### 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-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+1}n) - c(m^{r}n) = 0 (mod m^{r-1}/d^{r-2}), where d = gcd(2, m). partitions, m-non-squashing partitions, m-ary partitions, stacking boxes, congruences.

Original language | English (US) |
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Journal | Journal of Integer Sequences |

Volume | 8 |

Issue number | 5 |

State | Published - Oct 24 2005 |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

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## Cite this

*Journal of Integer Sequences*,

*8*(5).