Combinatorial matrices derived from generalized Motzkin paths

In this paper, we consider the generalized Motzkin paths whose step set consists of $$E = (1, 0), N= (0,1), U = (1,1)$$ E = ( 1 , 0 ) , N = ( 0 , 1 ) , U = ( 1 , 1 ) and $$D = (1,-1)$$ D = ( 1 , - 1 ) . In the general case, for the number of such paths running from (0, 0) to $$(k,n-2k)$$ ( k , n - 2 k ) , we define a number triangle, which turns out to be a common extension of Pascal triangle and Delannoy triangle. Under the restriction of above or below the x-axis, these paths can be seen as an unified generalization of the well-known Dyck paths, Motzkin paths, and Schröder paths. We also consider the counting of such paths above the main diagonal. In every condition, we treat with two classes of paths, which are restricted and unrestricted paths. For each class of paths, the corresponding counting array is a Riordan array. Numerous Combinatorial matrices such as the Catalan matrix, Motzkin matrix, and Schröder matrix are special cases of these Riordan arrays.