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A note on a combinatorial interpretation of the e-coefficients of the chromatic symmetric function

Timothy Y. Chow
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Abstract

Stanley has studied a symmetric function generalization X_G of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of X_G in terms of elementary symmetric functions has nonnegative coefficients if G is a clawfree incomparability graph. Here we give a combinatorial interpretation of these coefficients by combining Gasharov's work on the conjecture with Egecioglu and Remmel's combinatorial interpretation of the inverse Kostka matrix. This gives a new proof of a partial nonnegativity result of Stanley. As an interesting byproduct we derive a previously unnoticed result relating acyclic orientations to P-tableaux.

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Timothy Y. Chow. 2002-08-16. A note on a combinatorial interpretation of the e-coefficients of the chromatic symmetric function. https://arxiv.org/abs/math/9712230

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