Integrals, partitions and MacMahon's Theorem

In two previous papers, the study of partitions with short sequences has been developed both for its intrinsic interest and for a variety of applications. The object of this paper is to extend that study in various ways. First, the relationship of partitions with no consecutive integers to a theorem of MacMahon and mock theta functions is explored independently. Secondly, we derive in a succinct manner a relevant definite integral related to the asymptotic enumeration of partitions with short sequences. Finally, we provide the generating function for partitions with no sequences of length K and part exceeding N.

• Type: Journal
• Year: 2007
• Where: JCT
• Authors: George Andrews, Henrik Eriksson, Fedor Petrov, Dan Romik
• Keywords: Consecutive Integers, JCT 2007, Mock Theta Functions, Short Sequences