A029572 -id:A029572 - cp's OEIS frontend

A029588 a(n) = A029572(n) / 24.

0, 0, 0, 0, 0, 1, 6, 42, 336, 3024, 27216, 299376, 3592512, 46702656, 653837184, 9880206336, 158083301376, 2687416123392, 48373490221056, 919096314200064, 18375168222867456, 385878532680216576, 8489327718964764672, 195254537536189587456, 4686108900868550098944

Offset: 0

N. J. A. Sloane

nonn

Cf. A029572.

A293211 Triangle T(n,k) is the number of permutations on n elements with at least one k-cycle for 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 4, 3, 2, 15, 9, 8, 6, 76, 45, 40, 30, 24, 455, 285, 200, 180, 144, 120, 3186, 1995, 1400, 1260, 1008, 840, 720, 25487, 15855, 11200, 8820, 8064, 6720, 5760, 5040, 229384, 142695, 103040, 79380, 72576, 60480, 51840, 45360, 40320, 2293839, 1427895, 1030400, 793800, 653184, 604800, 518400, 453600, 403200, 362880

Offset: 1

Dennis P. Walsh, Oct 02 2017

T(n,k) is equivalent to n! minus the number of permutations on n elements with zero k-cycles (sequence A122974).

			T(n,k) (the first 8 rows):
:     1;
:     1,     1;
:     4,     3,     2;
:    15,     9,     8,    6;
:    76,    45,    40,   30,   24;
:   455,   285,   200,  180,  144,  120;
:  3186,  1995,  1400, 1260, 1008,  840,  720;
: 25487, 15855, 11200, 8820, 8064, 6720, 5760, 5040;
  ...
T(4,3)=8 since there are exactly 8 permutations on {1,2,3,4} with at least one 3-cycle: (1)(234), (1)(243), (2)(134), (2)(143), (3)(124), (3)(142), (4)(123), and (4)(132).

Dennis P. Walsh, The number of permutations with no k-cycles

Columns k=1-10 give: A002467, A027616, A027617, A029571, A029572, A029573, A029574, A029575, A029576, A029577.
Row sums give A132961.
T(n,n) gives A000142(n-1) for n>0.
T(2n,n) gives A052145.
Cf. A122974, A126074.

T:=(n,k)->n!*sum((-1)^(j+1)*(1/k)^j/j!,j=1..floor(n/k)); seq(seq(T(n,k),k=1..n),n=1..10);

Table[n!*Sum[(-1)^(j + 1)*(1/k)^j/j!, {j, Floor[n/k]}], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Oct 02 2017 *)

T(n,k) = n! * Sum_{j=1..floor(n/k)} (-1)^(j+1)*(1/k)^j/j!.
T(n,k) = n! - A122974(n,k).
E.g.f. of column k: (1-exp(-x^k/k))/(1-x). - Alois P. Heinz, Oct 11 2017

cp's OEIS Frontend

A029588 a(n) = A029572(n) / 24.

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A293211 Triangle T(n,k) is the number of permutations on n elements with at least one k-cycle for 1 <= k <= n.

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