On the combinatorics of lecture hall partitions

A lecture hall partition of length n is an integer sequence lambda=(lambda (1),...,lambda (n)) satisfying 0 less than or equal to lambda (1)/1 less than or equal to lambda (2)/2 less than or equal to...less than or equal to lambda (n)/n. (1) 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.