The Miracle of Integer Eigenvalues

For partially ordered sets \((X, \preccurlyeq)\), we consider the square matrices \(M^{X}\) with rows and columns indexed by linear extensions of the partial order on \(X\). Each entry \((M^{X})_{PQ}\) is a formal variable defined by a pedestal of the linear order \(Q\) with respect to linear order \(P\). We show that all eigenvalues of any such matrix \(M^{X}\) are \(\mathbb{Z}\)-linear combinations of those variables.