Counterexamples to the Neggers-stanley Conjecture

The Neggers-Stanley conjecture asserts that the polynomial counting the linear extensions of a labeled finite partially ordered set by the number of descents has real zeros only. We provide counterexamples to this conjecture. A finite partially ordered set (poset) P of cardinality p is said to be labeled if its elements are identified with the integers 1, 2, . . . , p. We will use the symbol ≺ to denote the partial order on P and < to denote the usual order on the integers. The Jordan-Hölder set L(P ) is the set of permutations π = (π1, . . . , πp) of [p] def = {1, 2, . . . , p} which encode the linear extensions of P . More precisely, π ∈ L(P ) if πi ≺ πj implies i < j. A descent in a permutation π is an index i such that πi > πi+1. Let des(π) denote the number of descents in π. The W -polynomial of a labeled poset P is defined by W (P, t) = ∑