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A223887
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Number of 4-colored labeled graphs on n vertices.
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9
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1, 4, 28, 340, 7108, 254404, 15531268, 1613235460, 284556079108, 85107970698244, 43112647751430148, 36955277740855136260, 53562598422461559373828, 131186989945696839128432644, 542676256323680030599454982148
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OFFSET
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0,2
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COMMENTS
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A simple graph G is a k-colorable graph if it is possible to assign one of k' <= k colors to each vertex of G so that no two adjacent vertices receive the same color. Such an assignment of colors is called a coloring function for the graph.
A k-colored graph is a k-colorable graph together with its coloring function. This sequence gives the number of labeled 4-colored graphs on n vertices. An example is given below.
See A047863 for labeled 2-colored graphs on n vertices and A191371 for labeled 3-colored graphs on n vertices. See A076316 for labeled 4-colorable graphs on n vertices and A224068 for the count of labeled graphs colored using exactly 4 colors.
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REFERENCES
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F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973.
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LINKS
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FORMULA
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a(n) = sum {k = 0..n} binomial(n,k)*2^(k*(n-k))*b(k)*b(n-k), where b(n) := sum {k = 0..n} binomial(n,k)*2^(k*(n-k)).
Let E(x) = sum {n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is E(x)^4 = sum {n >= 0} a(n)*x^n/(n!*2^C(n,2)) = 1 + 4*x + 28*x^2/(2!*2) + 340*x^3/(3!*2^3) + .... In general, for k = 1, 2, ..., E(x)^k is a generating function for labeled k-colored graphs (see Stanley).
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EXAMPLE
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a(2) = 28: There are two labeled 4-colorable graphs on 2 nodes, namely
A) 1 2 B) 1 2
o o o----o
Using 4 colors there are 16 ways to color the graph of type A and 4*3 = 12 ways to color the graph of type B so that adjacent vertices do not share the same color. Thus there are in total 28 labeled 4-colored graphs on 2 vertices.
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PROG
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(PARI)
N=66; x='x+O('x^N);
E=sum(n=0, N, x^n/(n!*2^binomial(n, 2)) );
tgf=E^4; v=concat(Vec(tgf));
v=vector(#v, n, v[n] * (n-1)! * 2^((n-1)*(n-2)/2) )
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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