Signature Catalan Combinatorics
Cesar Ceballos, Rafael S. Gonz\'alez D'Le\'on

TL;DR
This paper introduces a broad generalization of Catalan combinatorics using signatures derived from compositions and rational parameters, unifying and extending classical, Fuss-Catalan, and rational Catalan objects with explicit bijections and recurrences.
Contribution
It proposes a new framework for generalized Catalan objects indexed by compositions, connecting classical, Fuss-Catalan, and rational Catalan combinatorics with explicit bijections and recurrences.
Findings
Defined $s$-Catalan objects for compositions $s$
Established bijections between generalized Catalan objects
Derived a recurrence generalizing classical Catalan recurrence
Abstract
The Catalan numbers constitute one of the most important sequences in combinatorics. Catalan objects have been generalized in various directions, including the classical Fuss-Catalan objects and the rational Catalan generalization of Armstrong-Rhoades-Williams. We propose a wider generalization of these families indexed by a composition which is motivated by the combinatorics of planar rooted trees; when and we recover the classical Catalan and Fuss-Catalan combinatorics, respectively. Furthermore, to each pair of relatively prime numbers we can associate a signature that recovers the combinatorics of rational Catalan objects. We present explicit bijections between the resulting -Catalan objects, and a fundamental recurrence that generalizes the fundamental recurrence of the classical Catalan numbers. Our framework allows us to define…
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