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

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.