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arXiv · 2607.27753

Franklin's identity for $n$-color partitions and companion Beck-type identities

Cristina Ballantine·Roberto Tauraso
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Abstract

We show that some classical identities valid for ordinary partitions have precise analogues for $n$-color partitions, that is partitions in which a part of size $n\geq 1$ can occur in colors $1, 2, \ldots, n$. For $r \ge 2$ and $j \ge 0$, we write $\mathcal{O}_{j,r}(m)$ and $\mathcal{D}_{j,r}(m)$ for the sets of $n$-color partitions of $m$ with, respectively, exactly $j$ different parts whose size and color are divisible by $r$, and exactly $j$ different parts occurring at least $r$ times. We prove an $n$-color version of Franklin's theorem, $|\mathcal{O}_{j,r}(m)| = |\mathcal{D}_{j,r}(m)|$, along with two Beck-type identities. We give both analytic and combinatorial proofs for all theorems.

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Cristina Ballantine, Roberto Tauraso. 2026-07-30. Franklin's identity for $n$-color partitions and companion Beck-type identities. https://arxiv.org/abs/2607.27753

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