Representation functions and the Neggers-Stanley condition for weight-shaped-posets

Let P be a poset (partially ordered set), i.e., a set equipped with a relation<where x < y implies y 6< x and x < y, y < z implies x < z. The relation≤ as usual means x = y or x < y. For details on the theory of posets we refer the reader to [6,7]. In these texts further references are supplied as well. Let w : P → R be a weight function on the elements of P , where R is the set of all real numbers. Let α : E → R be a weight (length) function on the ‘‘edges’’ (x, y)where x < y, i.e., on the (strict) order relation denoted by<. Furthermore, let γ : G→ R be a weight (gap) function on the ‘‘gaps’’ {x, y}, where x and y are not comparable (free, parallel) denoted by x ◦ y (x ‖ y). For example, the antichain 3 has Hasse diagram: