Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials

Details Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials

2010-09-23

arxiv.org/abs/1009.4690v3

Mathematics

Combinatorics

16 pages, 7 figures, revised version

Scientific paper

We exhibit a canonical connection between maximal (0,1)-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between k-triangulations of the n-gon and k-fans of Dyck paths of length 2(n-2k). Using this, we translate a conjectured cyclic sieving phenomenon for k-triangulations with rotation to the language of k-flagged tableaux with promotion.

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