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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

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

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

Cf. A000088, A000110, A000569, A006125, A321750, A321751, A321895, A321911.

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).

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.

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.

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.

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.

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)

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 *)

Cf. A000088, A000110, A000569, A006125, A321750, A321751, A321895, A321911.

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