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

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