A174551 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

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

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!.

EXAMPLE

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;

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;

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