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A277203 Number of distinct chromatic symmetric functions realizable by a graph on n vertices. 17
1, 2, 4, 11, 33, 146, 939, 10932 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 augmented monomial symmetric functions (see A321895). - Gus Wiseman, Nov 21 2018

LINKS

Table of n, a(n) for n=1..8.

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.

EXAMPLE

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)

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A123435 A123409 A123472 * A123905 A123442 A123405

Adjacent sequences:  A277200 A277201 A277202 * A277204 A277205 A277206

KEYWORD

nonn,more

AUTHOR

Sam Heil and Caleb Ji, Oct 04 2016

STATUS

approved

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Last modified December 9 19:51 EST 2019. Contains 329879 sequences. (Running on oeis4.)