Lecture Hall Partitions

We prove a finite version of the well-known theorem that says that the number of partitions of an integer N into distinct parts is equal to the number of partitions of N into odd parts. Our version says that the number of “lecture hall partitions of length n ” of N equals the number of partitions of N into small odd parts: 1,3,5, ldots, 2n-1 . We give two proofs: one via Bott's formula for the Poincaré series of the affine Coxeter group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\tilde C_n $$ \end{document}, and one direct proof.