On the combinatorics of lecture hall partitions

A lecture hall partition of length n is an integer sequence λ = (λ₁,..., λ_n) satisfying 0 ≤ λ₁/1 ≤ λ₂/2 ≤ ... ≤ λ_n/n. (1) Bousquet-Mélou 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-Mélou and Eriksson.