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Hook formulas for skew shapes. I: \(q\)-analogues and bijections. (English) Zbl 1373.05026
Summary: The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In [“Schubert calculus and hook formula”, talk at the 73rd séminaire lotharingien de combinatoire, Strobl (Austria), 8–10 September 2014, http://tinyurl.com/z6paqzu], H. 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.

MSC:
05A30 \(q\)-calculus and related topics
05E10 Combinatorial aspects of representation theory
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