A145225 Triangle read by rows: T(n,k) is the number of odd permutations (of an n-set) with exactly k fixed points.

0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 6, 0, 6, 0, 0, 20, 30, 0, 10, 0, 0, 135, 120, 90, 0, 15, 0, 0, 924, 945, 420, 210, 0, 21, 0, 0, 7420, 7392, 3780, 1120, 420, 0, 28, 0, 0, 66744, 66780, 33264, 11340, 2520, 756, 0, 36, 0, 0

Offset: 0

Author

Abdullahi Umar, Oct 10 2008

Keywords

• nonn
• tabl

Examples

			Triangle starts:
   0;
   0,  0;
   1,  0, 0;
   0,  3, 0,  0;
   6,  0, 6,  0, 0;
  20, 30, 0, 10, 0, 0;
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
• Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830.

Crossrefs

Row sums are A001710 for n > 1.

Columns k=0..2 are A000387, A145222, A145223.

Cf. A000387, A008290, A055137, A145224.

Programs

Maple

A145225 := proc(n,k)
    binomial(n,k)*A000387(n-k) ; # re-use code of A000387
end proc:
seq(seq(A145225(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jul 06 2023

Mathematica

A145225[n_, k_] := Binomial[n, k]*Binomial[n - k, 2]*Subfactorial[n - k - 2];
Table[A145225[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 31 2025 *)

Formula

T(n,k) = C(n,k) * A000387(n-k).

E.g.f.: x^(k+2) * exp(-x) / (2*k!*(1-x)).

T(n,k) + A145224(n,k) = A008290(n,k). - R. J. Mathar, Jul 06 2023

T(n,k) = (A008290(n,k) - A055137(n,k)) / 2. - Julian Hatfield Iacoponi, Aug 08 2024