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.
Accepted: The Ramanujan Journal (2020)