Eulerian poset

Summary

An Eulerian poset is a finite graded poset with a least and a greatest element. In it, every interval of positive length has as many elements of even rank as of odd rank. Equivalently, its Möbius function is $\mu(x, y) = (-1)^{\rho(y) - \rho(x)}$, where $\rho$ is the rank. Face lattices of convex polytopes are Eulerian, and the condition encodes the Euler–Poincaré relation for their face numbers.