Hecke growth diagrams, and maximal increasing and decreasing sequences in fillings of stack polyominoes

TL;DR

This paper introduces a bijection and growth diagram techniques for 01-fillings of stack polyominoes, revealing symmetric distributions of chain lengths and offering new insights into related combinatorial structures.

Contribution

It provides a novel bijection and growth diagram approach that generalizes existing techniques, connecting chain length distributions with symmetric properties in stack polyomino fillings.

Findings

Establishes a bijection between 01-fillings and corner labelings

Shows symmetric joint distribution of chain lengths

Generalizes Rubey's growth diagram techniques

Abstract

We establish a bijection between $01$-fillings of stack polyominoes with at most one $1$ per column and labelings of the corners along the top-right border of stack polyominoes. These labellings indicate the lengths of the longest increasing and decreasing chains of the largest rectangular region below and to the left of the corners. Our results provide an alternative proof of Guo and Poznanovi\'c's theorem on the lengths of the longest increasing and decreasing chains have a symmetric joint distribution over $01$-fillings of stack polyomino. Moreover, our results offer new perspective to Chen, Guo and Pang's result on the crossing number and the nesting number have a symmetric joint distribution over linked partitions. In particular, our construction generalizes the growth diagram techniques of Rubey for the $01$-fillings of stack polyominoes with at most one $1$ per column and row.