Lenart's bijection via bumpless pipe dreams

TL;DR

This paper proves that the Gao-Huang bijection between pipe dreams and bumpless pipe dreams preserves associated tableaux for vexillary permutations, providing a new proof of Lenart's bijection and extending methods to record tableaux.

Contribution

It demonstrates that the Gao-Huang bijection preserves tableaux for vexillary permutations and extends to record tableaux for bumpless pipe dreams, offering new insights into these combinatorial objects.

Findings

Gao-Huang bijection preserves associated tableaux for vexillary permutations

New proof of Lenart's bijection using Gao-Huang bijection

Extension to recording tableaux for bumpless pipe dreams

Abstract

Pipe dreams and bumpless pipe dreams for vexillary permutations are each known to be in bijection with certain semistandard tableaux via maps due to Lenart and Weigandt, respectively. Recently, Gao and Huang have defined a bijection between the former two sets. In this note we show for vexillary permutations that the Gao-Huang bijection preserves the associated tableaux, giving a new proof of Lenart's result. Our methods extend to give a recording tableau for any bumpless pipe dream.