A Chung–Feller property for the generalized Schröder paths

Abstract In this paper, we consider the generalized Delannoy paths with steps E = ( 1 , 0 ) , D = ( 1 , 1 ) , N = ( 0 , 1 ) , and N ′ = ( 0 , 2 ) , where each step is labelled with weights 1, a , b , and d , respectively. By using Riordan array method to study enumeration of these paths in general case and with the restriction that no step goes above the main diagonal, we obtain three families of matrices. We consider the correlation between these matrices, and obtain a Chung–Feller type theorem for these paths. By way of illustration, we give several examples of Riordan arrays.