Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths

TL;DR

This paper explores the minors of matrices derived from weighted partial Motzkin paths, establishing new determinant identities and explicit formulas related to Catalan numbers through combinatorial bijections.

Contribution

It introduces novel determinant identities for matrices associated with weighted partial Motzkin paths and connects these to classical sequences like Catalan numbers.

Findings

Derived new determinant identities involving minors of the matrix mma;

Established explicit formulas for Catalan numbers in specific cases;

Connected weighted Motzkin path enumeration to classical combinatorial sequences.

Abstract

A partial Motzkin path is a path from $(0, 0)$ to $(n, k)$ in the $XOY$-plane that does not go below the $X$-axis and consists of up steps $U=(1, 1)$, down steps $D=(1, -1)$ and horizontal steps $H=(1, 0)$. A weighted partial Motzkin path is a partial Motzkin path with the weight assignment that all up steps and down steps are weighted by 1, the horizontal steps are endowed with a weight $x$ if they are lying on $X$-axis, and endowed with a weight $y$ if they are not lying on $X$-axis. Denote by $M_{n,k}(x, y)$ to be the weight function of all weighted partial Motzkin paths from $(0, 0)$ to $(n, k)$, and $\mathcal{M}=(M_{n,k}(x,y))_{n\geq k\geq 0}$ to be the infinite lower triangular matrices. In this paper, we consider the sums of minors of second order of the matrix $\mathcal{M}$, and obtain a lot of interesting determinant identities related to $\mathcal{M}$, which are proved by bijections using weighted partial Motzkin paths. When the weight parameters $(x, y)$ are specialized, several new identities are obtained related to some classical sequences involving Catalan numbers. Besides, in the alternating cases we also give some new explicit formulas for Catalan numbers.