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

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.