cp's OEIS Frontend

A321994 Number of different chromatic symmetric functions of hypertrees on n vertices.

%I A321994 #15 Apr 11 2019 20:59:46
%S A321994 1,1,2,4,9,22,59,165
%N A321994 Number of different chromatic symmetric functions of hypertrees on n vertices.
%C A321994 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 augmented monomial symmetric functions (see A321895).
%C A321994 Stanley conjectured that the number of distinct chromatic symmetric functions of trees with n vertices is equal to A000055, i.e., the chromatic symmetric function distinguishes between trees. It has been proven for trees with up to 25 vertices. If it is true in general, does the chromatic symmetric function also distinguish between hypertrees, meaning this sequence would be equal to A035053?
%H A321994 Jeremy L. Martin, <a href="http://jlmartin.faculty.ku.edu/~jlmartin/talks/LasVegas2.pdf">The uniqueness problem for chromatic symmetric functions of trees</a> (2015)
%H A321994 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 A321994 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.
%t A321994 spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];
%t A321994 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]]]];
%t A321994 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A321994 density[c_]:=Total[(Length[#]-1&)/@c]-Length[Union@@c];
%t A321994 hyall[n_]:=Select[stableSets[Select[Subsets[Range[n]],Length[#]>1&],Or[SubsetQ[#1,#2],Length[Intersection[#1,#2]]>1]&],And[Union@@#==Range[n],Length[csm[#]]==1,density[#]==-1]&];
%t A321994 chromSF[g_]:=Sum[m[Sort[Length/@stn,Greater]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}];
%t A321994 Table[Length[Union[chromSF/@If[n==1,{{{1}}},hyall[n]]]],{n,5}]
%Y A321994 Cf. A000055, A000569, A001187, A001349, A006125, A035053, A229048, A240936, A245883, A277203, A030019, A321750, A321895, A321911.
%K A321994 nonn,more
%O A321994 1,3
%A A321994 _Gus Wiseman_, Nov 24 2018