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arXiv · 2508.00074

On simultaneous $(s, s+t, s+2t, \dots)$-core partitions

William Keith·Rishi Nath·James Sellers
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Abstract

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.

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William Keith, Rishi Nath, James Sellers. 2025-07-31. On simultaneous $(s, s+t, s+2t, \dots)$-core partitions. https://arxiv.org/abs/2508.00074

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