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

We introduce the class of weight-shaped-posets and we define the representation functions of such wsws-posets as generalizations of the representation polynomials of finite posets. We define the Brylawski decomposition of wsws-posets and use it to construct the algorithm for the computation of the representation functions. The class of wsws-posets contains a class of fuzzy posets for example. The Neggers–Stanley conjecture for wsws-posets is also discussed.