A lecture hall theorem for $m$-falling partitions

Abstract

For an integer $m\ge 2$, a partition $\hat{I}»=(\hat{I}{»}_1,\hat{I}{»}_2,\dots )$ is called $m$-falling, a notion introduced by Keith, if the least nonnegative residues mod $m$ of $\hat{I}{»}_i$'s form a nonincreasing sequence. We extend a bijection originally due to the third author to deduce a lecture hall theorem for such $m$-falling partitions. A special case of this result gives rise to a finite version of Pak-Postnikov's $(m,c)$-generalization of Euler's theorem. Our work is partially motivated by a recent extension of Euler's theorem for all moduli, due to Keith and Xiong. We note that their result actually can be refined with one more parameter.

14 pages, 3 figures, 1 tables