A321751 -id:A321751 - cp's OEIS frontend

A277203 Number of distinct chromatic symmetric functions realizable by a graph on n vertices.

1, 2, 4, 11, 33, 146, 939, 10932

Offset: 1

Sam Heil and Caleb Ji, Oct 04 2016

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). - Gus Wiseman, Nov 21 2018

			For n = 3, under the p basis, the CSF's are: p_{1, 1, 1}, p_{1, 1, 1} - p_{2, 1}, p_{1, 1, 1} - 2p_{2, 1} + p_{3}, p_{1, 1, 1} - 3p_{2, 1} + 2p_{3}.
From _Gus Wiseman_, Nov 21 2018: (Start)
The a(4) = 11 chromatic symmetric functions (m is the augmented monomial symmetric function basis):
                                     m(1111)
                            m(211) + m(1111)
                           2m(211) + m(1111)
          m(22) +          2m(211) + m(1111)
                           3m(211) + m(1111)
          m(22) +          3m(211) + m(1111)
                   m(31) + 3m(211) + m(1111)
         2m(22) +          4m(211) + m(1111)
          m(22) +  m(31) + 4m(211) + m(1111)
         2m(22) + 2m(31) + 5m(211) + m(1111)
  m(4) + 3m(22) + 4m(31) + 6m(211) + m(1111)
(End)

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.

Cf. A000088, A000110, A000569, A006125, A229048, A240936, A245883, A277204, A277205, A321750, A321751, A321895, A321911.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
chromSF[g_]:=Sum[m[Sort[Length/@stn,Greater]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}];
simpleSpans[n_]:=simpleSpans[n]=If[n==0,{{}},Union@@Table[If[#=={},Union[ine,{{n}}],Union[Complement[ine,List/@#],{#,n}&/@#]]&/@Subsets[Range[n-1]],{ine,simpleSpans[n-1]}]];
Table[Length[Union[chromSF/@simpleSpans[n]]],{n,6}] (* Gus Wiseman, Nov 21 2018 *)

A321911 Number of distinct chromatic symmetric functions of simple connected graphs with n vertices.

Original entry on oeis.org

1, 1, 2, 6, 20, 103, 759

Offset: 1

Gus Wiseman, Nov 21 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 p 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).

			The a(4) = 6 connected chromatic symmetric functions (m is the augmented monomial symmetric function basis):
                    m(1111)
           m(211) + m(1111)
          2m(211) + m(1111)
  m(22) + 2m(211) + m(1111)
  m(22) + 3m(211) + m(1111)
  m(31) + 3m(211) + m(1111)

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.
Gus Wiseman, A partition of connected graphs, Electronic J. Combinatorics 12, N1 (2005), 8pp. arXiv:math/0505155 [math.CO].

First differs from A079457 at a(7) = 759, A079457(7) = 706.

Cf. A000110, A001349, A001187, A125702, A147878, A191970, A229048, A240936, A245883, A277203, A317631, A317632, A320921, A321750, A321751, A321895.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
chromSF[g_]:=Sum[m[Sort[Length/@stn,Greater]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}];
simpleSpans[n_]:=simpleSpans[n]=If[n==0,{{}},Union@@Table[If[#=={},Union[ine,{{n}}],Union[Complement[ine,List/@#],{#,n}&/@#]]&/@Subsets[Range[n-1]],{ine,simpleSpans[n-1]}]];
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[Length[Union[chromSF/@Select[simpleSpans[n],Length[csm[#]]==1&]]],{n,6}]

A321750 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in p(u), where H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 3, 6, 1, 2, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 6, 4, 12, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 20 2018

Row n has length A000041(A056239(n)).

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			Triangle begins:
   1
   1
   1   0
   1   2
   1   0   0
   1   1   0
   1   0   0   0   0
   1   3   6
   1   2   0   0   0
   1   0   1   0   0
   1   0   0   0   0   0   0
   1   2   2   2   0
   1   0   0   0   0   0   0   0   0   0   0
   1   1   0   0   0   0   0
   1   0   1   0   0   0   0
   1   6   4  12  24
   1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
   1   1   2   2   0   0   0
For example, row 18 gives: p(221) = m(5) + 2m(32) + m(41) + 2m(221).

Wikipedia, Symmetric polynomial

Row sums are A321751.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319182, A319191, A321742-A321765.

A321889 Sum of coefficients of forgotten symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, -1, 3, 1, -2, -1, 10, 3, 2, 1, -7, -1, -2, -2, 47, 1, 6

Offset: 1

Gus Wiseman, Nov 20 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sum of coefficients of p(211) = -f(4) - 2f(22) - 2f(31) - 2f(211) is a(12) = -7.

Wikipedia, Symmetric polynomial

Row sums of A321888. An unsigned version is A321751.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319182, A319191, A319193, A321742-A321765.

cp's OEIS Frontend

A277203 Number of distinct chromatic symmetric functions realizable by a graph on n vertices.

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A321911 Number of distinct chromatic symmetric functions of simple connected graphs with n vertices.

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A321750 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of m(v) in p(u), where H is Heinz number, m is monomial symmetric functions, and p is power sum symmetric functions.

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A321889 Sum of coefficients of forgotten symmetric functions in the power sum symmetric function of the integer partition with Heinz number n.

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