Geodesic
Maximal fillings of Moon polyominoes, simplicial complexes, and Schubert polynomials
The electronic journal of combinatorics, Tome 19 (2012) no. 1
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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. Moreover, for Ferrers shapes we 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.
DOI : 10.37236/1167
Classification : 05E45, 05A15, 05A05, 05E05, 05B50
Mots-clés : maximal \((0,1)\)-fillings of a moon polyomino, Schubert polynomials 
@article{10_37236_1167,
     author = {Luis Serrano and Christian Stump},
     title = {Maximal fillings of {Moon} polyominoes, simplicial complexes, and {Schubert} polynomials},
     journal = {The electronic journal of combinatorics},
     year = {2012},
     volume = {19},
     number = {1},
     doi = {10.37236/1167},
     zbl = {1244.05239},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1167/}
}
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Luis Serrano; Christian Stump. Maximal fillings of Moon polyominoes, simplicial complexes, and Schubert polynomials. The electronic journal of combinatorics, Tome 19 (2012) no. 1. doi: 10.37236/1167

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