An Extension of Seller’s Theorem on Partitions and its Application

Abstract

Let $\mathcal{K}=(K_{ij})$ be an infinite lower triangular matrix of non-negative integers such that $K_{i0}=1$ and $K_{ii}\ge 1$ for $i\ge 0$. Define a sequence $\{V_i(\mathcal{K}){\}}_{m\ge 0}$ by the recurrence $V_{i+1}(\mathcal{K})=\sum _{j=0}^m{K}_{mj}{V}_j(\mathcal{K})$ with $V_0(\mathcal{K})=1$. Let $P(n;\mathcal{K})$ be the number of partitions of $n$ of the form $n=p_1+p_2+p_3+p_4+\cdots$ such that $p_j\ge \sum _{i\ge j}{K}_{ij}{p}_{i+1}$ for $j\ge 1$ and let $P(n;V(\mathcal{K}))$ denote the number of partitions of $n$ into summands in the set $V(\mathcal{K})=\{V_1(\mathcal{K}),V_2(\mathcal{K}),\dots \}$. Based on the technique of MacMahon’s partitions analysis, we prove that $P(n;\mathcal{K})=P(n;V(\mathcal{K}))$ which generalizes a recent result of Sellers’. We also give several applications of this result to many classical sequences such as Bell numbers, Fibonacci numbers, Lucas numbers, and Pell numbers.