### Abstract

In recent work, M. Schneider and the first author studied a curious class of integer partitions called “sequentiallyc congruent” partitions: the mth part is congruent to the (m+ 1) th part modulo m, with the smallest part congruent to zero modulo the number of parts. Let p_{S}(n) be the number of sequentially congruent partitions of n, and let p_{□}(n) be the number of partitions of n wherein all parts are squares. In this note we prove bijectively, for all n≥ 1 , that p_{S}(n) = p_{□}(n). Our proof naturally extends to show other exotic classes of partitions of n are in bijection with certain partitions of n into kth powers.

Original language | English (US) |
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Journal | Ramanujan Journal |

DOIs | |

State | Accepted/In press - 2020 |

### Keywords

- Combinatorics
- Number theory
- Partitions
- Sums of squares

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## Cite this

Schneider, R., Sellers, J. A., & Wagner, I. (Accepted/In press). Sequentially congruent partitions and partitions into squares.

*Ramanujan Journal*. https://doi.org/10.1007/s11139-020-00294-7