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A145877
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Triangle read by rows: T(n,k) is the number of permutations of [n] for which the shortest cycle length is k (1<=k<=n).
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2
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1, 1, 1, 4, 0, 2, 15, 3, 0, 6, 76, 20, 0, 0, 24, 455, 105, 40, 0, 0, 120, 3186, 714, 420, 0, 0, 0, 720, 25487, 5845, 2688, 1260, 0, 0, 0, 5040, 229384, 52632, 22400, 18144, 0, 0, 0, 0, 40320, 2293839, 525105, 223200, 151200, 72576, 0, 0, 0, 0, 362880, 25232230
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Row sums are the factorials (A000142).
Sum(T(n,k),k=2..n)=A000166(n) (the derangement numbers).
T(n,1)=A002467(n).
T(n,n)=(n-1)! (A000142).
Sum(k*T(n,k),k=1..n)=A028417(n).
For the statistic "length of the longest cycle", see A126074.
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FORMULA
| E.g.f. for column k is [1-exp(-x^k/k)]*exp[ -sum(x^j/j,j=1..k-1)]/(1-x) (Vladeta Jovovic (vladeta(AT)eunet.rs).
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EXAMPLE
| T(4,2)=3 because we have 3412=(13)(24), 2143=(12)(34) and 4321=(14)(23).
Triangle starts:
1;
1, 1;
4, 0, 2;
15, 3, 0, 6;
76, 20, 0, 0, 24;
455,105,40, 0, 0, 120;
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MAPLE
| F:=proc(k) options operator, arrow: (1-exp(-x^k/k))*exp(-(sum(x^j/j, j = 1 .. k-1)))/(1-x) end proc: for k to 16 do g[k]:= series(F(k), x=0, 16) end do: T:= proc(n, k) options operator, arrow: factorial(n)*coeff(g[k], x, n) end proc: for n to 11 do seq(T(n, k), k=1..n) end do; # yields sequence in triangular form
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MATHEMATICA
| Rest[Transpose[Table[Range[0, 16]! CoefficientList[
Series[(Exp[x^n/n] -1) (Exp[-Sum[x^k/k, {k, 1, n}]]/(1 - x)), {x, 0, 16}], x], {n, 1, 8}]]] // Grid (* Geoffrey Critzer, 4 Mar 2011 *)
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CROSSREFS
| Cf. A000142, A000166, A002467, A028417, A126074.
Sequence in context: A058054 A011352 A134895 * A057075 A200682 A156788
Adjacent sequences: A145874 A145875 A145876 * A145878 A145879 A145880
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 27 2008
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