A052145 -id:A052145 - cp's OEIS frontend

A126074 Triangle read by rows: T(n,k) is the number of permutations of n elements that have the longest cycle length k.

1, 1, 1, 1, 3, 2, 1, 9, 8, 6, 1, 25, 40, 30, 24, 1, 75, 200, 180, 144, 120, 1, 231, 980, 1260, 1008, 840, 720, 1, 763, 5152, 8820, 8064, 6720, 5760, 5040, 1, 2619, 28448, 61236, 72576, 60480, 51840, 45360, 40320, 1, 9495, 162080, 461160, 653184, 604800, 518400, 453600, 403200, 362880

Offset: 1

Dan Dima, Mar 01 2007

Sum of the n-th row is the number of all permutations of n elements: Sum_{k=1..n, T(n,k)} = n! = A000142(n) We can extend T(n,k)=0, if k<=0 or k>n.
From Peter Luschny, Mar 07 2009: (Start)
Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -1, summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A102189.
Same partition product with length statistic is A008275.
Diagonal a(A000217(n)) = rising_factorial(1,n-1), A000142(n-1) (n > 0).
Row sum is A000142. (End)
Let k in {1,2,3,...} index the family of sequences A000012, A000085, A057693, A070945, A070946, A070947, ... respectively. Column k is the k-th sequence minus its immediate predecessor. For example, T(5,3)=A057693(5)-A000085(5). - Geoffrey Critzer, May 23 2009

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   3,    2;
  1,   9,    8,    6;
  1,  25,   40,   30,   24;
  1,  75,  200,  180,  144,  120;
  1, 231,  980, 1260, 1008,  840,  720;
  1, 763, 5152, 8820, 8064, 6720, 5760, 5040;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
S. W. Golomb and P. Gaal, On the number of permutations of n objects with greatest cycle length k, Adv. in Appl. Math., 20(1), 1998, 98-107.
IBM Research, Ponder This, December 2006.
Peter Luschny, Counting with Partitions. [From _Peter Luschny_, Mar 07 2009]
Peter Luschny, Generalized Stirling_1 Triangles. [From _Peter Luschny_, Mar 07 2009]
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.

Cf. A000142.
Cf. A071007, A080510, A028418.
Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395. - Peter Luschny, Mar 07 2009
T(2n,n) gives A052145 (for n>0). - Alois P. Heinz, Apr 21 2017

A:= proc(n,k) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*A(n-j,k), j=1..k)))
    end:
T:= (n, k)-> A(n, k) -A(n, k-1):
seq(seq(T(n,k), k=1..n), n=1..10);  # Alois P. Heinz, Feb 11 2013

Table[CoefficientList[ Series[(Exp[x^m/m] - 1) Exp[Sum[x^k/k, {k, 1, m - 1}]], {x, 0, 8}], x]*Table[n!, {n, 0, 8}], {m, 1, 8}] // Transpose // Grid (* Geoffrey Critzer, May 23 2009 *)

def A126074(n, k):
    f = factorial(n)
    P = Partitions(n, max_part=k, inner=[k])
    return sum(f // p.aut() for p in P)
for n in (1..9): print([A126074(n,k) for k in (1..n)]) # Peter Luschny, Apr 17 2016

T(n,1) = 1.
T(n,2) = n! * Sum_{k=1..[n/2]} 1/(k! * (2!)^k * (n-2*k)!).
T(n,k) = n!/k * (1-1/(n-k)-...-1/(k+1)-1/2k), if n/3 < k <= n/2.
T(n,k) = n!/k, if n/2 < k <= n.
T(n,n) = (n-1)! = A000142(n-1).
E.g.f. for k-th column: exp(-x^k*LerchPhi(x,1,k))*(exp(x^k/k)-1)/(1-x). - Vladeta Jovovic, Mar 03 2007
From Peter Luschny, Mar 07 2009: (Start)
T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n+1). (End)
Sum_{k=1..n} k * T(n,k) = A028418(n). - Alois P. Heinz, May 17 2016

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

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A126074 Triangle read by rows: T(n,k) is the number of permutations of n elements that have the longest cycle length k.

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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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A256880 n*n!/round(n/2).

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