Hook formulas for skew shapes IV. Increasing tableaux and factorial Grothendieck polynomials

TL;DR

This paper introduces new hook-length formulas for counting standard increasing tableaux related to factorial Grothendieck polynomials, generalizing classical and Naruse formulas for straight and skew shapes.

Contribution

It provides a novel family of hook-length formulas that extend classical and Naruse formulas to factorial Grothendieck polynomials and increasing tableaux.

Findings

Generalizes classical hook-length formula for straight shapes

Extends Naruse hook-length formula to skew shapes

Connects factorial Grothendieck polynomials with increasing tableaux

Abstract

We present a new family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes our formulas generalize the classical hook-length formula and Stanley's formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its $q$-analogues, which were studied in previous papers of the series.