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

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.

Fillings of skew shapes avoiding diagonal patterns

“…Backelin, West and Xin [1] have shown that the number of transversal fillings of a Ferrers shape that avoid δ k is the same as the number of such fillings that avoid ι k . This was later generalized to non-transversal fillings [13], and to more general shapes, such as stack polyominoes [8], moon polyominoes [15], or almost-moon polyominoes [14].…”