A lecture hall theorem for $m$-falling partitions

TL;DR

This paper introduces a new lecture hall theorem for $m$-falling partitions, extending existing bijections and connecting to generalizations of Euler's theorem, with implications for partition theory and combinatorics.

Contribution

It extends a bijection to establish a lecture hall theorem for $m$-falling partitions, linking to generalized Euler theorems and refining recent results.

Findings

Established a lecture hall theorem for $m$-falling partitions.

Derived a finite version of Pak-Postnikov's $(m,c)$-generalization of Euler's theorem.

Connected the new results to recent extensions of Euler's theorem for all moduli.

Abstract

For an integer $m\ge 2$, a partition $\lambda=(\lambda_1,\lambda_2,\ldots)$ is called $m$-falling, a notion introduced by Keith, if the least nonnegative residues mod $m$ of $\lambda_i$'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.