For n >= 1, let #psi#_n be the set of lecture hall partitions of length n, that is, the set of n-tuples of integers #lambda# = (#lambda#_1, ..., #lambda#_n) satisfying 0<= #lambda#_1/1 <= #lambda#_2/2 <= ... <= #lambda#_n. Let [#lambda#] be the partition ([#lambda#_1/1], ..., [#lambda#_n]), and let o([#lambda#]) denote the number of its odd parts. We show that the identity sum from #lambda# implied by #psi#_n of q~([#lambda#])u~(|[#lambda#]|)v~o(([#lambda#])) = (1 + uvq)(1 + uvq~2) ... (1 + uvq~n)/(1 - u~2q~(n + 1))(1 - u~2q~(n + 2)) ... (1 - u~2q~2n) is equivalent to a refinement of Bott's formula for the affine Coxeter group C_n, obtained by I. G. Macdonald (Math. Ann. 199 (1972), 161 174) and V. Reiner (Electron. J. Combin. 2 (1995), R25). The case u = 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, Birkhauser, 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 of q-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.
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