A211603 Triangular array read by rows: T(n,k) is the number of n-permutations that are pure cycles having exactly k fixed points; n>=2, 0<=k<=n-2.

1, 2, 3, 6, 8, 6, 24, 30, 20, 10, 120, 144, 90, 40, 15, 720, 840, 504, 210, 70, 21, 5040, 5760, 3360, 1344, 420, 112, 28, 40320, 45360, 25920, 10080, 3024, 756, 168, 36, 362880, 403200, 226800, 86400, 25200, 6048, 1260, 240, 45, 3628800, 3991680, 2217600, 831600, 237600, 55440, 11088, 1980, 330, 55

Offset: 2

Author

Geoffrey Critzer, Feb 10 2013

Keywords

• nonn
• tabl
• easy

Comments

Equivalently, T(n,k) is the number of n-permutations that are pure cycles of length n-k.

Row sums = A006231.

With a different row and column indexing, this triangle equals the infinitesimal generator of A008290. Equals the unsigned version of A238363, omitting its main diagonal. See also A092271. - Peter Bala, Feb 13 2017

Examples

			T(3,1) = 3 because we have (1)(2,3), (2)(1,3), (3)(1,2).
1;
2, 3;
6, 8, 6;
24, 30, 20, 10;
120, 144, 90, 40, 15;
720, 840, 504, 210, 70, 21;
5040, 5760, 3360, 1344, 420, 112, 28;
40320, 45360, 25920, 10080, 3024, 756, 168, 36;
362880, 403200, 226800, 86400, 25200, 6048, 1260, 240, 45;
		

Links

• Alois P. Heinz, Rows n = 2..150, flattened

Crossrefs

Cf. A006231 (row sums), A008290, A092271, A111492, A238363.

Programs

Maple

T:= (n, k)-> binomial(n, k)*(n-k-1)!:
seq(seq(T(n,k), k=0..n-2), n=2..12);  # Alois P. Heinz, Feb 10 2013

Mathematica

nn=10;f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[ Series[Exp[y x](Log[1/(1-x)]-x),{x,0,nn}],{x,y}]]//Grid

Formula

E.g.f.: exp(y*x)*(log(1/(1-x))-x).

T(n,k) = C(n,k)*(n-k-1)!. - Alois P. Heinz, Feb 10 2013

T(n,k) = A111492(n,n-k). - R. J. Mathar, Mar 07 2013