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 $q$-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.

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

References in corpus (4)

• Hook formulas for skew shapes I. $q$-analogues and bijections
• Schubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Lagrangian Grassmannian
• Probabilistic proofs of hook length formulas involving trees
• Outside nested decompositions of skew diagrams and Schur function determinants

Cited by in corpus (10)

• Hook formulas for skew shapes I. $q$-analogues and bijections
• On the Okounkov-Olshanski formula for standard tableaux of skew shapes
• Limit shapes of large skew Young tableaux and a modification of the TASEP process
• Reverse plane partitions of skew staircase shapes and $q$-Euler numbers
• Ratios of Hahn--Exton $q$-Bessel functions and $q$-Lommel polynomials
• Brändén's $(p,q)$-Eulerian polynomials, André permutations and continued fractions
• Hankel Continued fractions and Hankel determinants of the Euler numbers
• Sorting probability for large Young diagrams
• Hidden symmetries of weighted lozenge tilings
• Hook-length formula and applications to alternating permutations