Three-dimensional Catalan numbers and product-coproduct prographs

TL;DR

This paper introduces product-coproduct prographs, a new combinatorial class, and demonstrates their enumeration by 3-dimensional Catalan numbers, establishing their significance in combinatorial mathematics.

Contribution

The paper defines a novel class of planar graphs called product-coproduct prographs and proves they are counted by 3-dimensional Catalan numbers, linking them to existing combinatorial structures.

Findings

Product-coproduct prographs are enumerated by 3-dimensional Catalan numbers.

Bijections are established connecting these prographs to other combinatorial objects.

The work provides new insights into the combinatorial interpretation of 3D Catalan numbers.

Abstract

We present the new combinatorial class of product-coproduct prographs which are planar assemblies of two types of operators: products having two inputs and a single output and coproducts having a single input and two outputs. We show that such graphs are enumerated by the $3$-dimensional Catalan numbers. We present some combinatorial bijections positioning product-coproduct prographs as key objects to probe families of objects enumerated by the $3$-dimensional Catalan numbers.