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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. 3
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, 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

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

CROSSREFS

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

Sequence in context: A061312 A019130 A245348 * A239799 A305235 A120011

Adjacent sequences:  A174548 A174549 A174550 * A174552 A174553 A174554

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Mar 22 2010

STATUS

approved

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Last modified October 22 07:06 EDT 2021. Contains 348160 sequences. (Running on oeis4.)