A027617 -id:A027617 - cp's OEIS frontend

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

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

A071007 Number of permutations in the symmetric group S_n such that the maximal cycle has length exactly 3.

Original entry on oeis.org

0, 0, 0, 2, 8, 40, 200, 980, 5152, 28448, 162080, 979000, 6179360, 40575392, 279199648, 1997406320, 14825619200, 114365751040, 912510870272, 7521873125408, 64045101880960, 561615674345600, 5067769601121920, 47023128008540992, 447820056115824128

Offset: 0

Sharon Sela (sharonsela(AT)hotmail.com), May 19 2002

nonn

E.g.f.: exp( x + (x^2)/2 + (x^3)/3 ) - exp( x + (x^2)/2 ).

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A057693, A000085, A027617.

Column k=3 of A126074.

nn=20;Range[0,nn]!CoefficientList[Series[Exp[x+x^2/2+x^3/3]-Exp[x+x^2/2],{x,0,nn}],x]  (* Geoffrey Critzer, Jan 23 2013 *)

for(n=0,25,print1(polcoeff(serlaplace(exp(x+x^2/2+x^3/3)-exp(x+x^2/2)),n)","))

a(n) = A057693(n) - A000085(n).

More terms from Ralf Stephan, Apr 09 2003

cp's OEIS Frontend

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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