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

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 $n\ge 1$ can occur in colors $1,2,\dots ,n$. 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.