Anti-lecture hall compositions (Q1869230)

For a sequence \(\lambda= (\lambda_1, \lambda_2,\dots, \lambda_k)\) of integers, define the weight \(|\lambda|\) of \(\lambda\) by \(|\lambda|= \lambda_1+\cdots+ \lambda_k\). In this paper the authors introduce the set \(A_k\) of the anti-lecture hall compositions whose parts satisfy the inequalities \(\lambda_1/1\geq \lambda_2/2\geq\cdots\geq \lambda_k/k\geq 0\). They present bijective proofs of two identities which may be considered as analogs of the classical lecture hall theorems. It is shown, for instance, that \[ \sum_{\lambda\in A_k} q^{|\lambda|}= \prod^k_{i=1} {1+ q^i\over 1- q^{i+1}}. \] In the paper this identity is called anti-lecture hall theorem.