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 X O Y -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 ) the weight function of all weighted partial Motzkin paths from ( 0 , 0 ) to ( n , k ) , and M = ( M n , k ( x , y ) ) n ≥ k ≥ 0 the infinite lower triangular matrices. In this paper, we consider the sums of minors of second order of the matrix M , and obtain a lot of interesting determinant identities related to 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.