Counting pattern-avoiding integer partitions

Abstract

A partition $$\alpha $$ is said to contain another partition (or pattern) $$\mu $$ if the Ferrers board for $$\mu $$ is attainable from $$\alpha $$ under removal of rows and columns. We say $$\alpha $$ avoids $$\mu $$ if it does not contain $$\mu $$ . In this paper we count the number of partitions of n avoiding a fixed pattern $$\mu $$ , in terms of generating functions and their asymptotic growth rates. We find that the generating function for this count is rational whenever $$\mu $$ is (rook equivalent to) a partition in which any two part sizes differ by at least two. In doing so, we find a surprising connection to metacyclic p-groups. We further obtain asymptotics for the number of partitions of n avoiding a pattern $$\mu $$ . Using these asymptotics we conclude that the generating function for $$\mu $$ is not algebraic whenever $$\mu $$ is rook equivalent to a partition with distinct parts whose first two parts are positive and differ by 1.