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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.

%I A174551 #7 Dec 26 2012 20:13:46
%S A174551 1,1,1,4,3,2,27,19,12,6,256,175,110,60,24,3125,2101,1320,750,360,120,
%T A174551 46656,31031,19502,11340,5880,2520,720,823543,543607,341796,201726,
%U A174551 109200,52080,20160,5040,16777216,11012415,6927230,4131036,2298744,1164240,514080,181440,40320
%N 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.
%H A174551 Alois P. Heinz, <a href="/A174551/b174551.txt">Rows n = 0..140, flattened</a>
%F A174551 T(n,k) = Sum_{i=0..k} (-1)^i C(k,i) (n-i)^n; T(n,0) = n^n; T(n,n) = n!.
%e A174551 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}.
%e A174551 The triangle begins:
%e A174551 1;
%e A174551 1, 1;
%e A174551 4, 3, 2;
%e A174551 27, 19, 12, 6;
%e A174551 256, 175, 110, 60, 24;
%e A174551 3125, 2101, 1320, 750, 360, 120;
%e A174551 46656, 31031, 19502, 11340, 5880, 2520, 720;
%e A174551 823543, 543607, 341796, 201726, 109200, 52080, 20160, 5040;
%p A174551 T:= (n,k)-> add((-1)^i*binomial(k, i)*(n-i)^n, i=0..k):
%p A174551 seq(seq(T(n,k), k=0..n), n=0..10);  # _Alois P. Heinz_, Dec 26 2012
%t A174551 Table[Table[ Sum[(-1)^i Binomial[k, i] (n - i)^n, {i, 0, k}], {k, 0, n}], {n, 0, 7}] // Grid
%Y A174551 Columns are: A045531, A126778, A126779, A126780, A126232, A126781
%K A174551 nonn,tabl
%O A174551 0,4
%A A174551 _Geoffrey Critzer_, Mar 22 2010