On the combinatorics of lecture hall partitions

Abstract

A lecture hall partition of length n is an integer sequence \(\lambda = (\lambda _1 , \ldots ,\lambda _n )\) satisfying\(0 \leqslant \frac{{\lambda _1 }}{1} \leqslant \frac{{\lambda _2 }}{2} \leqslant \cdots \leqslant \frac{{\lambda _n }}{n}\)Bousquet-Melou and Eriksson showed that the number of lecture hall partitions of length n of a positive integer N whose alternating sum is k equals the number of partitions of N into k odd parts less than 2n. We prove the fact by a natural combinatorial bijection. This bijection, though defined differently, is essentially the same as one of the bijections found by Bousquet-Melou and Eriksson.