Counting Partitions Inside A Rectangle
SIAM JOURNAL ON DISCRETE MATHEMATICS(2020)
摘要
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 ((m+l)(m))(q). We obtain sharp asymptotics throughout the regime l = Theta(m) and n = Theta(m(2)), while previously sharp asymptotics were derived by Takacs [J. Statist. Plann. Inference, 14 (1986), pp. 123-142] only in the regime where vertical bar n - lm/2 vertical bar= O(root lm(l + m)) 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 significantly refining Sylvester's unimodality theorem and giving effective asymptotic estimates for related Kronecker and plethysm coefficients from representation theory.
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关键词
partitions, q-binomial coefficients, Kronecker coefficients, central limit theorem
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