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 \(K_{ii} \geq 1\) for \(i \geq 0\). Define a sequence \(\{V_i(\mathcal{K})\}_{m\geq0}\) 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 \(p_j \geq \sum_{i\geq j} K_{ij}p_{i+1}\) for \(j \geq 1\) and let \(P(n;V(\mathcal{K}))\) denote the number of partitions of \(n\) into summands in the set \(V(\mathcal{K}) = \{V_1(\mathcal{K}), V_2(\mathcal{K}), \ldots\}\). 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.