Séminaire Lotharingien de Combinatoire, 82B.64 (2019), 12 pp.

Stephen Melczer, Greta Panova, and Robin Pemantle

Counting partitions inside a rectangle

Abstract. We consider the number of partitions of n whose Young diagrams fit inside an m x l rectangle; equivalently, we study the coefficients of the q-binomial coefficient \binom{m+l}{m}_q. We obtain sharp asymptotics throughout the regime l = Θ(m) and n = Θ(m2) where a limit shape exists. Previously, sharp asymptotics were derived by Takács only in the regime where |n-lm/2| = O((l m (l + m)1/2) using a local central limit theorem. Our approach is to solve a related large deviation problem: we describe the tilted measure that produces configurations whose bounding rectangle has the given aspect ratio and is filled to the given proportion. Our results are sufficiently sharp to yield the first asymptotic estimates on the consecutive differences of these numbers when n is increased by one and m, l remain the same, hence quantifying and significantly refining Sylvester's unimodality theorem.


Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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