{Distribution of core sizes}
\label{sec:distrib}

In this section we will sketch the proof of \cref{thm:conv}. We prove the weak convergence by calculating moments.
As usual, we fix $t \geq 2$.
Let $\epsilon(n) \in \{0,1,\dots,t-1\}$ be the remainder of $n$ is divided by $t$. 
For convenience, define
\label{def-ln}
$\ell_n(y) = t\lfloor y\sqrt{n}\rfloor+\epsilon(n)$, and we assume that all the functions defined below are supported on $ y\ge 0.$
We will make crucial use of the functions
\begin{equation}
\label{def-psi-n}
\psi_n(y) = \frac{\numcore(\ell_n(y))}{n^{(t-3)/4}},
\end{equation}
and
\begin{equation}
\label{def-gnk}
g_{n,k}(y) = 
\ds \ell_n(y)^{k} n^{(t-1)/4-k/2} \frac{d_t(n - \ell_n(y))}{p(n)}.
\end{equation}
Recall that the function $f_n(x)$ is defined in \eqref{def-fn}.
The functions $g_{n,k}$ and $\psi_n$ are defined so that $\psi_n(y) g_{n,0}(y)=f_n(\ell_n(y)/\sqrt{n})$ is roughly like $f_n(ty)$ for large $n$.
We now define
\begin{equation}
\label{def:g_k}
g_k(y)= %\begin{cases}
\kappa y^{k}e^{-\beta t y} ,
%0 & y < 0,
%\end{cases}
\end{equation}
where $\kappa=t^{(t+2+2k)/2} 2^{-3(t-1)/4} 3^{-(t-1)/4}$ and we recall that we have defined $\beta = \pi/ \sqrt{6}$. 




\begin{lem}
	\label{lem:conv_of_integal}
	For any natural number $k$, the sequence of functions $g_{n,k}$ converges uniformly to $g_k$ as $n\to\infty$. Moreover, 
	\begin{equation}\label{integral_expectation}
	\lim_{n\to \infty} \int_{0}^{\infty} \;\text{d}y \; \psi_n(y)g_{n,k}(y) =\lim_{n\to \infty} \int_{0}^{\infty} \;\text{d}y \; \psi_n(y)g_k(y).
	\end{equation}
\end{lem}

\cref{lem:conv_of_integal} is lengthy and technical, and so we do not sketch the proof. A complete proof can be found in~\cite{1911.03135}.

  


\begin{lem}
	\label{lem:moments-conv}
	For all positive integer $k$, 
	\begin{equation}
	\lim_{n\to \infty}\mathbb{E}(X_n^k)= \left(\frac{\sqrt{6}}{\pi} \right)^k \prod_{i=1}^{k} \left(\frac{t-1}{2}+k-i \right).
	\end{equation}
\end{lem}
\begin{proof}[Sketch of proof]
	We first note that the required limit of expectation equals the expression in \eqref{integral_expectation}. Using \cref{lem:conv_of_integal} and integration by parts, we reduce the problem to finding the integral of $\psi_n(y)$, with other quantities well understood in terms of $g_k$. We use \cref{thm:sum} to obtain an estimate for the integral of $\psi_n(y)$, which enables us to obtain the required  expression for the $k$'th moment.
\end{proof}


\begin{proof}[Proof of \cref{thm:conv}]
	From the formula for the moments of the gamma distribution, one can verify that the moment generating function of the gamma distribution is determined by its moments (see \cite[Theorem 30.1]{billingsley-1995}, for example). Using \cref{lem:moments-conv}, we have shown that all the moments of $X_n$ exist and converge to those of $X$, which is gamma distributed. Again appealing to a standard result on weak convergence~\cite[Theorem 30.2]{billingsley-1995}, we see that $X_n$ converges weakly to $X$.
\end{proof}







\section{Hook lengths of random cells of random partitions}
\label{sec:hooks}

The main result of this section is that for large enough $n$, the modulo class of the hook length of a uniformly random cell of a uniformly random partition is approximately the uniform distribution on the modulo classes $\{0,1,\dots , t-1 \}$ . 




We obtain the following proposition as an easy application of \cref{cor:expectation} by noting that removing a $t$-rim hook results reduces the number of hooks $h_c$ with $t|h_c$ by exactly one. The same is not true for non-zero modulo classes.

\begin{prop}
	\label{prop:mod_zero_case}
	For a uniformly random cell $c$ of a uniformly random partition $\lambda$ of $n$, the probability that the hook length of $c$ in $\lambda$ is divisible by $t$ is $1/t+O(n^{-1/2})$.
\end{prop}


Using the results in \cref{sec:distrib}, we are only able to prove the results in \cref{prop:mod_zero_case}. To obtain the stronger result stated in \cref{thm:random hook length}, we will need to appeal to the theory of $t$-runner abaci.


We will now estimate the number of cells of the region with small hook lengths. The following lemma gives us a crucial bound in proving the main theorem of this section.

\begin{lem}
	\label{lem:small-hooks}
	For $\lambda\vdash n$ and an integer $m$, the cardinality of the set $\mathcal{B}=\{c\in \lambda \mid h_c< m \}$ is less than $m\sqrt{2n}$.
\end{lem}
\begin{proof}
	It is enough to construct an injection from the set of unordered
	pairs of distinct cells in $\mathcal{B}$ to $\lambda \times [m] \times [m]$. We describe such a construction below.
	
	Suppose $c_1$ and $c_2$ are two cells in $\mathcal{B}$, and say that $c_1$
	is to the west of $c_2$ and if both are in the same column let $c_1$ to be south of $c_2$. Let $a_2$ be the arm length of $c_2$ and $l_1$ be the leg length of $c_1$. Then map the pair $\{c_1,c_2\}$ to $(c,a_2,l_1)$, where $c$ is the unique cell in the intersection of the column containing $c_1$ and the row containing $c_2$. 
	
	This map is clearly injective since the cell $c$ describes the column (or row) in which the cell $c_1$ (resp. $c_2$) lives and $l_1$ (resp $a_2$) gives the exact location of $c_1$ (resp $c_2$) in the partition. 
\end{proof}

\subsection{A natural action of $S_t$}
We define the action of $S_t$ on  $\mathcal{D}_t$, the set of $t$-divisible partitions, as follows. For any $\sigma\in S_t$ and a $t$-divisible  partition $\nu$ with $t$-quotient $(\nu^0, \nu^1,\dots ,\nu^{t-1})$, define $\sigma \nu $ be the $t$-divisible partition corresponding to the $t$-quotient $(\nu^{\sigma_0}, \nu^{\sigma_1},\dots ,\nu^{\sigma_{t-1}})$.
Note that the above action preserves the size of the $t$-divisible partition. 

\begin{defn}
	The {\em $b$-smoothing} of a $t$-divisible partition $\nu$, denoted $C_\nu^b$, is the union of cells in the Young diagram of $\nu$ whose corresponding $(0,1)$ pairs are at least $(b+1)$ columns apart in the $t$-runner abacus of $\nu$.
\end{defn}



\begin{prop}
	\label{prop:smoothing-subpartition}
	The set of cells $C_\nu^b$ is a connected subpartition of $\nu$. Moreover $C_\nu^b=C_{\sigma\nu}^b$ for all $\sigma\in S_t$ and $b \geq 0$.
\end{prop}


For any cell $c\in C_\nu^b$, define $h_c^\nu$ to be the hook length of the cell $c$ in the partition $\nu$. For the corresponding $(0,1)$ pair in the $t$-runner abacus of $\nu$, where $0\in \nu^i$ and $1\in \nu^j$, we see that $h_c^\nu \equiv j-i\mod t$. Using the action of $S_t$ defined above, we get the following

\begin{lem} 
	\label{lem:hooklength_t-div}
	Let $\nu$ be a uniformly random $t$-divisible partition of $n$ and $b \geq 0$.
	For a uniformly random cell $c\in C_\nu^b$, the probability that the hook length $h_c^\nu$  is congruent to $i$ modulo $t$, where $i \neq 0$, is independent of $i$.
\end{lem}



\begin{defn}
	\label{def:action of S_t}
	We define the action of $\sigma \in S_t$ on $\mathcal{P}$ by 
	$\sigma \lambda = \Delta_t^{-1}(\rho, \sigma \nu)$,
	where $\Delta_t (\lambda) =(\rho, \nu)$ is defined in \cref{thm:partn-divn}. 
\end{defn}





Let $\lambda \in \mathcal{P}$ with $\Delta_t (\lambda) =(\rho, \nu)$, and $\rho$ be determined by  the $t$-tuple $(p_0,\dots, p_{t-1})$, where $p_i$ is the position of justification of abacus $\rho^i$ and let $b_\lambda = \max_{1 \leq i < j \leq t-1} |p_i-p_j|$.
We will denote $C_\nu^{b_\lambda}$ by $C_\lambda$ for brevity and call it the {\em canonical smoothing} of the $t$-quotient of $\lambda$. 



\begin{prop}
	\label{prop:injection_of_cells}
	Let $\lambda$ be a partition with $\Delta_t (\lambda) =(\rho, \nu)$.
	Then there exists an injective map $\phi$ that takes the cells in $C_\lambda$ to the cells in $\lambda$ such that for any cell $c\in C_\lambda$, the hook lengths
	$h^\lambda_{\phi(c)}\equiv h^\nu_c \mod t$.
\end{prop}

\begin{proof}
	Let $c \in C_\lambda\subset \nu$ be given by the pair $(\nu^a_i, \nu^b_j) =  (0,1)$ where $j-i > b_\lambda$ by definition. Let $\rho$ be determined by the justification positions $(p_0,\dots ,p_{t-1})$, where $\sum_{i=0}^{t-1}p_i=0$ as explained before. Then a required map can be defined as
	\[
	\phi(c) := (\nu^a_{i+p_a}, \nu^b_{j+p_b}).
	\]
\end{proof}




\subsection{Proof of the main theorem}

The main idea is to show that the image of the canonical smoothing $\phi(C_\lambda)$ comprise of most of the cells in a uniformly random partition $\lambda$ of size $n$. 
We first obtain a bound $|b_\lambda|\le 2\sqrt{|\rho|}$ using elementary counting. Then we apply \cref{lem:small-hooks} to get the following



\begin{prop}
	\label{prop:size nu-cnu}
	For any partition $\lambda \vdash n$ with $\Delta_t (\lambda) =(\rho, \nu)$, 
	\begin{equation}
	|\nu|-|C_\lambda|< \# \{c\in \nu \mid h_c<t(b_\lambda+1) \} = O( t (b_\lambda+1)\sqrt{n}).
	\end{equation}
\end{prop}


We are now in a position to prove the main result of this section.

\begin{proof}[{Proof of \cref{thm:random hook length}}]
	Let $x_{i}(n)$ be the probability that a uniformly random cell $c$ of a uniformly random partition of $n$ has hook length congruent to $i$ modulo $t$.
	Using \cref{prop:injection_of_cells} and \cref{lem:hooklength_t-div}, we obtain by conditioning for any nonzero classes $i$ and $j$ modulo $t$, the difference between $x_i(n)$ and $x_j(n)$ in absolute value is upper bounded by the probability that $c \notin \phi(C_\lambda)$. Therefore, by \cref{thm:partn-divn}, we have that
	\[
	|x_i(n)-x_j(n)|  \leq  \sum_{\substack{\lambda\vdash n \\ \Delta_t (\lambda) =(\rho, \nu)}} \frac{|\nu|-|C_\lambda|+|\rho|}{np(n)}.
	\]
	Now, by \cref{prop:size nu-cnu} and \cref{cor:expectation},
	\[
	|x_i(n)-x_j(n)| < O(n^{-1/2}\mathbb{E}(\sqrt{|\core_t(\lambda)|})) + O(n^{-1/2}).
	\]
	Using the standard fact that $\mathbb{E}\sqrt{Y}\le \sqrt{\mathbb{E} Y }$ for any nonnegative random variable $Y$, we see that the right hand side is $O(n^{-1/4})$.
	Since $x_0(n)+x_1(n)+\dots+x_{t-1}(n)=1$ and $x_0(n)=1/t+O(n^{-1/4})$, we obtain the required result.
\end{proof}


\acknowledgements{This research was driven by computer exploration using \texttt{Sage}~\cite{sagemath}. We thank D. Grinberg and an anonymous referee for useful comments.}

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