Hook length biases and general linear partition inequalities
Hook length biases and general linear partition inequalities
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DOI：
10.1007/s40687-023-00402-1
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发表时间：
2023-03
影响因子：
1.2
通讯作者：
C. Ballantine;Hannah E. Burson;William Craig;A. Folsom;Boya Wen
中科院分区：
文献类型：
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作者：
C. Ballantine;Hannah E. Burson;William Craig;A. Folsom;Boya Wen
部分出于钩内容公式的某些限制性分区表示论，我们认为在奇数与不同的分区固定长度的钩子的总数。我们证明了在n的所有奇数分区中，长度分别为2和3的钩子比在n的所有不同分区中更多，并对任意钩子长度作出类似的猜想。我们还建立了额外的偏见的结果上的差距的大小为1，分别为2，在所有奇数与不同的分区n。我们猜想类似的偏见和渐近性，以及在奇数不同的分区与自共轭分区的固定长度的钩子的数量的同余。证明我们的偏差结果长度为3的钩子的一个不可分割的组成部分是一个线性不等式，涉及q（n），n的不同分区的数量。本文还建立了q（n）的有效线性不等式，这是一个独立的结果。我们的方法是分析和组合，我们的结果和示意图相交的领域表示论，解析数论，分割理论，和q-系列。特别是，我们使用Rademacher型精确公式forq（n），赖特的圆方法，模块化，q系列变换，渐近方法，和组合参数。
Motivated in part by hook-content formulas for certain restricted partitions in representation theory, we consider the total number of hooks of fixed length in odd versus distinct partitions. We show that there are more hooks of length 2, respectively 3, in all odd partitions ofnthan in all distinct partitions ofn, and make the analogous conjecture for arbitrary hook length. We also establish additional bias results on the number of gaps of size 1, respectively 2, in all odd versus distinct partitions ofn. We conjecture similar biases and asymptotics, as well as congruences for the number of hooks of fixed length in odd distinct partitions versus self-conjugate partitions. An integral component of the proof of our bias result for hooks of length 3 is a linear inequality involvingq(n), the number of distinct partitions ofn. In this article we also establish effective linear inequalities forq(n) in great generality, a result which is of independent interest. Our methods are both analytic and combinatorial, and our results and conjectures intersect the areas of representation theory, analytic number theory, partition theory, andq-series. In particular, we use a Rademacher-type exact formula forq(n), Wright’s circle method, modularity,q-series transformations, asymptotic methods, and combinatorial arguments.