Four transformations on the Catalan triangle

TL;DR

This paper introduces four new transformations on the Catalan triangle using determinants and permanents, revealing new identities and offering fresh insights into combinatorial structures.

Contribution

It presents four novel transformations on the Catalan triangle, connecting determinants and permanents to generate new combinatorial identities.

Findings

Many known identities are recovered and new ones are discovered.

The transformations offer a new perspective on the structure of the Catalan triangle.

The methods unify determinant and permanent approaches in combinatorics.

Abstract

In this paper, we define four transformations on the classical Catalan triangle $\mathcal{C}=(C_{n,k})_{n\geq k\geq 0}$ with $C_{n,k}=\frac{k+1}{n+1}\binom{2n-k}{n}$. The first three ones are based on the determinant and the forth is utilizing the permanent of a square matrix. It not only produces many known and new identities involving Catalan numbers, but also provides a new viewpoint on combinatorial triangles.