On simultaneous $(s, s+t, s+2t, \dots)$-core partitions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917150002774016 |
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| author | Keith, William Nath, Rishi Sellers, James |
| author_facet | Keith, William Nath, Rishi Sellers, James |
| contents | We consider simultaneous $(s,s+t,s+2t,\dots,s+pt)$-core partitions in the large-$p$ limit, or (when $s<t$), partitions in which no hook may be of length $s \pmod{t}$. We study generating functions, containment properties, and congruences when $s$ is not coprime to $t$. As a boundary case of the general study made by Cho, Huh and Sohn, we provide enumerations when $s$ is coprime to $t$, and answer positively a conjecture of Fayers on the polynomial behavior of the size of the set of simultaneous $(s,s+t,s+2t,\dots,s+pt)$-core partitions when $p$ grows arbitrarily large. Of particular interest throughout is the comparison to the behavior of simultaneous $(s,t)$-cores. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00074 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On simultaneous $(s, s+t, s+2t, \dots)$-core partitions Keith, William Nath, Rishi Sellers, James Combinatorics 05A17, 11P81, 11P83 We consider simultaneous $(s,s+t,s+2t,\dots,s+pt)$-core partitions in the large-$p$ limit, or (when $s<t$), partitions in which no hook may be of length $s \pmod{t}$. We study generating functions, containment properties, and congruences when $s$ is not coprime to $t$. As a boundary case of the general study made by Cho, Huh and Sohn, we provide enumerations when $s$ is coprime to $t$, and answer positively a conjecture of Fayers on the polynomial behavior of the size of the set of simultaneous $(s,s+t,s+2t,\dots,s+pt)$-core partitions when $p$ grows arbitrarily large. Of particular interest throughout is the comparison to the behavior of simultaneous $(s,t)$-cores. |
| title | On simultaneous $(s, s+t, s+2t, \dots)$-core partitions |
| topic | Combinatorics 05A17, 11P81, 11P83 |
| url | https://arxiv.org/abs/2508.00074 |