A203092 Triangular array read by rows. T(n,k) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1 that have exactly k components.

1, 1, 1, 1, 4, 1, 1, 18, 9, 1, 1, 116, 78, 16, 1, 1, 1060, 810, 220, 25, 1, 1, 12702, 10335, 3260, 495, 36, 1, 1, 187810, 158613, 54740, 9835, 966, 49, 1, 1, 3296120, 2854908, 1046024, 209510, 24696, 1708, 64, 1

Offset: 0

Author

Geoffrey Critzer, Dec 29 2011

Keywords

• nonn
• tabl

Comments

Row sums = A088957.

T(n,0)= 1, the empty function.

T(n,n)= 1, the identity function.

T(n,n-1)= n^2 (apparently).

Examples

			T(2,1)= 4 because there are 4 such partial functions on {1,2}: 1->1, 2->2, 1->1 2->1, 1->2 2->2,
1
1     1
1     4     1
1     18    9     1
1     116   78    16    1
1     1060  810   220   25    1
1     12702 10335 3260  495   36    1
		

Crossrefs

Cf. A088956, A144289

Programs

Mathematica

nn = 8; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
f[list_] := Select[list, # > 0 &];
Map[f, Range[0, nn]! CoefficientList[ Series[Exp[x] Exp[y t], {x, 0, nn}], {x, y}]] // Flatten

Formula

E.g.f.: exp(x)*exp(y T(x)) where T(x) is the e.g.f. for A000169.