Counting Partitions inside a Rectangle
Counting Partitions inside a Rectangle
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DOI：
10.1137/20m1315828
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发表时间：
2018-05
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影响因子：
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通讯作者：
S. Melczer;G. Panova;Robin Pemantle
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作者：
S. Melczer;G. Panova;Robin Pemantle
我们考虑在$m \times \ell$矩形内适合Young图的$n$分区的数目；同样地，我们研究了$q$ -二项式系数$\binom{m+\ell}{m}_q$的系数。我们在整个体系$\ell = \Theta (m)$和$n = \Theta (m^2)$中得到了明显的渐近性。以前，只有在$|n - \ell m /2| = O(\sqrt{\ell m (\ell + m)})$使用局部中心极限定理的情况下，才能由Takács导出尖锐渐近性。我们的方法是解决一个相关的大偏差问题：我们描述了产生其边界矩形具有给定长宽比并填充到给定比例的配置的倾斜度量。我们的结果足够清晰，当$n$增加1且$m, \ell$保持不变时，我们给出了这些数的连续差的第一个渐近估计，从而显著改进了Sylvester的单模定理。
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 $n = \Theta (m^2)$. Previously, sharp asymptotics were derived by Tak\'acs 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.