SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2388-2410, January 2020.
We consider the number of partitions of $n$ whose Young diagrams fit inside an $m \times \ell$ rectangle; equivalently, we study the coefficients of the $q$-binomial coefficient $\binom{m+\ell}{m}_q$. We obtain sharp asymptotics throughout the regime $\ell = \Theta (m)$ and $n = \Theta (m^2)$, while previously sharp asymptotics were derived by Takács [J. Statist. Plann. Inference, 14 (1986), pp. 123--142] only in the regime where $|n - \ell m /2| = O(\sqrt{\ell m (\ell + m)})$ 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 $m, \ell$ remain the same, hence significantly refining Sylvester's unimodality theorem and giving effective asymptotic estimates for related Kronecker and plethysm coefficients from representation theory.

中文翻译：

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计算矩形内的分区

SIAM离散数学杂志，第34卷，第4期，第2388-2410页，2020年1月。
我们考虑$ n $的杨氏图适合于$ m \ times \ ell $矩形内的分区的数量；等效地，我们研究$ q $-二项式系数$ \ binom {m + \ ell} {m} _q $的系数。在整个过程中，我们获得了尖锐的渐近性，而以前尖锐的渐近性是由塔卡奇派生的[J. 统计员。计划 Inference，14（1986），pp。123--142]仅在$ | n-\ ell m / 2 |的情况下。= O（\ sqrt {\ ell m（\ ell + m）}）$，使用局部中心极限定理。我们的方法是解决一个相关的大偏差问题：我们描述一种倾斜度量，该度量产生的配置的边界矩形具有给定的宽高比并填充到给定的比例。