OFFSET
0,3
COMMENTS
A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895). A graph is e-positive if, in the expansion of its chromatic symmetric function in terms of elementary symmetric functions, all coefficients are nonnegative.
LINKS
Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.
Richard P. Stanley and John R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, Journal of Combinatorial Theory Series A 62-2 (1993), 261-279.
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.
Gus Wiseman, The a(4) = 60 e-positive simple labeled graphs.
EXAMPLE
The 4 non-e-positive simple labeled graphs on 4 vertices are:
{{1,2},{1,3},{1,4}}
{{1,2},{2,3},{2,4}}
{{1,3},{2,3},{3,4}}
{{1,4},{2,4},{3,4}}
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 23 2018
STATUS
approved