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.