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Lecture hall partition

In number theory and combinatorics, a lecture hall partition is a partition that satisfies additional constraints on its parts. Informally, a lecture hall partition is an arrangement of rows in a tiered lecture hall, with the condition that students on any row can see over the heads of the students in front of them. Bousquet-M\'elou and Eriksson introduced them in 1997.

Definitions
The lecture hall partitions $$\mathbf L_n$$ are defined by


 * $$\mathbf L_n = \left\{\lambda\in\mathbb Z^n : 0\le\frac{\lambda_1}{1}\le\frac{\lambda_2}{2}\le\cdots\le\frac{\lambda_n}{n}\right\},$$

where &lambda;i refers to the i-th component of &lambda;. A lecture hall partition of N is any $$\lambda\in\mathbf L_n$$ such that N = |&lambda;|, where $$|\lambda| = \lambda_1 + \cdots + \lambda_n.$$

The s-lecture hall partitions, denoted $$\mathbf L_n^{(s)},$$ are a generalization of $$\mathbf L_n.$$ Given a sequence s = (s1,..., sn), the s-lecture hall partitions are defined by


 * $$\mathbf L_n^{(s)} = \left\{\lambda\in\mathbb Z^n : 0\le\frac{\lambda_1}{s_1}\le\frac{\lambda_2}{s_2}\le\cdots\le\frac{\lambda_n}{s_n}\right\}.$$

In fact, $$\mathbf L_n = \mathbf L_n^{(1,2,\ldots,n)}.$$ The term s-lecture hall partition is a misnomer. A partition, strictly speaking, disregards the order of the parts &lambda;i. However, given an s-lecture hall partition &lambda; of N, there may be a permutation of &lambda; that is also an s-lecture hall partition of N; in this case, &lambda; is properly called a composition of N. If s is non-decreasing, then &lambda; is always a partition; for example, the lecture hall partitions $$\mathbf L_n$$ are named properly, because in this case s = (1, 2, ..., n).

The lecture hall theorem
The lecture hall theorem states that the number of lecture hall partitions of N in $$\mathbf L_n$$ is equal to the number of partitions of N into odd parts less than 2n. Euler's partition theorem, for comparison, equates the number of partitions with odd parts to the number of partitions with distinct parts. Therefore, in the limit $$n\to\infty$$, the number of lecture hall partitions of N in $$\mathbf L_n$$ equals the number of partitions of N with distinct parts.

The lecture hall theorem takes the form of a generating function as


 * $$\sum_{\lambda\in\mathbf L_n} q^{|\lambda|} = \prod_{i=1}^n \frac{1}{1-q^{2i-1}}.$$

Polynomic sequences
A sequence s is called polynomic if


 * $$\sum_{\lambda\in\mathbf L_n^{(s)}} q^{|\lambda|} = \prod_{i=1}^n\frac{1}{1-q^{d_i}},$$

where d1,..., dn are some positive integers. By the lecture hall theorem, s = (1,..., n) is a polynomic sequence with di = 2i-1.

(k, l) sequences
For positive integers k, l, define the (k, l) sequence a by


 * $$\begin{align}

a_{2i} &= la_{2i-1} - a_{2i-2} \\ a_{2i+1} &= ka_{2i} - a_{2i-1}, \end{align}$$

where a1 = 1 and a2 = l. Similarly, define the (l, k) sequence b by interchanging k and l:


 * $$\begin{align}

b_{2i} &= kb_{2i-1} - b_{2i-2} \\ b_{2i+1} &= lb_{2i} - b_{2i-1}, \end{align}$$

where b1 = 1 and b2 = k. Denote $$\mathbf G_n^{(k,l)} = \mathbf L_n^{(a)}$$ and define


 * $$G_n^{(k,l)}(x,y) = \sum_{\lambda\in\mathbf G_n^{(k,l)}} x^{|\lambda|_o}y^{|\lambda|_e},$$

where $$|\lambda|_o = \lambda_1 + \lambda_3 + \cdots$$ and $$|\lambda|_e = \lambda_2 + \lambda_4 + \cdots.$$