A322011 Number of distinct chromatic symmetric functions of spanning hypergraphs (or antichain covers) on n vertices.

1, 2, 5, 19, 121

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).

Examples

			The a(3) = 5 chromatic symmetric functions:
                  m(111)
          m(21) + m(111)
         2m(21) + m(111)
         3m(21) + m(111)
  m(3) + 3m(21) + m(111)
		

Crossrefs

Cf. A000569, A006125, A229048, A240936, A245883, A277203, A030019, A321750, A321895, A321911, A321994.

Programs

Mathematica

chromSF[g_]:=Sum[m[Sort[Length/@stn,Greater]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}];
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
hyps[n_]:=Select[stableSets[Rest[Subsets[Range[n]]],SubsetQ],Union@@#==Range[n]&];
Table[Length[Union[chromSF/@hyps[n]]],{n,5}]