# Set-valued Rothe Tableaux and Grothendieck Polynomials

## TL;DR

This paper introduces set-valued Rothe tableaux for permutations and demonstrates their use in generating double Grothendieck polynomials for specific permutation classes, extending previous tableau formulas.

## Contribution

It defines set-valued Rothe tableaux and shows they can generate double Grothendieck polynomials for 1432-avoiding permutations, generalizing existing formulas.

## Key findings

- Set-valued Rothe tableaux characterize double Grothendieck polynomials for 1432-avoiding permutations.
- The formula specializes to known results for 321-avoiding permutations.
- Two alternative tableau formulas are derived using tableau complex properties.

## Abstract

The notion of set-valued Young tableaux was introduced by Buch in his study of the Littlewood-Richardson rule for stable Grothendieck polynomials. Knutson, Miller and Yong showed that the double Grothendieck polynomials of 2143-avoiding permutations can be generated by set-valued Young tableaux. In this paper, we introduce the structure of set-valued Rothe tableaux of permutations. Given the Rothe diagram $D(w)$ of a permutation $w$, a set-valued Rothe tableau of shape $D(w)$ is a filling of finite nonempty subsets of positive integers into the squares of $D(w)$ such that the rows are weakly decreasing and the columns are strictly increasing. We show that the double Grothendieck polynomials of 1432-avoiding permutations can be generated by set-valued Rothe tableaux. When restricted to 321-avoiding permutations, our formula specializes to the tableau formula for double Grothendieck polynomials due to Matsumura. Employing the properties of tableau complexes given by Knutson, Miller and Yong, we obtain two alternative tableau formulas for the double Grothendieck polynomials of 1432-avoiding permutations.