Abstract

For a fixed integer m ≥ 2, we say that a partition n = p_{1}+p_{2}+⋯+ p_{k} of a natural number n is m-non-squashing if p_{1} ≥ 1 and (m−1)(p_{1}+ ⋯ + p_{j−1}) ≤ p_{j} for 2 ≤ j ≤ k. In this paper we give a new bijective proof that the number of m-non-squashing partitions of n is equal to the number of m-ary partitions of n. Moreover, we prove a similar result for a certain restricted m-non-squashing partition function c(n) which is a natural generalization of the function which enumerates non-squashing partitions into distinct parts (originally introduced by Sloane and the second author). Finally, we prove that for each integer r ≥ 2,

c(m^{r+1}n) − c(m^{r}n) ≡ 0 (mod m^{r−1}/d^{r−2}),

where d = gcd(2, m).