On the Neggers–Stanley Conjecture and the Eulerian Polynomials

✍ Scribed by Vesselin Gasharov

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 387 KB
• Volume: 82
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1997.2839

✦ Synopsis

We prove combinatorially that the W-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 the W-polynomials are symmetric. Combinatorial proofs that the Eulerian polynomials are log-concave and unimodal are given and we construct a simplicial complex 2 with the property that the Hilbert function of the exterior algebra modulo the Stanley Reisner ideal of 2 is the sequence of Eulerian numbers, thus providing a combinatorial proof of a result of Brenti.

1998 Academic Press

1. Introduction

Let P=(P, O) be a poset with n elements and *: P Ä [1, 2, ..., n] a labeling of the vertices of P. A permutation |=| 1 } } } | n # S n is said to be (P, *)-compatible if * &1 (| i ) O P * &1 (| j ) implies i< j. For i 1 let d i (P, *) be the number of (P, *)-compatible permutations with i descents. The W-polynomial of the pair (P, *) is defined by

( 1 )

Two labelings * and + of P are called equivalent if *(x)<*( y) +(x)< +( y) for all x, y # P such that y covers x. It can be shown that as a function of *, W(P, *, t) depends only on the equivalence class of *. If P is a graded poset of rank 0 (i.e., an antichain), then W(P, *, t)=A n (t), the n th Eulerian polynomial. It is well known that the Eulerian polynomials have only real zeros. There is a conjectured generalization of this fact due to Neggers and Stanley (see [2,12,13,18]).