A348075 Triangular array read by rows: T(n,k) is the number of derangements whose shortest cycle has exactly k nodes; n >= 1, 1 <= k <= n.

0, 0, 1, 0, 0, 2, 0, 3, 0, 6, 0, 20, 0, 0, 24, 0, 105, 40, 0, 0, 120, 0, 714, 420, 0, 0, 0, 720, 0, 5845, 2688, 1260, 0, 0, 0, 5040, 0, 52632, 22400, 18144, 0, 0, 0, 0, 40320, 0, 525105, 223200, 151200, 72576, 0, 0, 0, 0, 362880, 0, 5777090, 2522520, 1425600, 1330560, 0, 0, 0, 0, 0, 3628800

Offset: 1

Author

Steven Finch, Sep 27 2021

Keywords

• nonn
• tabl

Comments

For the statistic "length of the longest cycle", see A211871.

Examples

			Triangle begins:
  0;
  0,     1;
  0,     0,     2;
  0,     3,     0,     6;
  0,    20,     0,     0,   24;
  0,   105,    40,     0,    0,  120;
  0,   714,   420,     0,    0,    0,  720;
  0,  5845,  2688,  1260,    0,    0,    0, 5040;
  0, 52632, 22400, 18144,    0,    0,    0,    0, 40320;
  ...
		

Links

• Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
• D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.

Crossrefs

Row sums give A000166, n >= 1.

Right border gives A000142.

Column 1 gives A000004.

Column 2 gives A158243.

Cf. A145877, A211871.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, x^m, add((j-1)!*
      b(n-j, min(m, j))*binomial(n-1, j-1), j=2..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..12);  # Alois P. Heinz, Sep 27 2021

Mathematica

b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[(j - 1)!*
     b[n - j, Min[m, j]]*Binomial[n - 1, j - 1], {j, 2, n}]];
T[n_] := If[n == 1, {0}, CoefficientList[b[n, n], x] // Rest];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Oct 03 2021, after Alois P. Heinz *)

Formula

T(n,n) = A000142(n-1), n >= 2.

T(n,2) = A158243(n), n >= 2.

T(n,k) = A145877(n,k) for k >= 2.