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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.
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%I #26 Feb 14 2017 02:38:13

%S 1,2,3,6,8,6,24,30,20,10,120,144,90,40,15,720,840,504,210,70,21,5040,

%T 5760,3360,1344,420,112,28,40320,45360,25920,10080,3024,756,168,36,

%U 362880,403200,226800,86400,25200,6048,1260,240,45,3628800,3991680,2217600,831600,237600,55440,11088,1980,330,55

%N 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.

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

%C Row sums = A006231.

%C 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

%H Alois P. Heinz, <a href="/A211603/b211603.txt">Rows n = 2..150, flattened</a>

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

%F T(n,k) = C(n,k)*(n-k-1)!. - _Alois P. Heinz_, Feb 10 2013

%F T(n,k) = A111492(n,n-k). - _R. J. Mathar_, Mar 07 2013

%e T(3,1) = 3 because we have (1)(2,3), (2)(1,3), (3)(1,2).

%e 1;

%e 2, 3;

%e 6, 8, 6;

%e 24, 30, 20, 10;

%e 120, 144, 90, 40, 15;

%e 720, 840, 504, 210, 70, 21;

%e 5040, 5760, 3360, 1344, 420, 112, 28;

%e 40320, 45360, 25920, 10080, 3024, 756, 168, 36;

%e 362880, 403200, 226800, 86400, 25200, 6048, 1260, 240, 45;

%p T:= (n, k)-> binomial(n, k)*(n-k-1)!:

%p seq(seq(T(n,k), k=0..n-2), n=2..12); # _Alois P. Heinz_, Feb 10 2013

%t 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

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

%K nonn,tabl,easy

%O 2,2

%A _Geoffrey Critzer_, Feb 10 2013