A lecture hall theorem for $m$-falling partitions

For an integer $mge 2$, 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-Postnikovs $(m,c)$-generalization of Eulers theorem. Our work is partially motivated by a recent extension of Eulers theorem for all moduli, due to Keith and Xiong. We note that their result actually can be refined with one more parameter.