Jena.GregoryAffiliation: University of Texas Rio Grande Valley Title Of Talk: Combinatorial statistics witnessing an infinite family of congruences for a sum of partition functions. Abstract: In 2007, Kronholm established infinite families of congruences in arithmetic progression, modulo any prime $\ell$, for $p(n,m)$, the function enumerating the partitions of $n$ into parts whose sizes come from the set $\{1,2,\dots ,m\}$. In 2022, Eichhorn, Kronholm, and Larsen proved there are combinatorial statistics described in terms of the multiplicities of the part sizes, called ``Multiplicity Based Statistics" (MB) that witness Kronholm's congruences. Here, ``witness" means given a congruence of the form $$p(n,m)\equiv 0 \pmod \ell,$$ a combinatorial statistic classifies the set of partitions of $n$ into $\ell$ equally sized subsets by directly inspecting the partitions themselves. In this talk, we prove the same MB statistics witnessing Kronholm's congruences witness another infinite family of congruences of the form $$p(a,m) \pm p(b,m)\equiv 0 \pmod{\ell},$$ where $b$ is determined by $a$ and $a$ and $b$ are not necessarily the same numbers in Kronholm's original result. We conclude by showing that we can gently modify these MB statistics to witness a third infinite family of congruences. This modification answers a question posed by Eichhorn, Kronholm, and Larsen.
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Last update made Tue Mar 10 21:28:04 CDT 2026.
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