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

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Abstract

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.

Keywords

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

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