Anti-Lecture Hall Compositions

We study the set Ak of integer sequences λ=(λ1,…,λk), defined byλ11⩾λ22⩾⋯λkk⩾0and show that the generating function is∑λ∈Ak q|λ| = ∏i=1k 1+qi1−qi+1,where |λ| = λ1+⋯+λk. We establish this by giving a bijective proof of the following refinement:∑λ∈Ak q|λ|u|⌊λ⌋|vo(⌊λ⌋) = ∏i=1k 1+uvqi1−u2qi+1.To our knowledge this is a new result that complements the family of the Lecture Hall Theorems (Ramanujan J. 1(1,2) (1997) 101, 165; J. Combin. Theory Ser. A 86(1) (1999) 63).