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

0,8

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.

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

Example

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

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 *)

Crossrefs

Row sums are A001710 for n > 1.

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

Cf. A000387, A008290, A055137, A145224.

Sequence in context: A062688 A067181 A321429 * A332442 A061480 A220692

Adjacent sequences: A145222 A145223 A145224 * A145226 A145227 A145228

Keyword

nonn,tabl

Author

Abdullahi Umar, Oct 10 2008

Status

approved