On a restricted m-non-squashing partition function

For a fixed integer m ≥ 2, we say that a partition n = p₁ + p₂ + ⋯ + p_k of a natural number n is m-non-squashing if p₁ ≥ 1 and (m - 1)(p₁ + ⋯ + P_j-i) ≤ 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+1n) - c(m^rn) = 0 (mod m^r-1/d^r-2), where d = gcd(2, m). partitions, m-non-squashing partitions, m-ary partitions, stacking boxes, congruences.