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A329655
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Square array read by antidiagonals: T(n,k) is the number of relations between set A with n elements and set B with k elements that are both right unique and left unique.
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1
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1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 33, 20, 5, 6, 30, 72, 72, 30, 6, 7, 42, 135, 208, 135, 42, 7, 8, 56, 228, 500, 500, 228, 56, 8, 9, 72, 357, 1044, 1545, 1044, 357, 72, 9, 10, 90, 528, 1960, 4050, 4050, 1960, 528, 90, 10, 11, 110, 747, 3392, 9275, 13326, 9275, 3392, 747, 110, 11
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OFFSET
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1,2
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COMMENTS
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A relation R between set A with n elements and set B with k elements is a subset of the Cartesian product A x B. A relation R is right unique if (a, b1) in R and (a,b2) in R implies b1=b2. A relation R is left unique if (a1,b) in R and (a2,b) in R implies a1=a2.
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LINKS
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FORMULA
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T(n,k) = Sum_{j=1..k} binomial(n,j)*binomial(k,j)*j!.
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EXAMPLE
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The symmetric array T(n,k) begins:
1, 2, 3, 4, 5, 6, 7, 8, 9, ...
2, 6, 12, 20, 30, 42, 56, 72, 90, ...
3, 12, 33, 72, 135, 228, 357, 528, 747, ...
4, 20, 72, 208, 500, 1044, 1960, 3392, 5508, ...
5, 30, 135, 500, 1545, 4050, 9275, 19080, 36045, ...
6, 42, 228, 1044, 4050, 13326, 37632, 93288, 207774, ...
7, 56, 357, 1960, 9275, 37632, 130921, 394352, 1047375, ...
8, 72, 528, 3392, 19080, 93288, 394352, 1441728, 4596552, ...
9, 90, 747, 5508, 36045, 207774, 1047375, 4596552, 17572113, ...
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MAPLE
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T:= (n, k)-> value(Sum(binomial(n, j)*binomial(k, j)*j!, j=1..k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);
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MATHEMATICA
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T[n_, k_] := Sum[Binomial[n, j] * Binomial[k, j] * j!, {j, 1, k}]; Table[T[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)
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PROG
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(MuPAD) T:=(n, k)->_plus (binomial(n, j)*binomial(k, j)* j! $ j=1..k):
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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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