In this paper, we consider generalized Delannoy paths with steps \(E_i=(i, 0)\) , \(N_i=(0, i)\) , \(D_i=(i, i), i>0\) , and \(S_{j, i}=(j, i), j>i>0\) , where all steps are weighted by 1 for \(E_{i}\) , \(a_i\) for \(N_{i}\) , \(b_i\) for \(D_i\) and \(e_i\) for \(S_{j, i}\) , respectively. Under the restriction of below the main diagonal \(y=x\) , these paths can be viewed as a unified generalization of the well-known Dyck paths, Schröder paths, and Delannoy paths. Using the almost-Riordan array method, we introduce a new family of generalized Delannoy matrices associated with the generalized Delannoy paths such that the weight functions \(a(t)=\sum _{i\ge 1}a_it^i\) , \(b(t)=\sum _{i\ge 1}b_it^i\) and \(e(t)=\sum _{i\ge 1}e_it^i\) . When weight functions are specialized, we obtain numerous combinatorial matrices such as generalized Schröder matrices, generalized Catalan matrices, etc. We also study the correlations between these matrices and give several examples.
