On the Neggers-Stanley conjecture and the Eulerian polynomials
If \(P=(P,<)\) is an \(n\)-poset (i.e., \(P\) contains \(n\) elements) and \(\lambda: P\to\{1,\dots, n\}\) is a labeling of the vertices of \(P\), then a permutation \(w\in S_n\) is \((P, \lambda)\)-compatible if \(\lambda^{-1}(w(i))<\lambda^{-1}(w(j))\) implies \(i<j\). For \(i\geq 1\), let \(d_i(P,\lambda)\) be the number of \((P,\lambda)\)-compatible permutations with \(i\) descents. The \(W\)-polynomial of \((P,\lambda)\) is defined as \(W(P,\lambda,t)=\sum_{i\geq 0}d_i(P, \lambda) t^{i+1}\). The Neggers-Stanley conjecture claims that \(W(P,\lambda,t)\) has only real zeros and by now there is a considerably body of information in its support. Thus, if \(P\) is an \(n\)-antichain \((n\)-set) implies \(W(P,\lambda,t) =A_n(t)\), the \(n\)th Eulerian polynomial, then the conjecture holds. In further support the author deduces that for graded \(n\)-posets of rank 1 or 2, when \(\lambda\) is a natural labeling \((u<v\) in \((P,<)\) implies \(\lambda(u)< \lambda(v))\), then \(W(P,\lambda,t)\) is unimodal, which is quite close to having real zeros when coefficients are non-negative integers as in this case. Also, via the simplicial complex associated with bounded \(n\)-posets and other structures the author is able to produce combinatorial proofs of several interesting known facts, including the observation that the Eulerian polynomials are both log-concave and unimodal.
scientific article; zbMATH DE number 1212923