Partitions Excluding Specific Polygonal Numbers As Parts

A partition λ of the nonnegative integer n is a sequence of nonnegative integers λ1 ≥ λ2 ≥ · · · ≥ λr with λ1 + λ2 + · · · + λr = n. Each value λi, 1 ≤ i ≤ r, is called a part of the partition. In this note, we consider partitions of n with parts related to k-gonal numbers for some fixed integer k ≥ 3. Various works have appeared involving partitions into polygonal parts. For example, M. D. Hirschhorn and the author have written a number of papers on partitions into a specified number of triangular numbers or squares [3]–[5]. Also, Andrews [2, Theorem 4.1] considers partitions into an unlimited number of triangular numbers. (Several sequences related to such partitions appear in Sloane’s Online Encyclopedia of Integer Sequences [6], including A001156, A002635, A002636, and A007294.) In contrast, few results appear in the literature in which polygonal numbers are excluded as parts. Andrews [1, Corollary 8.5] highlights one such instance when he considers the number of partitions of n with no square parts. Interestingly enough, Sloane [6] recently published this sequence of values (A087153) in August 2003. Our goal here is two-fold. First, we consider Andrews’ result above and extend it to all 2k-gons for k ≥ 2. Afterwards, we prove a similar result for 2k + 1-gons.