Hecke insertion and maximal increasing and decreasing sequences in fillings of stack polyominoes

TL;DR

This paper establishes a combinatorial bijection involving Hecke insertion to analyze the distribution of certain chain patterns in fillings of stack polyominoes, revealing symmetry properties and dependence solely on row lengths.

Contribution

It introduces a novel bijection using Hecke insertion for stack polyomino fillings, linking chain pattern counts to row lengths and providing new proofs of symmetry results.

Findings

Number of specific fillings depends only on row lengths

Bijection relates fillings differing by one row position

Symmetric distribution of crossing and nesting numbers

Abstract

We prove that the number of 01-fillings of a given stack polyomino (a polyomino with justified rows whose lengths form a unimodal sequence) with at most one 1 per column which do not contain a fixed-size northeast chain and a fixed-size southeast chain, depends only on the set of row lengths of the polyomino. The proof is via a bijection between fillings of stack polyominoes which differ only in the position of one row and uses the Hecke insertion algorithm by Buch, Kresch, Shimozono, Tamvakis, and Yong and the jeu de taquin for increasing tableaux of Thomas and Yong. Moreover, our bijection gives another proof of the result by Chen, Guo, and Pang that the crossing number and the nesting number have a symmetric joint distribution over linked partitions.