We study lattice paths by combinatorial methods on the positive lattice. We give some identity that produces the functional equations and generating functions to counting the lattice paths on or below the main diagonal. Also, we consider the subdiagonal lattice paths in relation to lower triangular arrays. This presents a Riordan array in conjunction with the columns of the matrix of the coefficients of certain formal power series by implying an infinite lower triangular matrix \( F=(f_{x,y})_{x,y\geqslant 0} \). We derive new combinatorial interpretations in terms of restricted lattice paths for some Riordan arrays.

中文翻译：

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Riordan 阵列和次对角晶格路径的差分方程

我们通过正格上的组合方法来研究格路径。我们给出一些恒等式来产生函数方程和生成函数来计算主对角线上或下方的晶格路径。此外，我们还考虑与下三角阵列相关的次对角晶格路径。这通过暗示无限下三角矩阵\( F=(f_{x,y})_{x,y\geqslant 0} \呈现了与某些形式幂级数的系数矩阵的列相结合的 Riordan 数组）。我们根据某些 Riordan 阵列的受限晶格路径得出了新的组合解释。