Hook formulas for skew shapes I. q-analogues and bijections

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:154

Article

Morales, Alejandro H.¹  Pak, Igor¹  Panova, Greta² 

[1] UCLA, Dept Math, Los Angeles, CA 90095 USA

[2] UPenn, Dept Math, Philadelphia, PA 19104 USA

关键词: Hook-length formula;    Excited tableau;    Standard Young tableau;    Flagged tableau;    Reverse plane partition;    Hillman-Grassl correspondence;    Robinson-Schensted-Knuth correspondence;    Greene's theorem;    Grassmannian permutation;    Factorial Schur function;   

DOI  :  10.1016/j.jcta.2017.09.002

来源: Elsevier

【 摘 要 】

The celebrated hook-length formula gives 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 over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse's formula, by using factorial Schur functions and a generalization of the Hillman-Grassl correspondence, respectively. The main new results are two different q-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. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free