Unimodular matrices and lattice paths enumeration via Pascal's triangle

This article investigates a remarkable combinatorial identity involving a distinguished family of matrices whose entries are defined via binomial coefficients. Specifically, we consider a class of $n \times n$ matrices parameterized by a positive integer $m$, where each entry reflects a structured pattern derived from Pascal's triangle, particularly the diagonals corresponding to figurate numbers such as triangular, tetrahedral, and higher-dimensional simplex numbers. We establish, by means of a bijective argument, that the determinant of any such matrix is identically equal to $1$, independent of the specific values of $m$ and $n$, provided that $2 \leq m \leq n$. This result unveils a profound connection between classical binomial identities and the enumeration of lattice paths in grid graphs.