Séminaire Lotharingien de Combinatoire, 89B.75 (2023), 12 pp.
Theo Douvropoulos and Matthieu Josuat-Vergès
Recursions and Proofs in Cataland
Abstract.
We give the first type-independent proof of the Kreweras-style formulas for the enumeration of noncrossing partitions in a real reflection group W, with respect to parabolic type. This answers a central open question in Coxeter-Catalan combinatorics, originally asked by Athanasiadis-Reiner in 2003, special cases of which have been open even longer. Our proof also covers the m-Fuss version of the problem, as well as similar Loday-style formulas for the refined-by-type enumeration of faces of the m-cluster complex of W. It proceeds by developing a family of combinatorial recursions that completely determine the enumeration and proving their algebraic counterparts.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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