Euler characteristic reciprocity for chromatic, flow and order polynomials

Journal of Singularities
volume 16 (2017), 212-227

Received: 3 May 2017. Received in revised form: 9 October 2017.

DOI: 10.5427/jsing.2017.16k

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Abstract:

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.