ON THE NEGGERS-STANLEY CONJECTURE AND THE EULERIAN POLYNOMIALS

We prove combinatorially that the W-polynomials of naturally labeled g raded posets of rank 1 or 2 (an antichain has rank 0) are unimodal, th us providing further supporting evidence for the Neggers-Stanley conje cture. For such posets we also obtain a combinatorial proof that the W -polynomials are symmetric. Combinatorial proofs that the Eulerian pol ynomials are log-concave and unimodal are given and we construct a sim plicial complex Delta with the property that the Hilbert function of t he exterior algebra module the Stanley-Reisner ideal of Delta is the s equence of Eulerian numbers, thus providing a combinatorial proof of a result of Brenti. (C) 1998 Academic Press.