Abstract
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 a 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. Using this, we translate a conjectured cyclic sieving phenomenon for $k$-triangulations with rotation to $k$-flagged tableaux with promotion. Nous décrivons un lien canonique entre les $(0,1)$-remplissages maximaux d'un polyomino-lune évitant les chaînes Nord-Est d'une longueur donnée, et les "pipe dreams'' réduits d'une certaine permutation. En suivant cette approche nous montrons que le complexe simplicial de tels remplissages maximaux est une sphère "vertex-decomposable'' et donc "shellable''. En particulier, cela entraîne un résultat de positivité sur les polynômes de Schubert. De plus, nous construisons, dans le cas des diagrammes de Ferrers, une bijection vers les remplissages maximaux évitant les chaînes Sud-Est de même longueur, qui se spécialise en une bijection entre les $k$-triangulations d'un $n$-gone et les $k$-faisceaux de chemins de Dyck. A l'aide de celle-ci, nous traduisons une instance conjecturale du phénomène de tamis cyclique pour les $k$-triangulations avec rotation dans le cadre des tableaux $k$-marqués avec promotion.
Highlights
Fix positive integers n and k such that 2k < n
Theorem 1.1 There is an explicit bijection between k-triangulations of a convex n-gon and k-fans of Dyck paths of length 2(n − 2k)
It is well known that k-triangulations of the n-gon can be seen as k-north-east fillings of the staircase shape (n − 1, . . . , 2, 1), and k-fans of Dyck paths of length 2(n − 2k) can be seen as ksouth-east fillings of the same staircase
Summary
Fix positive integers n and k such that 2k < n. There exists a canonical bijection between k-north-east fillings of λ and reduced pipe dreams of a permutation depending on λ and k. There exists a canonical bijection between k-north-east fillings of M and reduced pipe dreams (of a given permutation) living inside M. Corollary 1.5 The simplicial complex with facets being k-north-east fillings of a moon polyomino M is the join of a vertex-decomposable, triangulated sphere with a full simplex. 1≤i≤j
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