  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.
