\documentclass{article}[12pt]
\usepackage{e-jc}

\usepackage{tikz}
\usepackage{amsmath, amssymb}

\usetikzlibrary{snakes, arrows}
\usepackage{graphicx}
\usepackage{mathtools} %for \mathclap, fixing long subscripts in a summation
\usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref}


%\usepackage{xcolor}
%\hypersetup{
%    colorlinks,
%    linkcolor={red!80!black},
%    citecolor={blue!80!black},
%    urlcolor={blue!80!black}
%}


\theoremstyle{definition}

\newtheorem*{theorem*}{Theorem}
%\newtheorem{dummy}{}[section]
%\newtheorem{theorem}[dummy]{Theorem}
%\newtheorem{lemma}[dummy]{Lemma}
%\newtheorem{example}[dummy]{Example}
%\newtheorem{corollary}[dummy]{Corollary}
%\newtheorem{definition}[dummy]{Definition}
%\newtheorem{remark}[dummy]{Remark}
%\newtheorem{proposition}[dummy]{Proposition}
%\newtheorem{observation}{Observation}
%\newtheorem{question}{Question}
%\newtheorem{conjecture}[dummy]{Conjecture}
\newtheorem*{FalseHope}{False Hope}


\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\proj}{\mathbb{P}}
\newcommand{\core}{\mathbf{core}}
\newcommand{\DP}{\mathcal{DP}}
\newcommand{\SC}{\mathbf{SC}}
\newcommand{\TD}{\mathbf{TD}}
\newcommand{\OC}{\mathcal{OC}}
\newcommand{\Cat}{\mathbf{Cat}}
\newcommand{\sk}{s\ell}
\newcommand{\vac}{\mathbf{vac}}
\newcommand{\inv}{\mathbf{inv}}
\newcommand{\asm}{\mathbf{asm}}
\newcommand{\aparts}{\mathbf{size}_a}
\newcommand{\invsum}{\mathbf{invsum}}
\newcommand{\INV}{\mathbf{INV}}
\newcommand{\DES}{\mathbf{DES}}
\newcommand{\des}{\mathbf{des}}
\newcommand{\maj}{\mathbf{maj}}
\newcommand{\siz}{\mathbf{siz}}
\newcommand{\sqin}{\mathbf{sqin}}
\newcommand{\LD}{\mathbf{LD}}
\newcommand{\VS}{\mathbf{VS}}
\newcommand{\cone}{\mathfrak{c}}
\newcommand{\polyh}{\mathfrak{c}}
\newcommand{\dominant}{\mathcal{D}}
\DeclareMathOperator{\Sym}{Sym}


\dateline{Nov 24, 2015}{Mar 5, 2018}{TBD}
\MSC{11P81, 52B20, 05A17}
\title{Lattice points and simultaneous core partitions}
\author{Paul Johnson \\
  \small School of Mathematics and Statistics \\[-0.8ex]
  \small University of Sheffield \\[-0.8ex]
  \small Sheffield, United Kingdom \\
  \small \tt{paul.johnson@sheffield.ac.uk}
}


\begin{document}
\maketitle

\begin{abstract}
 We apply lattice point techniques to the study of simultaneous core partitions.  Our central observation is that for $a$ and $b$ relatively prime, the abacus construction identifies the set of simultaneous $(a,b)$-core partitions with lattice points in a rational simplex.  We apply this result in two main ways: using Ehrhart theory, we reprove Anderson's theorem that there are $(a+b-1)!/a!b!$ simultaneous $(a,b)$-cores; and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an $(a,b)$-core is $(a+b+1)(a-1)(b-1)/24$.  Our methods also give new derivations of analogous formulas for the number and average size of self-conjugate $(a,b)$-cores.
\end{abstract}




\section{Introduction}

This paper introduces lattice point geometry to the study of simultaneous core partitions.  We first establish some basic notation.

A \emph{partition of $n$} is a nonincreasing sequence $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_{k}> 0$ of positive integers such that $\sum \lambda_i=n$. We call $n$ the $\emph{size}$ of $\lambda$ and denote it by $|\lambda|$; we call
$k$ the \emph{length} of $\lambda$ and denote it by $\ell(\lambda)$.


We frequently identify $\lambda$ with its Young diagram; there are many conventions for this.  We draw $\lambda$ in the first quadrant, with the parts of $\lambda$ as the columns of a collection of boxes.
\begin{definition}
The \emph{arm} $a(\square)$ of a cell $\square$ is the number of cells contained in $\lambda$ and above $\square$, and the \emph{leg} $l(\square)$ of a cell is the number of cells contained in $\lambda$ and to the right of $\square$.
The \emph{hook length} $h(\square)$ of a cell is $a(\square)+l(\square)+1$.
\end{definition}
\begin{example}
The cell $(2,1)$ of $\lambda=3+2+2+1$ is marked $s$; the cells in the
leg and arm of $s$ are labeled $a$ and $l$, respectively.
\begin{center}
\begin{tikzpicture}[scale=.5]
\draw[thin, gray] (0,0) grid (1,3);
\draw[thin, gray] (1,0) grid (3,2);
\draw[thick] (0,0)--(0,3)--(1,3)--(1,2)--(3,2)--(3,1)--(4,1)--(4,0)--cycle;
\draw (1.5,.5) node{$s$};
\draw (1.5,1.5) node{$a$};
\draw (2.5,.5) node{$l$};
\draw (3.5,.5) node {$l$};
\draw (8.5,1.5) node[align=left] {$a(s)=\# a=1$ \\ $l(s)=\# l=2$ \\ $h(s)=4$};
\end{tikzpicture}
\end{center}
\end{example}
\begin{definition}
An $a$-\emph{core} is a partition that has no hook lengths of size $a$. An $(a,b)$-\emph{core} is a partition that is simultaneously an $a$-core and a $b$-core.  We use $\mathcal{C}_{a,b}$ to denote the set of $(a,b)$-cores.
\end{definition}
\begin{example}
We have labeled each cell $\square$ of $\lambda=3+2+2+1$ with its hook length $h(\square)$.
\begin{center}
\begin{tikzpicture}[scale=.5]
\draw[thin, gray] (0,0) grid (1,3);
\draw[thin, gray] (1,0) grid (3,2);
\draw[thick] (0,0)--(0,3)--(1,3)--(1,2)--(3,2)--(3,1)--(4,1)--(4,0)--cycle;
\draw (.5,.5) node{$6$};
\draw (1.5,.5) node{$4$};
\draw (2.5,.5) node{$3$};
\draw (3.5,.5) node{$1$};
\draw (.5,1.5) node {$4$};
\draw (1.5,1.5) node{$2$};
\draw (2.5,1.5) node{$1$};
\draw (.5,2.5) node{$1$};
\end{tikzpicture}
\end{center}
We see that $\lambda$ is \emph{not} an $a$-core for $a\in \{1,2,3,4,6\}$;
but \emph{is} an $a$-core for all other $a$.
\end{example}

Partitions of $n$ are closely related to the representation theory of the symmetric group (see \cite{JK}).  For $p$ a prime, $p$-core partitions of $n$ are related to the $p$ modular representation theory of $S_n$.  This was the initiatial motivation for the study of $a$-cores, but they have taken on a combinatorial life of their own.

The study of simultaneous core partitions begins Anderson's result \cite{anderson} that they are counted by the rational Catalan numbers, a natural generalization of Catalan numbers.  Apart from their intrinsic combinatorial interest, rational Catalan numbers and their $q$ and $(q,t)$ analogs appear in the study of Hecke algebras \cite{GG}, affine Springer varieties \cite{LS}, and compactified Jacobians of singular curves \cite{EM1, EM2}.   

In this paper, we use lattice point techniques to reprove Anderson's result and to prove a conjecture of Armstrong about the average size of simultaneous core partitions.  Our techniques also give new proofs about analogous results for self-conjugate simultaneous core partitions.  A previous version of this paper contained extra sections that began to apply these ideas to $q$ and $(q,t)$ analogs, but this doubled the length of the paper for little gain, and we worried it diluted from our main point.  This material is still available in the first version of his paper on the arXiv.

We reprove Anderson's theorem by identifying simultaneous core partitions with lattice points in a rational simplex.  The connection between rational Catalan numbers and this simplex is not new; it appears for instance in \cite{LS, GMV}.    However, we are not aware of any work using this connection to apply lattice point techniques.  After this identification is made, the other results follow quite naturally.
  
\subsection{Statement of results}

Recall that the Catalan number $\Cat_n=\frac{1}{2n+1}\binom{2n+1}{n}$. Catalan numbers count hundreds of different combinatorial objects; for example, the number of lattice paths from $(0,n)$ to $(n+1,0)$ that stay strictly below the line connecting these two points.
Rational Catalan numbers are a natural two parameter generalization of $\Cat_n$.
\begin{definition}
For $a,b$ relatively prime, the \emph{rational Catalan number}, or \emph{$(a,b)$ Catalan number} $\Cat_{a,b}$ is
$$\Cat_{a,b}=\frac{1}{a+b}\binom{a+b}{a}$$
\end{definition}
The rational Catalan number $\Cat_{a,b}$ counts the lattice paths from $(0,a)$ to $(b,0)$ that stay beneath the line from $(0,a)$ to $(b,0)$. This is consistent with the specialization $\Cat_{n,n+1}=\Cat_n$.


Simultaneous cores and rational Catalan numbers are connected by:
\begin{theorem}[Anderson \cite{anderson}] \label{thm:anderson}
  If $a$ and $b$ are relatively prime, then the number of simultaneous $(a,b)$-cores is the corresponding rational Catalan number: $|\mathcal{C}_{a,b}|=\Cat_{a,b}$.
\end{theorem}


Our first result is a new proof of Theorem \ref{thm:anderson} using the geometry of lattice points in rational polyhedra.  This framework easily extends to prove other results chief among them a proof of Armstrong's conjecture:


\begin{theorem} \label{thm:armstrong}
The average size of an $(a,b)$-core is $(a+b+1)(a-1)(b-1)/24$.
\end{theorem}

\begin{remark}
 Armstrong conjectured Theorem \ref{thm:armstrong} in 2011; it first appeared in print in \cite{AHJ}.
Stanley and Zanello \cite{SZ} have proven the Catalan case ($a=b+1$) of Armstrong's conjecture by different methods, and building on their work Aggarwal \cite{Aggarwal} has proven the case $a=mb+1$.
\end{remark}


Our two main tools are the abacus construction and Ehrhart theory. We briefly recall these ideas before giving a high-level overview of the proofs of Theorems \ref{thm:anderson} and \ref{thm:armstrong}.







\subsection{The abacus construction}
The main tool used to study $a$-cores is the abacus construction; the canonical reference for the abacus is James and Kerber \cite{JK}. We review this construction in detail in Section \ref{sec:abacus}. For now, we note that there are at least two variants of the abacus construction in the literature.
The first construction, which we call the \emph{positive abacus}, gives a bijection between $a$-cores and $\N^{a-1}$. Anderson's original proof used the positive abacus as part of a bijection between $(a,b)$-cores and $(a,b)$-Dyck paths, which were already known to be counted by $\Cat_{a,b}$.
We use the second construction, which we call the \emph{signed abacus}. The signed abacus is a bijection between $a$-core partitions and points in the $a-1$ dimensional lattice
$$\Lambda_a=\left\{c_0,\dots,c_a\in\Z \mid\sum c_{a-1}=0\right\}.$$

Key to our proof of Armstrong's conjecture is the following:

\begin{theorem*}
Under the signed abacus bijection, the size of an $a$-core is given by the quadratic function
$$Q(c_1,\dots, c_a)=\frac{a}{2}\sum c_i^2+\sum ic_i$$
\end{theorem*}

We are not sure where exactly where this theorem originates; a stronger version is used, seemingly independently, in \cite{GKS} and \cite{DS} to prove generating functions counting certain partitions are modular forms.   For completeness, proof this result as Theorem \ref{thm:quadratic}.

\subsection{Ehrhart and Euler-Maclaurin}
The abacus construction has long been used to study core partitions.  The novelty of this paper is to bring lattice point techniques to bear on the subject.

The number of lattice points in a polytope can be viewed as a discrete version of the volume of a polytope; Ehrhart theory is the study of this analogy. A gentle introduction to Ehrhart theory may be found in \cite{BR}.
Let $V$ be an $n$ dimensional real vector space, and $\Lambda\subset V$ an $n$ dimensional lattice.  A lattice polytope $P\subset V$ is a polytope all of whose vertices are points of $\Lambda$.
For $t$ a positive integer, let $tP$ denote the polytope obtained by scaling $P$ by $t$. For $t\geq 0$, define $L(P,t)$ to the number of lattice points in $tP$:
$$L(P,t)=\#\{\Lambda\cap tP\}$$
Clearly, the volume of $tP$ is $t^n$ times the volume of $P$. The central result in Ehrhart theory is that, parallel to this fact, $L(P,t)$ is a degree $n$ polynomial in $t$. 

Other than the fact that $L(P,t)$ is a polynomial of degree $n$, the one fact from Ehrhart theory we use is Ehrhart reciprocity. If we scale a polytope by $-t$, then keeping track of orientation the volume changes by $(-t)^n$. The polynomial $L(P,t)$ is not in general even or odd, and so $L(P,-t)$ cannot be $(-1)^n$ times the number of lattice points in $-P$. Ehrhart reciprocity states that instead 
$$L(P,-t)=(-1)^nL(P^\circ, t)$$
 where $P^\circ$ denotes the interior of $P$.


Euler-Maclaurin theory is the extension of Ehrhart theory to an analogy between integrating a polynomial over a polytope and summing it over the lattice points in the polytope. At the most basic level, this is the familiar ``sum of the first $n$ cubes'' formulas.  Specifically, if $f$ is a polynomial of degree $d$ on $V$, then $\int_{tP} f$ is a polynomial of degree $d+n$. Euler-Maclaurin theory says that the discrete analog
$$L(f, P, t)=\sum_{x\in\Lambda\cap tP} f(x)$$
is also a polynomial of degree $d+ n$. Ehrhart reciprocity also extends : $$L(f,P,-t)=(-1)^nL(f, -P^\circ, t)$$
Although unsurprising to experts, apparently this extension was first used (without proof) in \cite{CJM}; a proof now appears in \cite{AB}.

\subsection{Overview of method}

To explain the method used to prove Theorems \ref{thm:anderson} and \ref{thm:armstrong}, we begin with the following


\begin{FalseHope}
Fix $a$. Under the signed abacus construction, the set of $(a,b)$-cores are exactly those lattice points in $bP$, for some integral polytope $P\subset V_a:=\Lambda_1\otimes\mathbb{R}$.
\end{FalseHope}



If the false hope were true, Ehrhart theory would imply that, for $b$ relatively prime to a fixed $a$, $|\mathcal{C}_{a,b}|$ would be a polynomial of degree $a-1$ in $b$. It is clear from the definition that this polynomiality property holds for $\Cat_{a,b}$. Thus, proving Anderson's theorem for a fixed $a$ would reduce to showing that two polynomials are equal, which only requires checking finitely many values.

Furthermore, since the size of an $a$-core is a quadratic function $Q$ on the charge lattice, we could apply Euler-Maclaurin theory to see that the total size of all $(a,b)$ cores was a polynomial of degree $a+1$ in $b$, and we could exploit this polynomiality to prove Armstrong's conjecture.

Although the False Hope is not quite true, the strategy outlined above is essentially the one we follow. The set of $b$-cores inside the lattice of $a$-cores is a simplex, which we call $\SC_a(b)$ for \emph{Simplex of Cores}.
One minor tweak needed to the False Hope is that as we vary $b$ $\SC_a(b)$ is not only scaled, but also changed by a linear transformation. These transformations preserve the number of lattice points and the quadratic function $Q$ giving the size of the partitions, and so do not pose any real difficulties.
More troubling is that the polytope $\SC_{a}(b)$ is not integral, but only rational. Recall that a polytope $P$ is \emph{rational} if there is some $k\in\Z$ such that $kP$ is a lattice polytope.

Ehrhart and Euler/Maclaurin theory can be extended to rational polytopes at the cost of replacing polynomials by \emph{quasipolynomials}.

\begin{definition}
A function $f:\Z\to\C$ is a quasipolynomial of degree $d$ and period $n$ if there exist $n$ polynomials $p_0,\dots, p_{n-1}$ of degree $d$, so that for $x\in k+n\Z$, we have $f(x)=p_k(x)$.
\end{definition}

\begin{example} Let $P$ be the polytope $x, y\geq 0, 2x+y\leq 1$. Then
$$\#\{tP\cap \Z^2\}=\left\{\begin{array}{rl} \frac{t^2+4t+4}{4} & \text{$t$ even} \\
\frac{t^2+4t+3}{4} & \text{$t$ odd}\end{array}\right.$$
\end{example}


Since $\Cat_{a,b}$ is defined only for $a$ and $b$ relatively prime, it fits nicely into the quasipolynomial framework. For $a$ fixed, and $b$ in a fixed residue class mod $a$, $\Cat_{a,b}$ is a polynomial. It just so happens that residue classes relatively prime to $a$ have identical polynomials. Such ``accidental'' equalities between the polynomials for different residue classes happen frequently in Ehrhart theory, but are mysterious in general. Perhaps the most studied manifestation of this is \emph{period collapse} (see \cite{Haase} and references), where the quasipolynomial is in fact a polynomial. In our case, symmetry considerations give an elementary explanation of the ``accidental'' equalities between the polynomials for different residue classes.

In Lemma \ref{lem:standardsimplex} we show that the the polytope $\SC_a(b)$ is isomorphic to a rational simplex we call $\TD_a(b)$ (for \emph{Trivial Determinant}) that we now describe. Let $L_k$ be the one dimensional representation of $\Z_a=\Z/a\Z$ where $1\in \Z_a$ acts as $\exp(2\pi ik/a)$. Then any $b$ dimensional representation $V$ of $\Z_a$ may be written as
$$V=\bigoplus_{k=0}^{a-1} L_k^{\oplus z_k}$$
for nonnegative integers $z_i$ satisfying $\sum z_i=b$. Thus, there is a bijection between the set of $b$ dimensional representations of $\Z_a$ and lattice points in the standard simplex $b\Delta_{a-1}$, which has $\binom{a-1+b}{b}$ lattice points.


The simplex $\TD_a(b)$ is obtained by considering only those representations that have trivial determinant (i.e., $\wedge^b V\cong L_0$). This is equivalent to restricting to the index $a$ sublattice given by $\sum iz_i=0\pmod a$.


More generally, consider the set of representations with determinant isomorphic to $L_k$ for any $k$. Tensoring $V$ by $L_1$ corresponds to the cyclic permutation of coordinates $z_k\mapsto z_{k+1}$, and changes the determinant of $V$ by tensoring by $L_1^{\otimes b}=L_b$ (where we are using periodic indices). Thus, the dual $\Z_a$ acts on the set of all $b$ dimensional representations of $\Z_a$, and when $b$ is relatively prime to $a$ this action is free, and each orbit contains exactly one representation with trivial determinant. Hence, the number of points in $\TD_a(b)$ is exactly one $a$th of the number of points in $b\Delta_a$, namely $\binom{a-1+b}{b}/a=\Cat_{a,b}$, reproving Anderson's theore.  Armstrong's conjecture follows with a little more work.

The situation is illustrated in Figure \ref{fig:sublattice}. The left hand picture shows $\TD_3(10)\cong \SC_3(10)$, while the right hand picture shows the standard simplex $10\Delta_2$. The large black dots are the representations with trivial determinant, while the small dots and circles are those representations with determinant $L_1$ and $L_2$, respectively. Rotating about the center of the triangle by 120 degrees corresponds to tensoring by $L_1$ and permutes the different style dots.
\begin{figure}[h!]
\centering
  \caption{$\Lambda_3$ and $\Lambda_3^\prime$ inside $\mathcal{C}_{3, 10}$}
\label{fig:sublattice}
\begin{tikzpicture}[scale=.8]
\begin{scope}
\clip (-3.5, -4*1.7320508/2)--(-3.5, 4*1.7320508/2)--(2.5,
4*1.7320508/2)--(2.5, -4*1.7320508/2)--cycle;
\foreach \x in {-7,-6,...,7}{% Two indices running over each
\foreach \y in {-7,-6,...,7}{% node on the grid we have drawn
\node[draw,circle,inner sep=1pt,fill] at (\x-\y/2,1.7320508*\y/2) {};}}
\draw[dashed] (2, 10*1.7320508/2)--(2, -5*1.7320508);
\draw[dashed] (-15-3, 5*1.7320508)--(15-3, -5*1.7320508);
\draw[dashed] (-15-3, -5*1.7320508)--(15-3, 5*1.7320508);
\end{scope}
\begin{scope}[xshift=9cm]
\clip (-3.5, -4*1.7320508/2)--(-3.5, 4*1.7320508/2)--(2.5,
4*1.7320508/2)--(2.5, -4*1.7320508/2)--cycle;


\foreach \x in {-7,-6,...,7}{% Two indices running over each
\foreach \y in {-7,-6,...,7}{% node on the grid we have drawn
\node[draw,circle,inner sep=1pt,fill] at (\x-\y/2,1.7320508*\y/2) {};}}
\begin{scope}[xshift=.5cm, yshift=.2886751cm]
\foreach \x in {-7,-6,...,7}{% Two indices running over each
\foreach \y in {-7,-6,...,7}{% node on the grid we have drawn
\node[draw,circle,inner sep=1pt] at (\x-\y/2,1.7320508*\y/2) {};}}
\end{scope}
\begin{scope}[ xshift=.5cm, yshift=-.2886751cm]
\foreach \x in {-7,-6,...,7}{% Two indices running over each
\foreach \y in {-7,-6,...,7}{% node on the grid we have drawn
\node[draw,circle,inner sep=.3pt,fill] at (\x-\y/2,1.7320508*\y/2) {};}}
\end{scope}
\draw[dashed] (2, 10*1.7320508/2)--(2, -5*1.7320508);
\draw[dashed] (-15-3, 5*1.7320508)--(15-3, -5*1.7320508);
\draw[dashed] (-15-3, -5*1.7320508)--(15-3, 5*1.7320508);
\end{scope}
\end{tikzpicture}
\end{figure}

\subsection{Self-conjugate simultaneous cores}
The lattice point technique easily adapts to treat the case of self-conjugate simultaneous cores.  Recall that the conjugate of a partition is obtained by reflecting it about the line $y=x$; a partition is self-conjugate if it is equal to its conjugate.

Ford, Mai and Sze have shown \cite{FMS} that self-conjugate $(a,b)$-core partitions are counted by
$$\binom{\big\lfloor \frac{a}{2} \big\rfloor +\big\lfloor \frac{b}{2} \big\rfloor}{\big\lfloor \frac{a}{2} \big\rfloor}$$

Armstrong conjectured, and Chen, Huang and Wang recently proved \cite{CHW}, that the average size of self-conjugate $(a,b)$-core partitions is the same as the average size of all $(a,b)$-core partitions, namely $(a+b+1)(a-1)(b-1)/24$.

In Section \ref{sec:conjugate} we give new proofs of both of these results. The key idea is that the action of conjugation on $\SC_a(b)$ corresponds to the action of taking dual representations on $\TD_a(b)$.

This paper initially included sections applying lattice point ideas to $q$ and $(q,t)$ enumerations of simultaneous core partitions.  This material greatly increased the length of the paper but the results in these later sections were weaker and more speculative, and so were cut to make the key point and simplicity of the results clearer.  This material is still available on the version of this paper found on the arXiv: \url{https://arxiv.org/abs/1502.07934}.  



\subsection{Acknowledgements}
I learned about Armstrong's conjecture over dinner after speaking in the MIT combinatorics seminar. I would like to thank Jon Novak for the invitation, Fabrizio Zanello for telling me about the conjecture, and funding bodies everywhere for supporting seminar dinners.  

The anonymous referee gave many comments that improved the exposition.



%Once we have polynomiality, fixing the particular polynomial is not difficult: as the only $1$-core partition is the empty partition, we have that $(b-1)$ must be a factor of this polynomial.  Ehrhart-reciprocity will tell us that $(a+b+1)$ must also be a factor of the polynomial.  Together with the obvious $(a,b)$ symmetry, we have thus fixed the polynomial as $(a+b+1)(a-1)(b-1)$ times some constant, and so we only have to compute one more average value to get our main result.



\section{Abaci and Electrons} \label{sec:abacus}

This section is a review of the fermionic viewpoint of partitions and the abacus model of $a$-cores.  It contains no original material.  The main results are that $a$-cores are in bijection with points on the ``charge lattice'' $\Lambda_a$, and that the size of a given $a$-core is given by a quadratic function on the lattice
.

%\begin{definition}
%Let $$V_a=\{ \mathbf{c}=(c_0,\dots, c_{a-1}) | c_i\in \R; \quad \sum c_i=0\},$$
%and the let the \emph{charge lattice} $\Lambda_a\subset V_a$ be
% $$\Lambda_a=\{ \mathbf{c}=(c_0\dots, c_{a-1}) | \sum c_i=0;\quad c_i\in \Z\}$$
%\end{definition}

\subsection{Fermions}
 We begin with a motivating fairy tale. It should not be mistaken for an attempt at accurate physics or accurate history.

According to quantum mechanics, the possible energies levels of an electron are quantized -- they can only be half integers (perhaps a better name is odd half integers), i.e. elements of $$\Z_{1/2}=\{a+1/2\mid a\in\Z\}.$$  In particular, basic quantum mechanics predicts electrons with negative energy.  Physically, it makes no sense to have negative energy electrons.


\emph{Dirac's electron sea} solves the problem of negative energy electrons by redefining the vacuum state $\vac$.  The Pauli exclusion principle states that each possible energy level can have at most one electron in it; thus, we can view any set of electrons as a subset $S\subset \Z_{1/2}$.  Intuitively, the vacuum state $\vac$ should consist of empty space with no electrons at all, and hence correspond to the empty set $\emptyset\subset\Z_{1/2}$.

Dirac suggested instead to take $\vac$ to be an
infinite ``sea'' of negative energy electrons.   Specifically, in Dirac's vacuum state every negative energy level is filled with an electron, but none of the positive energy states are filled. Then by Pauli's
exclusion principle we cannot add a negative energy electron
to $\vac$, but positive energy electrons can be added as usual.  Thus, Dirac's electron sea solves the problem of negative energy electrons.

As an added benefit, Dirac's electron sea predicts the positron, a particle
that has the same energy levels as an electron, but positive charge.  Namely, a positron corresponds to a ``hole'' in the electron sea -- a negative
energy level \emph{not} filled with an electron.  Removing a negative energy electron results in adding positive charge and positive energy, and hence can be interpreted as adding a positron.

Our fairy tale leads us to the following definitions:

\begin{definition} Let $\Z_{1/2}^\pm$ denote the set of all positive/negative half integers, respectively.

The vacuum $\vac\subset \Z_{1/2}$ is the set $\Z_{1/2}^-$.

A \emph{state} $S$ is a set $S\subset\Z+1/2$ such that the symmetric
difference 
$$S \triangle \vac=(S\cap\Z_{1/2}^+)\cup (S^c\cap\Z_{1/2}^-)$$
 is finite.  States should be interpreted as a finite collection of electrons -- 
the elements of $S\cap \Z^+_{1/2}$, which we will denote by $S^+$ -- and a finite collection of positrons -- the elements of $S^c\cap \Z^-_{1/2}$, which we will denote by $S^-$.

The \emph{charge} $c(S)$ of a state $S$ is the number of positrons minus the number of electrons:
$$c(S)=| S^-| -| S^+|$$

The $\emph{energy}$ $e(S)$ of a state $S$ is the sum of all the energies of the positrons and the electrons:
$$e(S)=\sum_{k\in S^+}\; k +\sum_{k\in S^-}\; -k$$

\end{definition}

It is convenient to represent a state $S$ as a Maya diagram.

\begin{definition} The \emph{Maya diagram} of $S$ is an infinite sequence of circles on the $x$-axis, one circle centered at each element of $\Z_{1/2}$, with the positive circles extending to the left and the negative direction to the right.  A black ``stone'' is placed on the
circle corresponding to $k\in\Z_{1/2}$ if and only if $k\in S$. \end{definition}

\begin{example}
The Maya diagram corresponding to the vacuum vector $\vac$ is shown below.
\begin{center}
\begin{tikzpicture}[scale=.6]
\draw (-5.5,0) node{$\cdots$};
\draw (-4.5,0) circle (.3) node[below=3pt]{$\frac{9}{2}$};
\draw (-3.5,0) circle (.3) node[below=3pt]{$\frac{7}{2}$};
\draw (-2.5,0) circle (.3) node[below=3pt]{$\frac{5}{2}$};
\draw (-1.5,0) circle (.3) node[below=3pt]{$\frac{3}{2}$};
\draw (-.5,0) circle (.3) node[below=3pt]{$\frac{1}{2}$};
\filldraw (.5,0) circle (.3) node[below=3pt]{$\frac{-1}{2}$};
\filldraw (1.5,0) circle (.3) node[below=3pt]{$\frac{-3}{2}$};
\filldraw (2.5,0) circle (.3) node[below=3pt]{$\frac{-5}{2}$};
\filldraw (3.5,0) circle (.3) node[below=3pt]{$\frac{-7}{2}$};
\filldraw (4.5,0) circle (.3) node[below=3pt]{$\frac{-9}{2}$};
\draw (5.5,0) node{$\cdots$};
\draw (0,-.5) to (0,.5);
\end{tikzpicture}
\end{center}
\end{example}


\begin{example} \label{ex:particles}
The following Maya diagram illustrates the state $S$ consisting of an
electron with energy $3/2$, and two positrons, with energy $1/2$ and
$5/2$.

\begin{center}
\begin{tikzpicture}[scale=.6]
\draw (0,-.5) to (0,.5);
\draw (-5.5,0) node{$\cdots$};
\draw (-4.5,0) circle (.3) node[below=3pt]{$\frac{9}{2}$};
\draw (-3.5,0) circle (.3) node[below=3pt]{$\frac{7}{2}$};
\draw (-2.5,0) circle (.3) node[below=3pt]{$\frac{5}{2}$};
\filldraw (-1.5,0) circle (.3) node[below=3pt]{$\frac{3}{2}$};
\draw (-.5,0) circle (.3) node[below=3pt]{$\frac{1}{2}$};
\draw (.5,0) circle (.3) node[below=3pt]{$\frac{-1}{2}$};
\filldraw (1.5,0) circle (.3) node[below=3pt]{$\frac{-3}{2}$};
\draw (2.5,0) circle (.3) node[below=3pt]{$\frac{-5}{2}$};
\filldraw (3.5,0) circle (.3) node[below=3pt]{$\frac{-7}{2}$};
\filldraw (4.5,0) circle (.3) node[below=3pt]{$\frac{-9}{2}$};
\draw (5.5,0) node{$\cdots$};
\end{tikzpicture}
\end{center}
\end{example}


\subsection{Boundary Paths}

We now describe a bijection from the set of partitions $\mathcal{P}$ to the
set of charge 0 states, that sends a partition
$\lambda\in\mathcal{P}_n$ of size $n$ to a state $S_\lambda$ with
energy $e(S_\lambda)=n$.  This bijection can be understood in two ways:
as recording the boundary path of $\lambda$, or recording the modified
Frobenius coordinates of $\lambda$.

We draw partitions in ``Russian
notation'' -- rotated $\pi/4$ radians counterclockwise and scaled up
by a factor of $\sqrt{2}$, so that each segment of the border path of
$\lambda$ is centered above a half integer on the $x$-axis.  We traverse the boundary path of $\Lambda$ from left to right.  For each segment of the border path, we place an electron in the corresponding energy level if that segment of the border slopes up, and we leave the energy state empty if that segment of border path slopes down.

\begin{example} \label{ex:electronstopartitions}

We illustrate the bijection in the case of $\lambda=3+2+2$.  The corresponding state $S_\lambda$ consists of two
electrons with energy $5/2$ and $1/2$, and two positrons with energy
$3/2$ and $5/2$.

\begin{center}
\begin{tikzpicture}
\begin{scope}[gray, very thin, scale=.6]
\clip (-5.5, 5.5) rectangle (5.5, -5);
\draw[rotate=45, scale=1.412] (0,0) grid (6,6);
\end{scope}

\begin{scope}[rotate=45, very thick, scale=.6*1.412]
\draw (0,5.5) -- (0, 3) -- (1,3) -- (1,2) -- (3,2) -- (3,0) -- (5.5,0);
\end{scope}

\begin{scope}[scale=.6, dotted]

\draw (-4.5,0) -- (-4.5, 4.5);
\draw (-3.5,0) -- (-3.5, 3.5);
\draw (-2.5,0) -- (-2.5, 3.5);
\draw (-1.5,0) -- (-1.5, 3.5);
\draw (-.5,0) -- (-.5, 3.5);
\draw (.5,0) -- (.5, 4.5);
\draw (1.5,0) -- (1.5, 4.5);
\draw (2.5,0) -- (2.5, 3.5);
\draw (3.5,0) -- (3.5, 3.5);
\draw (4.5,0) -- (4.5, 4.5);
\end{scope}


\begin{scope}[scale=.6, yshift=-.5cm]
\draw[solid] (0,-.5) -- (0,.5);
\draw (-5.5,0) node{$\cdots$};
\draw (-4.5,0) circle (.3) node[below=3pt]{$\frac{9}{2}$};
\draw (-3.5,0) circle (.3) node[below=3pt]{$\frac{7}{2}$};
\filldraw (-2.5,0) circle (.3) node[below=3pt]{$\frac{5}{2}$};
\draw (-1.5,0) circle (.3) node[below=3pt]{$\frac{3}{2}$};
\filldraw (-.5,0) circle (.3) node[below=3pt]{$\frac{1}{2}$};
\filldraw (.5,0) circle (.3) node[below=3pt]{$\frac{-1}{2}$};
\draw (1.5,0) circle (.3) node[below=3pt]{$\frac{-3}{2}$};
\draw (2.5,0) circle (.3) node[below=3pt]{$\frac{-5}{2}$};
\filldraw (3.5,0) circle (.3) node[below=3pt]{$\frac{-7}{2}$};
\filldraw (4.5,0) circle (.3) node[below=3pt]{$\frac{-9}{2}$};
\draw (5.5,0) node{$\cdots$};
\end{scope}
\end{tikzpicture}
\end{center}

\end{example}

The energies of the electrons and the positrons of $\lambda$ are sometimes called the \emph{modified Frobenius coordinates}, and can also be seen as follows.  The $y$-axis dissects the partition $\lambda$ into two pieces.  The left side of $\lambda$ consists of $c$ rows, where $c$ is the number
of electrons.  The number of boxes in the $i$th row is the energy of
the $i$th electron.  The right half of $\lambda$ also consists of $c$ rows, withthe number of boxes in a given row the energy of the corresponding positron.  



\begin{example} Consider Example \ref{ex:electronstopartitions}.  If the $y$-axis was drawn in, left of the $y$-axis would be two rows, the bottom row having 2.5 boxes and the top row .5 boxes, which are the energies of the electrons in $S$.  Similarly, the right hand side has two rows with 2.5 and 1.5 boxes, the energies of the positrons in $S$.


\end{example}

The bijection between partitions and states of charge zero may be
modified to give a bijection between partitions and states of charge $c$ for any $c\in\Z$, simply translate the partition to the right by $c$.

\subsection{The Abacus construction}

Rather than view the Maya diagram as a series of stones in a line, we
now view it as beads on the runner of an abacus.  Sliding the beads
to be right justified allows the charge of a state to be read off,
as it is easy to see how many electrons have been added or are missing
from the vacuum state.  In what follows, we mix our metaphors and talk about electrons and positrons on runners of an abacus.

\begin{example} \label{ex:mayabijection}
Consider Example \ref{ex:particles}, where the Maya diagram consists of
two positrons and an electron.  Pushing the beads to be right
justified, we see the first bead is one step to the right of zero, and
hence the original state had charge 1.

\begin{center}
\begin{tikzpicture}[scale=.6]
\draw (0,-.5) to (0,.5);
\draw (-5.5,0) node{$\cdots$};
\draw (-4.5,0) circle (.3) node[below=3pt]{$\frac{9}{2}$};
\draw (-3.5,0) circle (.3) node[below=3pt]{$\frac{7}{2}$};
\draw (-2.5,0) circle (.3) node[below=3pt]{$\frac{5}{2}$};
\filldraw (-1.5,0) circle (.3) node[below=3pt]{$\frac{3}{2}$};
\draw (-.5,0) circle (.3) node[below=3pt]{$\frac{1}{2}$};
\draw (.5,0) circle (.3) node[below=3pt]{$\frac{-1}{2}$};
\filldraw (1.5,0) circle (.3) node[below=3pt]{$\frac{-3}{2}$};
\draw (2.5,0) circle (.3) node[below=3pt]{$\frac{-5}{2}$};
\filldraw (3.5,0) circle (.3) node[below=3pt]{$\frac{-7}{2}$};
\filldraw (4.5,0) circle (.3) node[below=3pt]{$\frac{-9}{2}$};
\draw (5.5,0) node{$\cdots$};

\begin{scope}[yshift=-2.5cm]
\draw[-triangle 45, snake=snake,line after snake=1mm] (-4,0)--(4,0) node [above,  text centered,midway] {Push beads};

\end{scope}

\begin{scope}[yshift=-4cm]
\draw (0,-.5) to (0,.5);
\draw (-5.5,0) node{$\cdots$};
\draw (-4.5,0) circle (.3) node[below=3pt]{$\frac{9}{2}$};
\draw (-3.5,0) circle (.3) node[below=3pt]{$\frac{7}{2}$};
\draw (-2.5,0) circle (.3) node[below=3pt]{$\frac{5}{2}$};
\draw (-1.5,0) circle (.3) node[below=3pt]{$\frac{3}{2}$};
\draw (-.5,0) circle (.3) node[below=3pt]{$\frac{1}{2}$};
\draw (.5,0) circle (.3) node[below=3pt]{$\frac{-1}{2}$};
\filldraw (1.5,0) circle (.3) node[below=3pt]{$\frac{-3}{2}$};
\filldraw (2.5,0) circle (.3) node[below=3pt]{$\frac{-5}{2}$};
\filldraw (3.5,0) circle (.3) node[below=3pt]{$\frac{-7}{2}$};
\filldraw (4.5,0) circle (.3) node[below=3pt]{$\frac{-9}{2}$};
\draw (5.5,0) node{$\cdots$};
\end{scope}


\end{tikzpicture}
\end{center}
\end{example}

To understand how to read the hooklengths of $\lambda$ off the Maya diagram for $\lambda$, we introduce the notion of \emph{inversions}.  The cells $\square\in\lambda$ are in bijection with the
\emph{inversions} of the boundary path; that is, with pairs of segments
$(\text{step}_1, \text{step}_2)$, where $\text{step}_1$ occurs before $\text{step}_2$,
but $\text{step}_1$ is traveling NE and $\text{step}_2$ is traveling SE.  The bijection
sends $\square$ to the segments at the end of its arm and leg.

Translating to the fermionic viewpoint, cells of $\lambda$ are in
bijection with pairs 
$$\left\{(e, e-k)\mid e\in\Z_{1/2}, k>0\right\}$$ of a filled energy level $e$ and an
empty energy level $e-k$ of lower energy; we call such a pair an \emph{inversion}.  We call the first entry the \emph{hand} of the inversion, and the second entry the \emph{foot} of the inversion, as they are the portions of the boundary path that lie at the end of the arm and the leg of the corresponding cell, respectively.  The hook length $h(\square)$ of the corresponding cell is $k$.

If $(e,e-k)$ is such a pair, reducing the energy of the electron from
$e$ to $e-k$ changes $\lambda$ by removing the
rim hook corresponding to the cell $\square$.  This rim-hook has
length $k$.



\begin{example}
The cell $\square=(2,1)$ of $\lambda=3+2+2$ (See Example \ref{ex:electronstopartitions}).
Here, $h(\square)=3$, and corresponds to the electron in energy state $1/2$
and the empty energy level $-5/2$; which are three apart.

\end{example}

To connect to cores and quotients, rather than place the electrons corresponding to $\lambda$ on one runner, we place them on $a$ different runners labelled from $0$ to $a-1$, putting the energy levels $ka-i-1/2$ on runner $i$.

If the hook length $h(\square)=ka$ is divisible by $a$, then the two
energy levels of $\textrm{inversion}(\square)$ lie on the same
runner.  Similarly, any inversion of energy states on the same
runner corresponds to a cell with hook length divisible by $a$.

\begin{example} The example above shows the partition $\lambda=10+7+6+4+4+3+1+1$ drawn in Russian notation above its 3-abacus.  The numbers on the boundary path illustrate which runner of the abacus that step belongs to.  The cells marked with a $\star$ are those with hook lengths divisible by three; they can be seen to correspond to inversions whose hand and foot are on the same runner.
\begin{center}
\begin{tikzpicture}

\begin{scope}[gray, very thin, scale=.4]
\clip (-11.5, 11.5) rectangle (11.5, -10);
\draw[rotate=45, scale=1.412, gray!20!white, fill] (0,0)--(0,10)--(1,10)--(1,7)--(2,7)--(2,6)--(3,6)--(3,4)--(5,4)--(5,3)--(6,3)--(6,1)--(8,1)--(8,0)--cycle;
\draw[rotate=45, scale=1.412] (0,0) grid (12,12);
\end{scope}

\begin{scope}[rotate=45, ultra thick, scale=.4*1.412]


\begin{scope}
\draw (.5,7.5) node{$\star$};
\draw (1.5,5.5) node{$\star$};
\draw (1.5,2.5) node{$\star$};
\draw (4.5,2.5) node{$\star$};

\draw (1,9.5) node{$1$};
\draw (1.5,7)node{$1$};
\draw (3,5.5)node{$1$};
\draw (4.5,4)node{$1$};
\draw (6,2.5)node{$1$};
\draw (7.5,1)node{$1$};
\draw (9.5,0)node{$1$};
\end{scope}
\begin{scope}
\draw (0,10.5)node{$2$};
\draw (1,8.5)node{$2$};
\draw (2,6.5)node{$2$};
\draw (3,4.5)node{$2$};
\draw (5,3.5)node{$2$};
\draw (6,1.5)node{$2$};
\draw (8,.5)node{$2$};
\draw (10.5,0)node{$2$};
\end{scope}
\begin{scope}
\draw (.5,10)node{$3$};
\draw (1, 7.5)node{$3$};
\draw (2.5,6)node{$3$};
\draw (3.5,4)node{$3$};
\draw (5.5,3)node{$3$};
\draw (6.5,1)node{$3$};
\draw (8.5,0)node{$3$};
\end{scope}

\end{scope}


\begin{scope}[scale=.4, yshift=-.5cm]
\draw(-11.5,0) circle(.3);
\draw(-8.5, 0)  circle (.3);
\filldraw(-5.5,0)  circle (.3);
\draw(-2.5,0)  circle (.3);
\filldraw(.5,0)  circle (.3);
\draw(3.5,0)  circle (.3);
\filldraw(6.5,0)  circle (.3);
\filldraw(9.5,0)  circle (.3);
\end{scope}

\begin{scope}[scale=.4, yshift=-1.5cm]
\draw(-10.5,0) circle(.3);
\draw(-7.5, 0)  circle (.3);
\draw(-4.5,0)  circle (.3);
\draw(-1.5,0)  circle (.3);
\draw(1.5,0)  circle (.3);
\draw(4.5,0)  circle (.3);
\draw(7.5,0)  circle (.3);
\filldraw(10.5,0)  circle (.3);
\end{scope}

\begin{scope}[scale=.4, yshift=-2.5cm]
\filldraw(-9.5,0) circle(.3);
\draw(-6.5, 0)  circle (.3);
\filldraw(-3.5,0)  circle (.3);
\filldraw(-.5,0)  circle (.3);
\filldraw(2.5,0)  circle (.3);
\filldraw(5.5,0)  circle (.3);
\filldraw(8.5,0)  circle (.3);
\filldraw(11.5,0)  circle (.3);
\end{scope}

\begin{scope}[scale=.4]
\draw (0,-3)--(0,0);
\end{scope}


\end{tikzpicture}
\end{center}


  \end{example}


Thus, $\lambda$ is an $a$-core if and only if the beads on each runner of the $a$-abacus are right justified.  Sliding the beads of an $a$-abacus of a partition to be right justified corresponds to ``taking the $a$-core'' of the partition, and the abacus construction was originally introduce to show that this process was well defined.

More important to us, the abacus construction gives another description of $a$-core partitions in terms of a lattice.  Although the total charge of all the runners in an $a$-abacus must be zero, the charge need not be evenly divided among the runners.  Let
$c_i$ be the charge on the $i$th runner; then we have $\sum c_i=0$, and the $c_i$ determine $\lambda$.

Similarly, given any $\mathbf{c}=(c_0,\dots,c_{a-1})\in\Z^a$ with $\sum c_i=0$, there is a unique right justified abacus with charge  $c_i$ on the $i$th runner.  The corresponding partition is an $a$-core which we denote $\core_a(\mathbf{c})$.

We have shown:

\begin{lemma}
There is a bijection $$\core_a:\{(c_0,\dots,c_{a-1}\mid c_i\in\Z, \sum c_i=0\}\to \{\lambda \mid \lambda \text{ is in $a$-core} \}$$
\end{lemma}


\begin{example}
We illustrate that $\core_3(0,3, -3)=7+5+3+3+2+2+1+1$; note that this is the $3$-core of the previous partition.  
\begin{center}
\begin{tikzpicture}

\begin{scope}[gray, very thin, scale=.4]
\clip (-11.5, 11.5) rectangle (11.5, -10);
\draw[rotate=45, scale=1.412, gray!20!white, fill] (0,0)--(0,7)--(1,7)--(1,5)--(2,5)--(2,3)--(4,3)--(4,2)--(6,2)--(6,1)--(8,1)--(8,0)--cycle;
\draw[rotate=45, scale=1.412] (0,0) grid (12,12);
\end{scope}

\begin{scope}[rotate=45, ultra thick, scale=.4*1.412]
\begin{scope}

\draw (0,8.5) node{$1$};
\draw (1,6.5)node{$1$};
\draw (2,4.5)node{$1$};
\draw (3.5,3)node{$1$};
\draw (5.5,2)node{$1$};
\draw (7.5,1)node{$1$};
\draw (9.5,0)node{$1$};
\end{scope}
\begin{scope}
\draw (0,10.5)node{$2$};
\draw (0,7.5)node{$2$};
\draw (1,5.5)node{$2$};
\draw (2,3.5)node{$2$};
\draw (4,2.5)node{$2$};
\draw (6,1.5)node{$2$};
\draw (8,.5)node{$2$};
\draw (10.5,0)node{$2$};
\end{scope}
\begin{scope}
\draw (0,9.5)node{$3$};
\draw (0.5, 7)node{$3$};
\draw (0.5,7)node{$3$};
\draw (1.5,5)node{$3$};
\draw (2.5,3)node{$3$};
\draw (4.5,2)node{$3$};
\draw (6.5,1)node{$3$};
\draw (8.5,0)node{$3$};
\end{scope}

\end{scope}






\begin{scope}[scale=.4, yshift=-.5cm]
\draw(-11.5,0) circle(.3);
\draw(-8.5, 0)  circle (.3);
\draw(-5.5,0)  circle (.3);
\draw(-2.5,0)  circle (.3);
\filldraw(.5,0)  circle (.3);
\filldraw(3.5,0)  circle (.3);
\filldraw(6.5,0)  circle (.3);
\filldraw(9.5,0)  circle (.3);
\end{scope}

\begin{scope}[scale=.4, yshift=-1.5cm]
\draw(-10.5,0) circle(.3);
\draw(-7.5, 0)  circle (.3);
\draw(-4.5,0)  circle (.3);
\draw(-1.5,0)  circle (.3);
\draw(1.5,0)  circle (.3);
\draw(4.5,0)  circle (.3);
\draw(7.5,0)  circle (.3);
\filldraw(10.5,0)  circle (.3);
\end{scope}

\begin{scope}[scale=.4, yshift=-2.5cm]
\draw(-9.5,0) circle(.3);
\filldraw(-6.5, 0)  circle (.3);
\filldraw(-3.5,0)  circle (.3);
\filldraw(-.5,0)  circle (.3);
\filldraw(2.5,0)  circle (.3);
\filldraw(5.5,0)  circle (.3);
\filldraw(8.5,0)  circle (.3);
\filldraw(11.5,0)  circle (.3);
\end{scope}

\begin{scope}[scale=.4]
\draw (0,-3)--(0,0);
\end{scope}


\end{tikzpicture}
\end{center}
\end{example}


\begin{theorem} \label{thm:quadratic}
$$|\core_a(\mathbf{c})|=\frac{a}{2}\sum_{k=0}^{a-1} c_k^2+ kc_k$$
\end{theorem}

\begin{proof}
 If  $c_k>0$ the $k$th runner has $c_k$ positrons, with
 energies 
\begin{align*}
(k+1/2)&,\\
(k+1/2)&+a, \\
(k+1/2)&+2a,\\
\vdots\quad &\quad \vdots \\
 (k+1/2)&+(c_k-1)a
\end{align*}
 and so the
 particles on the $k$th runner have total energy
 $$\frac{a}{2}(c_k^2-c_k)+(k+1/2)c_k.$$

 If $c_k<0$, the $k$th runner has $-c_k$ electrons, and a similar calculation shows they have a total energy of $$\frac{a}{2}(c_k^2+c_k)-c_k(a-k-1/2)=\frac{a}{2}(c_k^2-c_k)+(k+1/2)c_k.$$

Since $\sum c_k=0$, the total energy of all particles simplifies to $\frac{a}{2}\sum (c_k^2+kc_k)$.
\end{proof}


\section{Simultaneous Cores} \label{sec:numberandsize}

We now turn to studying the set of $b$-cores within the lattice
$\Lambda_a$ of $a$-cores.
\subsection{Under the abacus construction, \texorpdfstring{$(a,b)$}{(a,b)}-cores form a simplex}

First, some notation and conventions.

Let $r_a(x)$ be the remainder when $x$ is divided by $a$, and $q_a(x)$ be the integer part of $x/a$, so that $x=aq_a(x)+r_a(x)$ for all $x$.
Furthermore, we use cyclic indexing for $\mathbf{c}\in\Lambda_a$;
that is, for $k\in\Z$, we set $c_k=c_{r_a(k)}$.

\begin{lemma}
Within the lattice of $a$ cores, the set of $b$ cores are the
lattice points satisfying the inequalities $$c_{i+b}-c_{i}\leq q_a(b+i)$$
for $i\in\{0,\dots,a-1\}$.
\end{lemma}

\begin{proof}

Fix $\mathbf{c}\in\Lambda_a$, let $\lambda=\core_a(\mathbf{c})$ be the corresponding $a$-core, and consider the $a$-abacus of $\lambda$.

Let $e_i$ denote the highest energy level of an electron on the $i$th runner.  We claim that $\core_a(\mathbf{c})$ is a $b$-core if an only if for each $i$, the energy state $e_i-b$ is filled.

Certainly this condition is necessary.  To see that it is sufficient, suppose that $\lambda$ is an $a$-core, and that for each $i$, the energy level $e_i-b$ is filled.  To see $\lambda$ is a $b$ core, we must show that for any filled energy level $L$, the energy level $L-b$ is also filled.  

If $L$ is on the $i$th runner, then $L=e_i-aw$ for some $w\geq 0$, and so $L-b=(e_i-b)-aw$.  But by supposition $e_i-b$ is a filled state, and $e_i-b-aw$ is to the right of it and on the same runner, and so it must be filled since $\lambda$ is an $a$-core.  Thus, we have seen that $\core_a(\mathbf{c})$ is also a $b$-core.

Using this, we now derivative the inequalities governing the set of $b$-cores within the lattice of $a$-cores.  The energy level $e_{i}-b$ is on runner $r_a(i+b)$, and so $\lambda$ is $b$-core if and only if $e_{i}-b\leq e_{i+b}$ (recall that we are using cyclic indexing).

Substituting $e_k=-ac_k-r(k)-1/2$ and simplifying gives that our inequality is equivalent to
\begin{align*}
a(c_{i+b}-c_i)& \leq b+i-r_a(i+b)
\end{align*}
and hence to
$$c_{i+b}-c_i \leq q_a(b+i).$$

\end{proof}

We have $a$ hyperplanes in an $a-1$ dimensional space; they either
form a simplex or an unbounded polytope.  It is easy to see the form a simplex when $a$ and $b$ are relatively prime, and an unbounded space when $a, b$ have a common factor.



\begin{remark} Generating functions of $(a,b)$-cores when $a$ and $b$ are not relatively prime have been studied in  \cite{AKS}, where they are shown to be modular forms.  We briefly indicate how our viewpoint of $(a,b)$-cores sheds light on this result.

First, it is easy to see there are infinitely many $(a,b)$ cores, as if $d=\gcd(a,b)$, then any $d$-core is also an $(a,b)$-core.

The inequalities given for $\SC_a(b)$ still describe the space of
$(a,b)$-cores when $a,b$ are no longer relatively prime, but these
inequalities no longer describe a simplex.  The inequalities no longer
relate all the $c_i$ to each other; rather, they decouple into $d$ sets of $a/d$ of variables

\begin{align*}
S_0 & =\{c_0, c_d, c_{2d},\dots, c_{a-d}\} \\
 S_1& =\{c_1,c_{d+1},\dots, c_{a-d+1}\} \\
& \quad \quad \cdots \\
S_{d-1}&=\{c_{d-1}, c_{2d-1},\dots, c_{a-1}\} \\
\end{align*}
If $\core_a(\mathbf{c})$ is also a $b$-core, then the inequalities described force the charges $c_i$ in a given group to be close together. However, for any
vector $(v_0,\dots, v_{d-1})$ with $\sum v_i=0$, we may shift each
element of $S_i$ by $v_i$ and all inequalities will still be satisfied.

These shifts generate a lattice isomorphic to the lattice of $d$-cores for $d=\gcd(a,b)$. The remaining choices of the $c_i$ in each group form simplex. So we see the set of $(a,b)$ cores is a finite number of translates of a lattice.  The sum over $Q$ over the points in a lattice will be a theta function, and so we see the generating function of $(a,b)$ cores will be a finite sum of theta functions, and hence modular.

\end{remark}


%\begin{remark}
%The space of $(a,b)$-Dyck paths can also be viewed as the lattice points in a polytope $\DP_{a}(b)$.  We now describe this polytope and the geometry of a bijection between the lattice points in $\SC_a(b)$ and $\DP_a(b)$.

%Again, we fix $a$, and a residue class $r\mod a$, and let $b=ak+r$, with $k$ variable.  Let $h_i$ be the furthest right position of the Dyck path on the $i$th runner from the top, and set $h_0=0$.  We have $h_{i-1}\leq h_i \leq ki+\delta_i$, where $0\leq \delta_i\leq r$ being.

%The step of making the abacus ``flush'' in the bijection $(a,b)$-Dyck paths and $(a,b)$-cores can be seen as doing a case analysis depending on which of the $c_i$ is smallest; the shape of the polytope $\DP_a(b)$ is one of the $a$ pieces that the simplex of cores is cut into by the chambers .

%In the other direction, the looking at number of boxes missing from the optimal statistic $\mod a$ determines an index $a$ coset of the lattice; the $a$ cosets correspond precisely to the $a$ chambers of which dependence of the $c_i$ are maximal.

%PICTURE(S?)
%\end{remark}


In the charge coordinates $\mathbf{c}$, neither the hyperplanes defining the set of $b$ cores nor the quadratic form $Q$ are symmetrical about the origin.  We shift coordinates to remedy this.
\begin{definition}
Define $\mathbf{s}=(s_0,\dots, s_a)\in V_{a-1}$ by
$$s_i=\frac{i}{a}-\frac{a-1}{2a}$$
\end{definition}

The $i/a$ term ensures $s_{i+1}-s_i=1/a$; subtracting $\frac{a-1}{2a}$ ensures that  $\mathbf{s}\in V_a$, i.e. $\sum s_i=0$.
 
\begin{lemma}
In the shifted charge coordinates
$$x_i=c_i+s_i$$
the inequalities defining the set of $b$ cores become
$$x_{i+b}-x_{i}\leq b/a$$
and the size of an $a$-core is given by
$$Q(\mathbf{x})=-\frac{a^2-1}{24}+\frac{a}{2}\sum_{i=0}^{a-1} x_i^2$$
\end{lemma}

\begin{proof}
That the linear term of $Q$ vanishes in the $\mathbf{x}$ coordinates follows immediately from the definition of $\mathbf{s}$.  The constant term of $Q$ in the $\mathbf{x}$ coordinates is $-\frac{a}{2}\sum_{i=0}^{a-1} s_i^2$, which a short computation shows is $-\frac{a^2-1}{24}$.

The statement about the set of $b$-cores follows from the computation

\begin{align*}
x_{i+b}-x_{i}&=c_{i+b}-c_i+s_{i+b}-s_i \\
&\leq  q_a(i+b)+r_a(i+b)/a-i/a \\
&=(b+i)/a-i/a \\
&=b/a
\end{align*}
\end{proof}

Although we often use the $x$ coordinates, to show that the simplex of $\SC_a(b)$ is isomorphic to the simplex $\TD_a(b)$ of trivial determinant representations another change of variables is needed.
 
\begin{lemma} \label{lem:standardsimplex}
Let $a$ and $b$ be relatively prime, and let 
$$k=-\frac{b+1}{2}\pmod a$$
  Then the change of variables
$$z_i=x_{ib+k}-x_{(i+1)b+k}+b/a$$
gives an isomorphism between the rational simplices $\SC_a(b)$ and $\TD_a(b)$. 
\end{lemma}

\begin{proof}
It is immediate that the $z_i$ satisfy $\sum z_i=b$ and $z_i\geq 0$.  The integrality of the $z_i$ follows from the fact that the fractional part of $x_i-x_j$ is $(i-j)/a$.  We must show $\sum iz_i=0\mod a$.

One computes:
$$\sum_{i=0}^{a-1} iz_i=-ax_k+\sum_{i=0}^{a-1} x_i+\frac{b}{a}\sum_{i=0}^{a-1} i$$
Since the fractional part of $x_k$ is $s_k=k/a-(a-1)/2a$, plugging in the definition of $k$ gives that $ax_k=-b/2\pmod a$.  Since $\sum x_i=0$ and $\sum i=(a-1)a/2$, we see $\sum iz_i=0\pmod a$.

A further computation shows this change of variables is invertible.
\end{proof}


\begin{corollary}[Anderson \cite{anderson}] \label{cor:anderson} The number of simultaneous $(a,b)$-cores is $\Cat_{a,b}$.
\end{corollary}

\begin{proof} This follows quickly from Lemma \ref{lem:standardsimplex}, and was outlined in the introduction; we reproduce it here for completeness.


The scaled simplex $b\Delta_a$ has $\binom{a+b-1}{a-1}$
  usual lattice points.  Cyclically permuting the variables preserves
  $b\Delta_a$ and the standard lattice, and when $b$ is relatively prime to $a$ it cyclical permutes the $a$ cosets of the charge lattice.   

Thus the standard lattice points in
  $b\Delta_a$ are equidistributed among the $a$-cosets of the
  charge lattice, and hence each one contains $\frac{1}{a}\binom{a+b-1}{a-1}=\Cat_{a,b}$.


\end{proof}

\subsection{The size of simultaneous cores}

We now have all the ingredients needed to prove Armstrong's conjecture.  We derive it as a consequence of:

\begin{theorem} \label{thm:polynomial}
For fixed $a$, and $b$ relatively prime to $a$,  the average size of an $(a,b)$-core is a polynomial of degree 2 in $b$.
\end{theorem}

\begin{proof}
For fixed $a$, the number of $(a,b)$-cores is $1/a$ times the
number of lattice points in $b\Delta_{a-1}$, which is a polynomial
$F_a(b)$ of degree $a-1$.  In the $\mathbf{x}$-coordinates $Q(\mathbf{x})=|\core_a(\mathbf{x})|$ is invariant under $S_a$, and in particular rotation, so the sum of the sizes of all $(a,b)$-cores is $1/a$ times the sum of $Q$ over the lattice points in $b\Delta_{a-1}$.   By Euler-Maclaurin theory, the sum of a quadratic function over the lattice points in $b\Delta_{a-1}$ is a polynomial $G_a(b)$ of degree $a+1$.

Thus, the average value of an $(a,b)$-core is $G_a(b)/F_a(b)$, the quotient of a polynomial of degree $a+1$ in $b$ by a polynomial of degree $a-1$ in $b$.  To show this is a polynomial of degree two in $b$, we need to show that every root of $F_a$ is a root of $G_a$.

Corollary \ref{cor:anderson} says that the roots of $F_a$ are $-1,
-2,\dots, -(a-1)$.  We now give another derivation of this fact, using Ehrhart reciprocity, that easily adapts to shown these are also roots of $G_a$. 

Ehrhart reciprocity says that $F_a(-x)$ is, up to a sign, the number of
points in the \emph{interior} of $x\Delta_{a-1}$.  The interior
consists of the points in $x\Delta_{a-1}$ none of whose coordinates
are zero, and so the first interior point in $x\Delta_{a-1}$ is  $(1,1,\dots,1)\in a\Delta_{a-1}$.   Thus, $F_a(b)$ vanishes at $b=-1,\dots, -(a-1)$, and as it has degree $a-1$ it has no other roots.

Ehrhart reciprocity extends to Euler-Maclaurin theory, to say that up to a sign $Q_a(-x)$ is the sum of $F$ of the interior points of
$x\Delta_{a-1}$. Thus $Q_a(-x)$ also vanishes at
$b=-1,\dots,-(a-1)$, and so $P_a/Q_a$ is a polynomial of degree 2.

\end{proof}


\begin{corollary} \label{cor:averagesize}
When $(a,b)$ are relatively prime, the average size of an $(a,b)$ core is $(a+b+1)(a-1)(b-1)/24$.
\end{corollary}

\begin{proof}

Fix $a$, and let $P_a(b)=G_a(b)/F_a(b)$ be the degree two polynomial that
gives the average value of the $(a,b)$-cores when $a$ and $b$ are
relatively prime.  As we know $P_a(b)$ is a polynomial of degree 2, we can determine it by computing only three values.

First, we find the two roots of $P_a(b)$.  As the only $1$ core is the empty partition, we have $F_a(1)=1$ and $G_a(1)=0$, and so $P_a(1)=0$.

Ehrhart reciprocity gives that $G_a(-a-b)$ is, up to a sign, the sum of $Q$ over the lattice points in the interior of $(a+b)\Delta_a$, which are just the lattice points contained in $b\Delta_a$, and hence equal to $G_a(b)$.  In particular, $P_a(-a-1)=0$.  

Finally, we compute $P_a(0)$.  It is clear that $\mathcal{S}_a(0)=\{0\}$.  Although this is not a point of $\Lambda_a$, it is in $\Lambda_a^\prime$, and so $P_a(0)=Q(0)=-(a^2-1)/24$.
\end{proof}

\subsection{Self-conjugate \texorpdfstring{$(a,b)$}{(a,b)}-cores} \label{sec:conjugate}

In Lemma \ref{lem:conjugatedual}, we show that under the bijection between $(a,b)$-cores and $b$-dimensional representations of $\Z_a$ with trivial determinant, conjugating a partition corresponds to sending a representation $V$ to its dual $V^*$.  In the lattice point of view, this is a linear map $T$, and hence the self-dual $(a,b)$-cores correspond to the lattice points in the fixed point locus of $T$.  

We show in Lemma \ref{lem:conjugatecount}, that the $T$-fixed lattice points in $\SC_a(b)$ are the lattice points in the $\lfloor a/2\rfloor$ dimensional simplex $\lfloor b/2\rfloor \Delta_{\lfloor a/2\rfloor}$, hence rederiving the count of simultaneous $(a,b)$-core partitions.  

Once we have done this, an analogous application of Euler-Maclaurin theory reproves the statement about the average value.

Let $T:V_a\to V_a$ be the linear map given by
$$T(c_i)=-c_{-1-i}$$
It is easy to check that when translated to core partitions, $T$ corresponds to taking the conjugate, that is: $$\core_a(c)^T=\core_a(T(c))$$
Thus the set of self-conjugate $(a,b)$-cores is the $T$ fixed locus of $\SC_a(b)$.  

Since $T(s)=s$, the same formula holds in the shifted coordinates $\bf{x}$. 



\begin{lemma} \label{lem:conjugatedual}
Under the isomorphism between $\SC_a(b)$ and $\TD_a(b)$ established in Lemma \ref{lem:standardsimplex}, the map $T$ sending a partition to its conjugate corresponds to taking the dual $\Z_a$ representation.
\end{lemma}

\begin{proof}
We want to show $T(z_i)=z_{-i}$.  We compute:
\begin{align*}
T(z_i)&=T(x_{ib+k}-x_{(i+1)b+k}) \\
&=-x_{-ib-k-1}+x_{-ib-b-k-1} \\
& =x_{-ib+k-(b+1+2k)}-x_{(-i+1)b+k-(b+1+2k)}
\end{align*}
And so we need $b+1+2k=0\pmod a$, but this is exactly the definition of $k$ in Lemma \ref{lem:standardsimplex}.
\end{proof}



\begin{lemma} \label{lem:conjugatecount}
The number of $b$-dimensional, self-dual $\Z_a$ representations with trivial determinant is $$\binom{\big\lfloor\frac{a}{2}\big\rfloor+\big\lfloor\frac{b}{2}\big\rfloor}{\big\lfloor\frac{a}{2}\big\rfloor}$$
\end{lemma}

\begin{proof}
  Let $a=2k$ or $2k+1$.  We give a bijection between the representations in question and $k$-tuples of non-negative integers $(z_1,\dots, z_k)$ with $2\sum z_i\leq b$.  The set of such $z_i$ are the lattice points in $\lfloor b/2\rfloor\Delta_{\lfloor a/2\rfloor}$, which are counted by the given binomial coefficient.

First, suppose that $a=2k+1$.  Then the only irreducible self-conjugate representation is the identity, and $T$ has a $k$ dimensional fixed point set consisting of points of the form $(u_0, u_1,\dots, u_k,u_k,\dots, u_1)$.  Thus, $\sum_{i=1}^k 2u_k\leq b$, and value of $u_0$ is fixed by $u_0+2\sum_{i=1}^k u_i=b$.

When $a=2k$, there are two irreducible self-conjugate representations, the identity and the sign representation induced by the surjection $\Z_a\to\Z_2$.  Again, $T$ has a $k$ dimensional fixed point set, this time consisting of points of the form $(u_0, u_1,\dots u_{k-1}, w_{k}, u_{k-1},\dots,u_1)$.  Now for such a representation, having trivial determinant is equivalent to $w_k$ being even, say $w_k=2u_k$.  Then again we have $\sum_{i=1}^k 2u_k\leq b$, with $u_0$ being determined by $u_0+2\sum_{i=1}^{k}u_i=b$. 
\end{proof}



\begin{proposition} \label{prop:conjugatesize}
Let $a$ and $b$ be relatively prime.  Then the average size of a self-conjugate $(a,b)$-core is $(a-1)(b-1)(a+b+1)/24$.
\end{proposition}


\begin{proof}

Since $a$ and $b$ are relatively prime, at most one is even, so we may assume $a$ is odd.  

The proof is essentially the same as that for all $(a,b)$-cores.  One complication is that it seems we must treat odd and even values of $b$ separately.   In each each case, an argument identical to Lemma \ref{thm:polynomial} gives that the average size is a polynomial of degree 2 in $b$.  A priori, we may have different polynomials for $b$ odd and $b$ even; however, the symmetry $(a,b)\leftrightarrow (a, -a-b)$ coming from Ehrhart reciprocity still holds and interchanges odd and even values of $b$, and so if we can compute three values of either polynomial (that don't get identified by this symmetry), we identify both polynomials.

All $1$ and $2$ cores are self conjugate, and thus if $b$ is 1 or 2, the average value is the same.  The arguments made in Corollary \ref{cor:averagesize} for the value of the polynomial at $b=0$ holds for self-conjugate partitions as well, giving a third value.
\end{proof}


\def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{Aggarwal}
Amol Aggarwal.
\newblock Armstrong's conjecture for {$(k,mk+1)$}-core partitions.
\newblock {\em European J. Combin.}, 47:54--67, 2015.
\newblock \url{http://arxiv.org/pdf/1407.5134v1.pdf}.

\bibitem{anderson}
Jaclyn Anderson.
\newblock Partitions which are simultaneously {$t_1$}- and {$t_2$}-core.
\newblock {\em Discrete Math.}, 248(1-3):237--243, 2002.

\bibitem{AB}
Federico Ardila and Erwan Brugall\'e.
\newblock The double {G}romov-{W}itten invariants of {H}irzebruch surfaces are
  piecewise polynomial.
\newblock {\em Int. Math. Res. Not. IMRN}, (2):614--641, 2017.
\newblock \url{http://arxiv.org/abs/1412.4563}.

\bibitem{AHJ}
Drew Armstrong, Christopher R.~H. Hanusa, and Brant~C. Jones.
\newblock Results and conjectures on simultaneous core partitions.
\newblock {\em European J. Combin.}, 41:205--220, 2014.
\newblock \url{http://arxiv.org/abs/1308.0572}.

\bibitem{AKS}
David Aukerman, Ben Kane, and Lawrence Sze.
\newblock On simultaneous {$s$}-cores/{$t$}-cores.
\newblock {\em Discrete Math.}, 309(9):2712--2720, 2009.

\bibitem{BR}
Matthias Beck and Sinai Robins.
\newblock {\em Computing the continuous discretely}.
\newblock Undergraduate Texts in Mathematics. Springer, New York, 2007.
\newblock \url{http://math.sfsu.edu/beck/ccd.html}.

\bibitem{CJM}
Renzo Cavalieri, Paul Johnson, and Hannah Markwig.
\newblock Wall crossings for double {H}urwitz numbers.
\newblock {\em Adv. Math.}, 228(4):1894--1937, 2011.
\newblock \url{http://arxiv.org/abs/1003.1805}.

\bibitem{CHW}
William Y.~C. Chen, Harry H.~Y. Huang, and Larry X.~W. Wang.
\newblock Average size of a self-conjugate {$(s,t)$}-core partition.
\newblock {\em Proc. Amer. Math. Soc.}, 144(4):1391--1399, 2016.
\newblock \url{http://arxiv.org/abs/1405.2175}.

\bibitem{DS}
Robbert Dijkgraaf and Piotr Su{\l}kowski.
\newblock Instantons on {ALE} spaces and orbifold partitions.
\newblock {\em J. High Energy Phys.}, (3):013, 24, 2008.
\newblock \url{http://arxiv.org/abs/0712.1427}.

\bibitem{FMS}
Ben Ford, Ho{\`a}ng Mai, and Lawrence Sze.
\newblock Self-conjugate simultaneous {$p$}- and {$q$}-core partitions and
  blocks of {$A_n$}.
\newblock {\em J. Number Theory}, 129(4):858--865, 2009.

\bibitem{GKS}
Frank Garvan, Dongsu Kim, and Dennis Stanton.
\newblock Cranks and {$t$}-cores.
\newblock {\em Invent. Math.}, 101(1):1--17, 1990.

\bibitem{GG}
Iain~G. Gordon and Stephen Griffeth.
\newblock Catalan numbers for complex reflection groups.
\newblock {\em Amer. J. Math.}, 134(6):1491--1502, 2012.
\newblock \url{http://arxiv.org/abs/0912.1578}.

\bibitem{GMV}
Eugene Gorsky, Mikhail Mazin, and Monica Vazirani.
\newblock Affine permutations and rational slope parking functions.
\newblock {\em Trans. Amer. Math. Soc.}, 368(12):8403--8445, 2016.
\newblock \url{http://arxiv.org/abs/1403.0303}.

\bibitem{EM1}
Evgeny Gorsky and Mikhail Mazin.
\newblock Compactified {J}acobians and {$q,t$}-{C}atalan numbers, {I}.
\newblock {\em J. Combin. Theory Ser. A}, 120(1):49--63, 2013.
\newblock \url{http://arxiv.org/abs/1105.1151}.

\bibitem{EM2}
Evgeny Gorsky and Mikhail Mazin.
\newblock Compactified {J}acobians and {$q,t$}-{C}atalan numbers, {II}.
\newblock {\em J. Algebraic Combin.}, 39(1):153--186, 2014.
\newblock \url{http://arxiv.org/abs/1204.5448}.

\bibitem{Haase}
Christian Haase and Tyrrell~B. McAllister.
\newblock Quasi-period collapse and {${\rm GL}_n(\Bbb Z)$}-scissors congruence
  in rational polytopes.
\newblock In {\em Integer points in polyhedra---geometry, number theory,
  representation theory, algebra, optimization, statistics}, volume 452 of {\em
  Contemp. Math.}, pages 115--122. Amer. Math. Soc., Providence, RI, 2008.
\newblock \url{http://arxiv.org/abs/0709.4070}.

\bibitem{JK}
Gordon James and Adalbert Kerber.
\newblock {\em The representation theory of the symmetric group}, volume~16 of
  {\em Encyclopedia of Mathematics and its Applications}.
\newblock Addison-Wesley Publishing Co., Reading, Mass., 1981.
\newblock With a foreword by P. M. Cohn, With an introduction by Gilbert de B.
  Robinson.

\bibitem{LS}
G.~Lusztig and J.~M. Smelt.
\newblock Fixed point varieties on the space of lattices.
\newblock {\em Bull. London Math. Soc.}, 23(3):213--218, 1991.

\bibitem{SZ}
R.~Stanley and P.~Zanello.
\newblock The {C}atalan case of {A}rmstrong's conjecture on simultaneous core
  partitions.
\newblock \url{http://arxiv.org/abs/1312.4352}.

\end{thebibliography}


\end{document}

