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A145877 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). 5
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; text; internal format)
OFFSET

1,4

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.

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

E.g.f. for column k is (1-exp(-x^k/k))*exp( -sum(j=1..k-1, x^j/j ) ) / (1-x). - Vladeta Jovovic

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;

:  3186,  714,  420,    0,  0,   0, 720;

: 25487, 5845, 2688, 1260,  0,   0,   0, 5040;

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

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, Mar 04 2011 *)

CROSSREFS

Cf. A000142, A000166, A002467, A028417, A126074.

T(2n,n) gives A110468(n-1) (for n>0). - Alois P. Heinz, Apr 21 2017

Sequence in context: A208748 A134895 A318468 * A283572 A057075 A281653

Adjacent sequences:  A145874 A145875 A145876 * A145878 A145879 A145880

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Oct 27 2008

STATUS

approved

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Last modified December 6 19:22 EST 2019. Contains 329809 sequences. (Running on oeis4.)