An extension of Seller's theorem on partitions and its application

作者：Sun Yidong^*

来源：Ars Combinatoria, 2008, 86: 3-11.

Let K = (K-ij) be an infinite lower triangular matrix of nonnegative integers such that K-i0 = 1 and K-ii >= 1 for i >= 0. Define a sequence {V-m(K)}(m >= 0) by the recurrence Vm+1 (K) = Sigma(m)(j=0) KmjVj(K) with V-0(K) = 1. Let P(n; K) be the number of partitions of n of the form n = p(1) + p(2) + p(3) + p(4) + ... such that p(j) >= Sigma(i >= j) K(ij)p(i+1) for j >= 1 and let P(n;V(K)) denote the number of partitions of n into summands in the set V(K) = {V-1(K), V-2(K),...}. Based on the technique of MacMahon's partitions analysis, we prove that P(n; K) = P(n; V(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.

• 出版日期2008-1
• 单位大连理工大学