Two deformed Pascal's triangles and its new properties

TL;DR

This paper explores deformed Pascal's triangles through determinant methods, providing new combinatorial proofs and uncovering four novel properties of Pascal's triangle related to Dyck paths and matrix determinants.

Contribution

It introduces determinant-based approaches to analyze deformed Pascal's triangles, offering new proofs and properties not previously documented.

Findings

Determinant of deformed Pascal's triangle counts Dyck paths.

New properties of Pascal's triangle derived from Toeplitz-Hessenberg matrices.

Recurrence formulas for Catalan number determinants.

Abstract

In this paper, firstly, by a determinant of deformed Pascal's triangle, namely the normalized Hessenberg matrix determinant, to count Dyck paths, we give another combinatorial proof of the theorems which are of Catalan numbers determinant representations and the recurrence formula. Secondly, a determinant of normalized Toeplitz-Hessenberg matrix, whose entries are binomials, arising in power series, we derive new four properties of Pascal's triangle.