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A partial Motzkin path is a path from $(0, 0)$ to $(n, k)$ in the $XOY$-planethat does not go below the $X$-axis and consists of up steps $U=(1, 1)$, downsteps $D=(1, -1)$ and horizontal steps $H=(1, 0)$. A weighted partial Motzkinpath is a partial Motzkin path with the weight assignment that all up steps anddown 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 arenot lying on $X$-axis. Denote by $M_{n,k}(x, y)$ to be the weight function ofall 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 lowertriangular matrices. In this paper, we consider the sums of minors of secondorder of the matrix $\mathcal{M}$, and obtain a lot of interesting determinantidentities related to $\mathcal{M}$, which are proved by bijections usingweighted partial Motzkin paths. When the weight parameters $(x, y)$ arespecialized, several new identities are obtained related to some classicalsequences involving Catalan numbers. Besides, in the alternating cases we alsogive some new explicit formulas for Catalan numbers.

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