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.
Bibliographical noteFunding 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.
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- Number theory
- Sums of squares