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
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
Programs
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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
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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!.