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A322012
Number of s-positive simple labeled graphs with n vertices.
1
1, 2, 8, 60, 1009
OFFSET
1,2
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 the augmented monomial symmetric function basis (see A321895). A graph is s-positive if, in the expansion of its chromatic symmetric function in terms of Schur 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.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 24 2018
STATUS
approved