cp's OEIS Frontend

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

%I A322066 #5 Nov 26 2018 17:04:33
%S A322066 1,1,2,8,64,1299
%N A322066 Number of e-positive antichains of sets spanning n vertices.
%C A322066 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.
%H A322066 Richard P. Stanley, <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/100.pdf">A symmetric function generalization of the chromatic polynomial of a graph</a>, Advances in Math. 111 (1995), 166-194.
%H A322066 Richard P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/taor.pdf">Graph colorings and related symmetric functions: ideas and applications</a>, Discrete Mathematics 193 (1998), 267-286.
%H A322066 Richard P. Stanley and John R. Stembridge, <a href="https://doi.org/10.1016/0097-3165(93)90048-D">On immanants of Jacobi-Trudi matrices and permutations with restricted position</a>, Journal of Combinatorial Theory Series A 62-2 (1993), 261-279.
%e A322066 The a(3) = 8 e-positive antichains:
%e A322066   {{1},{2,3}}
%e A322066   {{2},{1,3}}
%e A322066   {{3},{1,2}}
%e A322066   {{1,2},{1,3}}
%e A322066   {{1,2},{2,3}}
%e A322066   {{1,3},{2,3}}
%e A322066   {{1},{2},{3}}
%e A322066   {{1,2},{1,3},{2,3}}
%e A322066 The antichain {{1,2,3}} is not e-positive, as its chromatic symmetric function is -3e(3) + 3e(21).
%Y A322066 Cf. A006125, A229048, A240936, A277203, A321895, A321914, A321918, A321931, A321979, A321980, A321981, A321982, A321994, A322012.
%K A322066 nonn,more
%O A322066 0,3
%A A322066 _Gus Wiseman_, Nov 25 2018