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

1, 2, 4, 9, 23, 73, 304, 1954, 23075, 607507

Offset: 1

Author

Eric M. Schmidt, Sep 25 2013

Keywords

• nonn
• hard
• more

Comments

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

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

Examples

			From _Gus Wiseman_, Nov 24 2018: (Start)
The a(4) = 9 chromatic polynomials:
  -6x + 11x^2 - 6x^3 + x^4
  -4x +  8x^2 - 5x^3 + x^4
  -2x +  5x^2 - 4x^3 + x^4
  -3x +  6x^2 - 4x^3 + x^4
         2x^2 - 3x^3 + x^4
   -x +  3x^2 - 3x^3 + x^4
          x^2 - 2x^3 + x^4
                -x^3 + x^4
                       x^4
(End)
		

References

• F. M. Dong, K. M. Koh, and K. L. Teo. Chromatic Polynomials and Chromaticity of Graphs, World Scientific Publishing Company, 2005.

Links

• MathWorld, Chromatic Polynomial
• Eric M. Schmidt, The 304 polynomials for n=7

Crossrefs

Cf. A000088, A001187, A006125, A137568, A240936, A245883, A277203, A321911, A321994, A322011.

Programs

Mathematica

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
falling[x_,k_]:=Product[(x-i),{i,0,k-1}];
chromPoly[g_]:=Expand[Sum[falling[x,Length[stn]],{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[chromPoly/@simpleSpans[n]]],{n,5}] (* Gus Wiseman, Nov 24 2018 *)

Sage

def A229048(n):
    return len({g.chromatic_polynomial() for g in graphs(n)})

Sage

sorted({g.chromatic_polynomial() for g in graphs(n)})

Extensions

a(10) added by Eric M. Schmidt, Mar 20 2015