Variations of the Catalan numbers from some nonassociative binary operations

TL;DR

This paper explores nonassociative binary operations that generalize the associative law, leading to new variations of Catalan numbers and their connections to various combinatorial structures.

Contribution

It introduces a four-parameter generalization of associativity and derives new Catalan number variations linked to multiple combinatorial objects.

Findings

New variations of Catalan numbers derived from nonassociative operations

Connections established between these variations and combinatorial structures like trees and paths

Generalization of associative law through four parameters

Abstract

We investigate certain nonassociative binary operations that satisfy a four-parameter generalization of the associative law. From this we obtain variations of the ubiquitous Catalan numbers and connections to many interesting combinatorial objects such as binary trees, plane trees, lattice paths, and permutations.