For a fixed integer m ≥ 2, we say that a partition n = p1 + p2 + ⋯ + pk of a natural number n is m-non-squashing if p1 ≥ 1 and (m - 1)(p1 + ⋯ + Pj-i) ≤ pj 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(mr+1n) - c(mrn) = 0 (mod mr-1/dr-2), where d = gcd(2, m). partitions, m-non-squashing partitions, m-ary partitions, stacking boxes, congruences.
|Original language||English (US)|
|Journal||Journal of Integer Sequences|
|State||Published - Oct 24 2005|
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics