Recall that if $\lambda = (\lambda_1,\lambda_2,\dots \lambda_N)$ is a partition of the positive integer $n$, i.e. $$\begin{equation}\label{parteq1} n=\lambda_1+\lambda_2+\dots +\lambda_N, \end{equation}$$ then the convention is to arrange the parts so that $\lambda_1\geq\lambda_2\geq\dots \geq\lambda_N\geq 1$.

For a given positive integer $N$ (the number of "rows" in the "lecture hall"), a lecture-hall partition of the integer $n$ is a collection of non-negative integers $\lambda =(\lambda_1,\lambda_2,\dots \lambda_N)$ such that $\eqref{parteq1}$ holds, and $$\begin{equation}\label{lecparteq1} \frac{\lambda_1}{N}\geq\frac{\lambda_2}{N-1}\geq\dots \geq\frac{\lambda_N}{1}\geq 0. \end{equation}$$ Note in particular that for lecture hall partitions, some of the $\lambda_i$ may be 0.

Let $\mathcal{L}_N$ denote the set of all such lecture hall partitions, i.e., $$\begin{equation}\label{Lneq1} \mathcal{L}_N:=\left \{(\lambda_1,\lambda_2,\dots \lambda_N)\bigg | \frac{\lambda_1}{N}\geq\frac{\lambda_2}{N-1}\geq\dots \geq\frac{\lambda_N}{1}\geq 0 \right \}, \end{equation}$$ and for any positive integer, let $l_N(n)$ denote the number of lecture hall partitions of $n$ in $\mathcal{L}_N$: $$\begin{equation}\label{Lneq2} l_N(n):=\left |\left \{(\lambda_1,\lambda_2,\dots \lambda_N)\in \mathcal{L}_N\bigg | \lambda_1+\lambda_2+\dots +\lambda_N=n \right \}\right|. \end{equation}$$

Lecture hall partitions were introduced by Bousquet-Mélou and Eriksson in [1], where they showed that the generating function for the sequence $\{l_N(n)\}_{n=0}^{\infty}$ (by convention, $l_N(0)=1$) had closed form given by $$\begin{equation}\label{Lneq3} \sum_{n=0}^{\infty} l_N(n)q^n=\prod_{j=1}^{N}\frac{1}{1-q^{2j-1}}. \end{equation}$$

A survey of extensions and developments in the theory may be found in the paper [2] by Carla Savage.

[1] Mireille Bousquet-Mélou and Kimmo Eriksson. Lecture hall partitions. Ramanujan J., 1(1), 101–111, 1997.

[2] Carla D. Savage, The mathematics of lecture hall partitions. J. Combin. Theory Ser. A 144 (2016), 443–475.

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