Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications
Abstract
The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In 2015 we gave two different -analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. In this paper we give an elementary proof of Naruse's formula based on the case of border strips. For special border strips, we obtain curious new formulas for the Euler and -Euler numbers in terms of certain Dyck path summations.
Cite
@article{arxiv.1610.04744,
title = {Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications},
author = {Alejandro H. Morales and Igor Pak and Greta Panova},
journal= {arXiv preprint arXiv:1610.04744},
year = {2020}
}
Comments
33 pages, 10 figures. This is the second paper of the series "Hook formulas for skew shapes". Most of Sections 8 and 9 in this paper used to be part of arxiv:1512.08348 (v1,v2); v2 fixed several typos; v3 made precision in definition of flagged tableaux in Section 3.2 and fixed typo in Example 3.3; v4 fixed small typos in proof of Corollary 7.6