Chromatic statistics for Catalan and Fu{\ss}-Catalan numbers

TL;DR

This paper introduces color-based refinements of Catalan and Fu{ ext}ss-Catalan numbers for polygon triangulations and their higher-dimensional analogues, providing explicit formulas for distributions using advanced combinatorial techniques.

Contribution

It develops new color statistics for triangulations and Fu{ ext}ss-Catalan complexes, deriving closed-form formulas for their distributions.

Findings

Derived explicit formulas for colored triangulations and complexes.

Applied Lagrange-Good inversion formula in combinatorial enumeration.

Extended classical Catalan and Fu{ ext}ss-Catalan numbers with new refinements.

Abstract

We refine Catalan numbers and Fu{\ss}-Catalan numbers by introducing colour statistics for triangulations of polygons and $d$-dimensional generalisations there-of which we call Fu{\ss}-Catalan complexes. Our refinements consist in showing that the number of triangulations, respectively Fu{\ss}-Catalan complexes, with a given colour distribution of its vertices is given by closed product formulae. The crucial ingredient in the proof is the Lagrange-Good inversion formula.