Hook formulas for skew shapes I. q-analogues and bijections
Hook formulas for skew shapes I. q-analogues and bijections
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DOI：
10.1016/j.jcta.2017.09.002
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发表时间：
2015-12
期刊：
影响因子：
--
通讯作者：
A. Morales;I. Pak;G. Panova
中科院分区：
文献类型：
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作者：
A. Morales;I. Pak;G. Panova
著名的钩长公式给出了一个直线形标准杨氏图数的乘积公式。2014年，成濑宣布了一个更通用的公式，用于计算斜形状的标准杨图的数量，作为钩长乘积的正和。本文分别用阶乘Schur函数和Hillman-Grassl对应的推广给出了Naruse公式的代数和组合证明，主要结果是Naruse公式的两个不同的q-类似：斜Schur函数的q-类似和斜形的反平面分拆的q-类似。我们建立明确的双射这些对象和家庭的整数数组与某些非零项目，这也证明了第二个公式。
The celebratedhook-length formulagives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum overexcited diagramsof products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse's formula, by usingfactorial Schur functionsand a generalization of theHillman–Grassl correspondence, respectively.The main new results are two differentq-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. We establish explicit bijections between these objects and families of integer arrays with certain nonzero entries, which also proves the second formula.