MacMahon’s Partition Analysis: I. The Lecture Hall Partition Theorem

Abstract

In this paper, we analyze the beautiful theorem of M. Bousquet-Mélon and K. Eriksson, the Lecture Hall Partition Theorem, via MacMahon’s Partition Analysis. Their theorem asserts that the number of partitions of n of the form b _j + b _j−1 +…+ b ₁ wherein

$$\frac{{b_j }} {j} \geqq \frac{{b_j - 1}} {{j - 1}} \geqq \cdots \geqq \frac{{b_\text{1} }} {\text{1}} \geqq 0$$

equals the number of partitions of n into odd parts each ≦ 2j − 1. As they have noted the theorem reduces to Euler’s classical partition theorem when j → ∞.

The central object of the paper is to demonstrate the power of the method of Partition Analysis, developed by P. A. Maahon over 80 years ago. In the early 1970’s, Richard Stanley successfully utilized Partition Analysis in his monumental treatment of magic labelings of graphs. Apart from this one shining moment, Partition Analysis has lain dormant. In this paper we hope to point to its further utility by proving the deep theorem of Bousquet-Mélon and Eriksson.

Partially supported by National Science Foundation Grant DMS-8702695-04 and by the Australian Research Council. Concerning the latter, the gracious support and interest of Omar Foda made possible my visit to Melbourne University and my introduction to Mireille Bousquet-Mélou and Lecture Hall Partitions.