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Séminaire Lotharingien de Combinatoire, 78B.24 (2017), 12 pp.

# Emerson León

# Stapledon Decompositions and Inequalities for Coefficients of Chromatic Polynomials

**Abstract.**
We use a polynomial decomposition result by Stapledon to show that the
numerator polynomial of the Ehrhart series of an open polytope is the
difference of two symmetric polynomials with nonnegative integer
coefficients. We obtain a related decomposition for order polytopes
and for the numerator polynomial of the corresponding series for
chromatic polynomials. The nonnegativity of the coefficients in such
decompositions provide inequalities satisfied by the coefficients of
chromatic polynomials for any simple graph.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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