A174551 Triangular array T(n,k): functions f:{1,2,...,n}-> {1,2,...,n} such that each of k fixed (but arbitrary) elements are in the image of f.

1, 1, 1, 4, 3, 2, 27, 19, 12, 6, 256, 175, 110, 60, 24, 3125, 2101, 1320, 750, 360, 120, 46656, 31031, 19502, 11340, 5880, 2520, 720, 823543, 543607, 341796, 201726, 109200, 52080, 20160, 5040, 16777216, 11012415, 6927230, 4131036, 2298744, 1164240, 514080, 181440, 40320

Offset: 0

Author

Geoffrey Critzer, Mar 22 2010

Keywords

• nonn
• tabl

Examples

			Letting the k arbitrary elements be {1,2}, T(3,2) = 12 because there are 12 such functions from [3] into [3]. {1, 1, 2}, {1, 2, 1}, {1, 2, 2}, {1, 2, 3}, {1, 3, 2}, {2, 1, 1}, {2,1, 2}, {2, 1, 3}, {2, 2, 1}, {2, 3, 1}, {3, 1, 2}, {3, 2, 1}.
The triangle begins:
1;
1, 1;
4, 3, 2;
27, 19, 12, 6;
256, 175, 110, 60, 24;
3125, 2101, 1320, 750, 360, 120;
46656, 31031, 19502, 11340, 5880, 2520, 720;
823543, 543607, 341796, 201726, 109200, 52080, 20160, 5040;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns are: A045531, A126778, A126779, A126780, A126232, A126781

Programs

Maple

T:= (n,k)-> add((-1)^i*binomial(k, i)*(n-i)^n, i=0..k):
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 26 2012

Mathematica

Table[Table[ Sum[(-1)^i Binomial[k, i] (n - i)^n, {i, 0, k}], {k, 0, n}], {n, 0, 7}] // Grid

Formula

T(n,k) = Sum_{i=0..k} (-1)^i C(k,i) (n-i)^n; T(n,0) = n^n; T(n,n) = n!.