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

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

1997-12-04

arxiv.org/abs/math/9712230v2

Mathematics

Combinatorics

Minor error corrected

Scientific paper

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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