The miracle of integer eigenvalues

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