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A174551
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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.
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3
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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T(n,k) = Sum_{i=0..k} (-1)^i C(k,i) (n-i)^n; T(n,0) = n^n; T(n,n) = n!.
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EXAMPLE
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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;
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MAPLE
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T:= (n, k)-> add((-1)^i*binomial(k, i)*(n-i)^n, i=0..k):
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MATHEMATICA
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Table[Table[ Sum[(-1)^i Binomial[k, i] (n - i)^n, {i, 0, k}], {k, 0, n}], {n, 0, 7}] // Grid
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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