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A201013
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Triangular array read by rows: T(n,k) is the number of 2-regular labeled graphs on n nodes that have exactly k connected components (cycles); n>=3, 1<=k<=floor(n/3).
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1
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1, 3, 12, 60, 10, 360, 105, 2520, 987, 20160, 9576, 280, 181440, 99144, 6300, 1814400, 1104840, 107415, 19958400, 13262040, 1708245, 15400, 239500800, 171119520, 27042444, 600600, 3113510400, 2366076960, 437729292, 16186170, 43589145600, 34941291840, 7335055728, 382056675, 1401400
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OFFSET
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3,2
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COMMENTS
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A 2-regular labeled graph is a simple labeled graph such that every vertex has degree 2.
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LINKS
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FORMULA
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E.g.f.: exp(-xy/2-x^2y/4)/(1-x)^(y/2).
T(n,1) = (n-1)!/2, T(n,k) = Sum_{j=3..n-3} C(n-1,j-1)*T(j,1)*T(n-j,k-1) for k>1. - Alois P. Heinz, Nov 25 2011
Sum_{k=1..floor(n/3)} T(n,k)*2^k = A038205(n) the number of permutations with minimum cycle size of 3. - Geoffrey Critzer, Nov 05 2012
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EXAMPLE
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1;
3;
12;
60, 10;
360, 105;
2520, 987;
20160, 9576, 280;
181440, 99144, 6300;
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MAPLE
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T:= proc(n, k) option remember; `if`(k=1, (n-1)!/2,
add(binomial(n-1, j-1) *T(j, 1) *T(n-j, k-1), j=3..n-3))
end:
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MATHEMATICA
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f[list_]:=Select[list, #>0&]; Flatten[Drop[Map[f, a = Log[1/(1 - x)]/2 - x/2 - x^2/4; Range[0, 20]! CoefficientList[Series[Exp[y a], {x, 0, 20}], {x, y}]], 3]]
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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