Anti-lecture hall compositions

We study the set A(k) of integer sequences lambda = (lambda(1),...,lambda(k)), defined by (1)/(lambda1) greater than or equal to (2)/(lambda2) greater than or equal to ... (k)/(lambdak) greater than or equal to 0 and show that the generating function is Sigma(lambdais an element ofAk)q\lambda\ = Pi(i=1)(k) (1-qi+1)/(1+qi), where \lambda\ = lambda(1) + ... +lambda(k). We establish this by giving a bijective proof of the following refinement: Sigma(lambdais an element ofAk)q(\lambda\)u\([lambda]\)v(o([lambda])) = Pi(i=1)(k) (1-u2qi+1)/(1+uvqi). To our knowledge this is a new result that complements the family of the Lecture Hall Theorems (C) 2002 Elsevier Science B.V. All rights reserved.