Hook Formulas for Skew Shapes

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. Our main results are two q-analogues of Naruse’s formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations. In particular, we obtain new summation formulas for the number of alternating permutations in terms of certain Dyck paths, and their q-analogues.