Sequentially congruent partitions and partitions into squares
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866914815884132352 |
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| author | Schneider, Robert Sellers, James A. Wagner, Ian |
| author_facet | Schneider, Robert Sellers, James A. Wagner, Ian |
| contents | In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the number of parts. Let $p_{\mathcal S}(n)$ be the number of sequentially congruent partitions of $n,$ and let $p_{\square}(n)$ be the number of partitions of $n$ wherein all parts are squares. In this note we prove bijectively, for all $n\geq 1,$ that $p_{\mathcal S}(n) = p_{\square}(n).$ Our proof naturally extends to show other exotic classes of partitions of $n$ are in bijection with certain partitions of $n$ into $k$th powers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1911_10236 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Sequentially congruent partitions and partitions into squares Schneider, Robert Sellers, James A. Wagner, Ian Number Theory Combinatorics In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the number of parts. Let $p_{\mathcal S}(n)$ be the number of sequentially congruent partitions of $n,$ and let $p_{\square}(n)$ be the number of partitions of $n$ wherein all parts are squares. In this note we prove bijectively, for all $n\geq 1,$ that $p_{\mathcal S}(n) = p_{\square}(n).$ Our proof naturally extends to show other exotic classes of partitions of $n$ are in bijection with certain partitions of $n$ into $k$th powers. |
| title | Sequentially congruent partitions and partitions into squares |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/1911.10236 |