Sequentially congruent partitions and partitions into squares

Robert Schneider; James A. Sellers; Ian Wagner

doi:10.1007/s11139-020-00294-7

Sequentially congruent partitions and partitions into squares

Robert Schneider, James A. Sellers, Ian Wagner

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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)
Pages (from-to)	645-650
Number of pages	6
Journal	Ramanujan Journal
Volume	56
Issue number	2
DOIs	https://doi.org/10.1007/s11139-020-00294-7
State	Published - Nov 2021

Bibliographical note

Funding Information:
The authors are grateful to Maxwell Schneider for conversations that improved our work, and to the anonymous referee for suggestions that strengthened this paper.

Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

Combinatorics
Number theory
Partitions
Sums of squares

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Sequentially congruent partitions and partitions into squares

Abstract

Bibliographical note

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