Sequentially congruent partitions and partitions into squares

Robert Schneider, James A. Sellers, Ian Wagner

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)645-650
Number of pages6
JournalRamanujan Journal
Issue number2
StatePublished - 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.


  • Combinatorics
  • Number theory
  • Partitions
  • Sums of squares


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