The determinants of certain matrices arising from the Boolean lattice (Q998467)

Let \(B\) be the Boolean lattice on an \(n\)-set with \(B=\bigsqcup B_i\) the rank decomposition and denote by \(M(n,i)\) the incidence matrix between \(B_i\) and \(B_{n-i}\). The main goal of this note is to give a recursive formula for the determinant of the matrix \(M(n,i)\) (see Theorem 4). Several interesting applications are also presented in the end of the paper.