A Refinement of the Lecture Hall Theorem

Forn⩾1, let Lnbe the set of lecture hall partitions of lengthn, that is, the set ofn-tuples of integersλ=(λ1, …, λn) satisfying 0⩽λ1/1⩽λ2/2⩽…⩽λn/n. Let ⌈λ⌉ be the partition (⌈λ1/1⌉, …, ⌈λn/n⌉), and leto(⌈λ⌉) denote the number of its odd parts. We show that the identity∑λ∈Ln q|λ|u|⌈λ⌉|vo(⌈λ⌉)=(1+uvq)(1+uvq2)…(1+uvqn)(1−u2qn+1)(1−u2qn+2)…(1−u2q2n)is equivalent to a refinement of Bottʹs formula for the affine Coxeter groupCn, obtained by I. G. Macdonald (Math. Ann.199(1972), 161–174) and V. Reiner (Electron. J. Combin.2(1995), R25). The caseu=v=1 of the above identity, called the lecture hall theorem, was proved by us in (Ramanujan J.1(1997), 101–111), and then by Andrews (“Mathematical Essays in Honor of G.-C. Rota,” pp. 1–22, Birkhäuser, Cambridge, MA, 1998). In the present paper, we give two direct proofs of the above identity. The first one is rather short, but requires a bit ofq-calculus; the second one is the first truly bijective proof ever found in the domain of lecture hall partitions. Although we describe our bijection in completely combinatorial terms, it finds its origin in the algebraic context of Coxeter groups. Both proofs are completely independent of all earlier proofs of the Lecture Hall Theorem.