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arXiv · 2310.18275

The Pak--Postnikov and Naruse skew hook length formulas: a new proof

Darij Grinberg·Nazar Korniichuk·Kostiantyn Molokanov·Severyn Khomych
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Abstract

The classical hook length formula of enumerative combinatorics expresses the number of standard Young tableaux of a given partition shape as a single fraction. In recent years, two generalizations of this formula have emerged: one by Pak and Postnikov, replacing the number by a (rational) generating function, and one by Naruse, which generalizes the setting from a partition to a skew partition. Both generalizations appear to lie significantly deeper, with no simple proofs known. We combine them into a generating-function identity for skew partitions, and prove it in a fairly elementary way using recursion, determinants and simple combinatorics.

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Darij Grinberg, Nazar Korniichuk, Kostiantyn Molokanov, Severyn Khomych. 2023-10-27. The Pak--Postnikov and Naruse skew hook length formulas: a new proof. https://arxiv.org/abs/2310.18275

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