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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10.1007/s11139-020-00294-7

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title = "Sequentially congruent partitions and partitions into squares",

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 pS(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 pS(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.",

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author = "Robert Schneider and Sellers, \{James A.\} and Ian Wagner",

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: {\textcopyright} 2020, Springer Science+Business Media, LLC, part of Springer Nature.",

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AU - Schneider, Robert

AU - Sellers, James A.

AU - Wagner, Ian

N1 - 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.

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N2 - 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 pS(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 pS(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.

AB - 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 pS(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 pS(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.

KW - Combinatorics

KW - Number theory

KW - Partitions

KW - Sums of squares

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DO - 10.1007/s11139-020-00294-7

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JF - Ramanujan Journal

IS - 2

ER -

Sequentially congruent partitions and partitions into squares

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