On the Neggers–Stanley Conjecture and the Eulerian Polynomials

We prove combinatorially that theW-polynomials of naturally labeled graded posets of rank 1 or 2 (an antichain has rank 0) are unimodal, thus providing further supporting evidence for the Neggers–Stanley conjecture. For such posets we also obtain a combinatorial proof that theW-polynomials are symmetric. Combinatorial proofs that the Eulerian polynomials are log-concave and unimodal are given and we construct a simplicial complexΔwith the property that the Hilbert function of the exterior algebra modulo the Stanley–Reisner ideal ofΔis the sequence of Eulerian numbers, thus providing a combinatorial proof of a result of Brenti.