A322066 Number of e-positive antichains of sets spanning n vertices.

1, 1, 2, 8, 64, 1299

Offset: 0

Author

Gus Wiseman, Nov 25 2018

Keywords

• nonn
• more

Comments

A stable partition of a hypergraph or set system is a set partition of the vertices where no non-singleton edge has all its vertices in the same block. The chromatic symmetric function is given by X_G = Sum_pi m(t(pi)) where the sum is over all stable partitions pi of G, t(pi) is the integer partition whose parts are the block-sizes of pi, and m is the basis of augmented monomial symmetric functions (see A321895). A hypergraph or set system is e-positive if, in the expansion of its chromatic symmetric function in terms of elementary functions, all coefficients are nonnegative.

Examples

			The a(3) = 8 e-positive antichains:
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
The antichain {{1,2,3}} is not e-positive, as its chromatic symmetric function is -3e(3) + 3e(21).
		

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

Cf. A006125, A229048, A240936, A277203, A321895, A321914, A321918, A321931, A321979, A321980, A321981, A321982, A321994, A322012.