A322012 Number of s-positive simple labeled graphs with n vertices.

1, 2, 8, 60, 1009

Offset: 1

Author

Gus Wiseman, Nov 24 2018

Keywords

• nonn
• more

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.

Crossrefs

a(n) >= A321979(n).

Cf. A000569, A006125, A229048, A240936, A277203, A321895, A321924, A321925, A321931, A321994.