  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.
