A Lecture Hall Theorem for m -Falling Partitions

For an integer m≥ 2 , a partition λ= (λ₁, λ₂, …) is called m-falling, a notion introduced by Keith, if the least non-negative 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 Xiong and Keith. We note that their result actually can be refined with one more parameter.