Arithmetic properties of 1--shell totally symmetric plane partitions

Authors

  • M. D. Hirschhorn UNSW
  • J. A. Sellers Penn State University

Keywords:

partition, totally symmetric plane partition, TSPP, generating function, congruence

Abstract

Blecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’, Util. Math. \(88\) (2012), 223–235] defined the combinatorial objects known as 1--shell totally symmetric plane partitions of weight \(n.\) He also proved that the generating function for \(f(n),\) the number of 1--shell totally symmetric plane partitions of weight \(n,\) is given by $$ \sum_{n\geq 0} f(n)q^n = 1+\sum_{n\geq 1} q^{3n-2}\prod_{i=0}^{n-2} (1+q^{6i+3}). $$ In this brief note, we prove a number of arithmetic properties satisfied by \(f(n)\) using elementary generating function manipulations and well--known results of Ramanujan and Watson. DOI: 10.1017/S0004972713000865

Published

2014-03-25

Issue

Vol. 89 No. 3 (2014)

Section

Articles
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