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A229048 Number of different chromatic polynomials of a simple graph with n nodes.

%I A229048 #66 Feb 16 2025 08:33:20
%S A229048 1,2,4,9,23,73,304,1954,23075,607507
%N A229048 Number of different chromatic polynomials of a simple graph with n nodes.
%C A229048 Partial sums of A245883. This may be proved using two facts: (i) the number of connected components of a graph is the multiplicity of the root 0 of the chromatic polynomial (thus the chromatic polynomial determines whether a graph is connected) and (ii) a disconnected graph is chromatically equivalent to some graph with an isolated vertex. The first statement is well known. For the latter statement, see p. 65 of [Dong]. - _Eric M. Schmidt_, Mar 20 2015
%C A229048 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 polynomial is given by chi_G(x) = Sum_p (x)_k, where the sum is over all stable partitions of G, k is the length (number of blocks) of p, and (x)_k is the falling factorial x(x-1)(x-2)...(x-k+1). - _Gus Wiseman_, Nov 24 2018
%D A229048 F. M. Dong, K. M. Koh, and K. L. Teo. Chromatic Polynomials and Chromaticity of Graphs, World Scientific Publishing Company, 2005.
%H A229048 MathWorld, <a href="https://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>
%H A229048 Eric M. Schmidt, <a href="/A229048/a229048_1.txt">The 304 polynomials for n=7</a>
%e A229048 From _Gus Wiseman_, Nov 24 2018: (Start)
%e A229048 The a(4) = 9 chromatic polynomials:
%e A229048   -6x + 11x^2 - 6x^3 + x^4
%e A229048   -4x +  8x^2 - 5x^3 + x^4
%e A229048   -2x +  5x^2 - 4x^3 + x^4
%e A229048   -3x +  6x^2 - 4x^3 + x^4
%e A229048          2x^2 - 3x^3 + x^4
%e A229048    -x +  3x^2 - 3x^3 + x^4
%e A229048           x^2 - 2x^3 + x^4
%e A229048                 -x^3 + x^4
%e A229048                        x^4
%e A229048 (End)
%t A229048 spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];
%t A229048 falling[x_,k_]:=Product[(x-i),{i,0,k-1}];
%t A229048 chromPoly[g_]:=Expand[Sum[falling[x,Length[stn]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}]];
%t A229048 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]}]];
%t A229048 Table[Length[Union[chromPoly/@simpleSpans[n]]],{n,5}] (* _Gus Wiseman_, Nov 24 2018 *)
%o A229048 (Sage)
%o A229048 def A229048(n):
%o A229048     return len({g.chromatic_polynomial() for g in graphs(n)})
%o A229048 (Sage) sorted({g.chromatic_polynomial() for g in graphs(n)})
%Y A229048 Cf. A000088, A001187, A006125, A137568, A240936, A245883, A277203, A321911, A321994, A322011.
%K A229048 nonn,hard,more
%O A229048 1,2
%A A229048 _Eric M. Schmidt_, Sep 25 2013
%E A229048 a(10) added by _Eric M. Schmidt_, Mar 20 2015