A229048 -id:A229048 - cp's OEIS frontend

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

1, 2, 5, 19, 121

Offset: 1

Gus Wiseman, Nov 24 2018

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).

			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)

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

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}]

A322063 Number of ways to choose a stable partition of an antichain of sets spanning n vertices.

Original entry on oeis.org

1, 1, 3, 25, 773, 160105

Offset: 0

Gus Wiseman, Nov 25 2018

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 a(3) = 25 stable partitions of antichains on 3 vertices. The antichain is on top, and below is a list of all its stable partitions.
  {1}{2}{3}      {1,2,3}        {1}{2,3}       {1,3}{2}       {1,2}{3}
  --------       --------       --------       --------       --------
  {{1,2,3}}      {{1},{2,3}}    {{1,2},{3}}    {{1},{2,3}}    {{1},{2,3}}
  {{1},{2,3}}    {{1,2},{3}}    {{1,3},{2}}    {{1,2},{3}}    {{1,3},{2}}
  {{1,2},{3}}    {{1,3},{2}}    {{1},{2},{3}}  {{1},{2},{3}}  {{1},{2},{3}}
  {{1,3},{2}}    {{1},{2},{3}}
  {{1},{2},{3}}
.
  {1,3}{2,3}     {1,2}{2,3}     {1,2}{1,3}     {1,2}{1,3}{2,3}
  --------       --------       --------       --------
  {{1,2},{3}}    {{1,3},{2}}    {{1},{2,3}}    {{1},{2},{3}}
  {{1},{2},{3}}  {{1},{2},{3}}  {{1},{2},{3}}

Cf. A000110, A000569, A006125, A006126, A229048, A240936, A277203, A321979, A322064, A322065.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
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]]]];
Table[Sum[Length[stableSets[Complement[Subsets[Range[n]],Union@@Subsets/@stn],SubsetQ]],{stn,sps[Range[n]]}],{n,5}]

A322065 Number of ways to choose a stable partition of a connected antichain of sets spanning n vertices.

Original entry on oeis.org

1, 1, 1, 11, 525, 146513

Offset: 0

Gus Wiseman, Nov 25 2018

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 a(3) = 11 stable partitions. The connected antichain is on top, and below is a list of all its stable partitions.
{1,2,3}        {1,3}{2,3}     {1,2}{2,3}     {1,2}{1,3}     {1,2}{1,3}{2,3}
--------       --------       --------       --------       --------
{{1},{2,3}}    {{1,2},{3}}    {{1,3},{2}}    {{1},{2,3}}    {{1},{2},{3}}
{{1,2},{3}}    {{1},{2},{3}}  {{1},{2},{3}}  {{1},{2},{3}}
{{1,3},{2}}
{{1},{2},{3}}

Cf. A000110, A001187, A006125, A048143, A229048, A240936, A245883, A277203, A321911, A321979, A322063, A322064.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
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]]]];
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]]]]]]]]];
Table[Sum[Length[Select[stableSets[Complement[Subsets[Range[n]],Union@@Subsets/@stn],SubsetQ],And[Union@@#==Range[n],Length[csm[#]]==1]&]],{stn,sps[Range[n]]}],{n,5}]

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

Original entry on oeis.org

1, 2, 8, 60, 1009

Offset: 1

Gus Wiseman, Nov 24 2018

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.

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.

a(n) >= A321979(n).

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

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

Original entry on oeis.org

1, 1, 2, 8, 64, 1299

Offset: 0

Gus Wiseman, Nov 25 2018

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.

			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).

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.

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

A380197 Number of ways to choose a simple labeled graph on [n] and properly color the vertices using the minimum number of colors.

Original entry on oeis.org

1, 1, 3, 25, 423, 16261, 1266843, 200830225, 65750156223, 42834021462061, 55174125327583923

Offset: 0

Geoffrey Critzer, Jan 22 2025

a(n) is the chromatic polynomial evaluated at the chromatic number of a graph summed over all simple labeled graphs on [n].

Eric Weisstein's World of Mathematics, Chromatic Number.
Eric Weisstein's World of Mathematics, Chromatic Polynomial.

Cf. A006125, A372920, A229048, A084268.

(* this code only returns a(0) - a(7) using precomputed data in Mathematica 12 *)
nog = {1, 2, 4, 11, 34, 156, 1044}; Prepend[Table[Total[n!/Table[GraphData[{n, i},AutomorphismCount"], {i, 1, nog[[n]]}]*Table[GraphData[{n, i},ChromaticPolynomial"][GraphData[{n, i}, "ChromaticNumber"]], {i, 1, nog[[n]]}]], {n, 1, 7}],1]

a(8)-a(10) from Andrew Howroyd, Jan 22 2025

cp's OEIS Frontend

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

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A322063 Number of ways to choose a stable partition of an antichain of sets spanning n vertices.

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A322065 Number of ways to choose a stable partition of a connected antichain of sets spanning n vertices.

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

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

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A380197 Number of ways to choose a simple labeled graph on [n] and properly color the vertices using the minimum number of colors.

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