OFFSET
1,2
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
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
EXAMPLE
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)
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 *)
PROG
(Sage)
def A229048(n):
return len({g.chromatic_polynomial() for g in graphs(n)})
(Sage) sorted({g.chromatic_polynomial() for g in graphs(n)})
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Eric M. Schmidt, Sep 25 2013
EXTENSIONS
a(10) added by Eric M. Schmidt, Mar 20 2015
STATUS
approved