Euler characteristic reciprocity for chromatic, flow and order   polynomials
  The Euler characteristic of a semialgebraic set can be considered as a generalization of the cardinality of a finite set. An advantage of semialgebraic sets is that we can define "negative sets" to be the sets with negative Euler characteristics. Applying this idea to posets, we introduce the notion of semialgebraic posets. Using "negative posets", we establish Stanley's reciprocity theorems for order polynomials at the level of Euler characteristics. We also formulate the Euler characteristic reciprocities for chromatic and flow polynomials. 
Takahiro Hasebe, Toshinori Miyatani and Masahiko Yoshinaga
math.CO math.AG math.AT