A PROOF OF ANDREWS’ CONJECTURE ON PARTITIONS WITH NO SHORT SEQUENCES

Abstract

Our main result establishes Andrews’ conjecture for the asymptotic of the generating function for the number of integer partitions of$n$without$k$consecutive parts. The methods we develop are applicable in obtaining asymptotics for stochastic processes that avoid patterns; as a result they yield asymptotics for the number of partitions that avoid patterns.Holroyd, Liggett, and Romik, in connection with certain bootstrap percolation models, introduced the study of partitions without$k$consecutive parts. Andrews showed that when$k=2$, the generating function for these partitions is a mixed-mock modular form and, thus, has modularity properties which can be utilized in the study of this generating function. For$k>2$, the asymptotic properties of the generating functions have proved more difficult to obtain. Using$q$-series identities and the$k=2$case as evidence, Andrews stated a conjecture for the asymptotic behavior. Extensive computational evidence for the conjecture in the case$k=3$was given by Zagier.This paper improved upon early approaches to this problem by identifying and overcoming two sources of error. Since the writing of this paper, a more precise asymptotic result was established by Bringmann, Kane, Parry, and Rhoades. That approach uses very different methods.