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arXiv · 1902.00228

A lecture hall theorem for $m$-falling partitions

Shishuo Fu·Dazhao Tang·Ae Ja Yee
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Abstract

For an integer $m\ge 2$, a partition $λ=(λ_1,λ_2,\ldots)$ is called $m$-falling, a notion introduced by Keith, if the least nonnegative residues mod $m$ of $λ_i$'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.

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Shishuo Fu, Dazhao Tang, Ae Ja Yee. 2019-02-01. A lecture hall theorem for $m$-falling partitions. https://arxiv.org/abs/1902.00228

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