Generalized triangulations, pipe dreams, and simplicial spheres. (English. French summary) Zbl 1355.05078

Serrano, Luis; Stump, Christian

Proceedings of the 23rd international conference on formal power series and algebraic combinatorics, FPSAC 2011, Reykjavik, Iceland, June 13–17, 2011. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science. Proceedings, 885-896 (2011).

Summary: 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.
For the entire collection see [Zbl 1239.05002].

MSC:

05B50	Polyominoes
05A15	Exact enumeration problems, generating functions
05E10	Combinatorial aspects of representation theory

Keywords:

\(k\)-triangulation; enumerative combinatorics; pipe dream; fans of Dyck paths; flagged Schur function; Schubert polynomial; Edelman-Greene insertion