1. Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications
- Author
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Greta Panova, Alejandro H. Morales, and Igor Pak
- Subjects
Discrete mathematics ,Mathematics::Combinatorics ,General Mathematics ,010102 general mathematics ,Skew ,Combinatorial proof ,Hook length formula ,05A05, 05A15, 05E05, 05A19 ,0102 computer and information sciences ,01 natural sciences ,Catalan number ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,Elementary proof ,Euler's formula ,symbols ,FOS: Mathematics ,Mathematics - Combinatorics ,Young tableau ,Combinatorics (math.CO) ,0101 mathematics ,Alternating permutation ,Mathematics - 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 $q$-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 $q$-Euler numbers in terms of certain Dyck path summations., Comment: 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
- Published
- 2016
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