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A329940
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Square array read by antidiagonals upwards: T(n,k) is the number of right unique relations between set A with n elements and set B with k elements.
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1
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1, 3, 2, 7, 8, 3, 15, 26, 15, 4, 31, 80, 63, 24, 5, 63, 242, 255, 124, 35, 6, 127, 728, 1023, 624, 215, 48, 7, 255, 2186, 4095, 3124, 1295, 342, 63, 8, 511, 6560, 16383, 15624, 7775, 2400, 511, 80, 9, 1023, 19682, 65535, 78124, 46655, 16806, 4095, 728, 99, 10
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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. T(n,k) is the number of right unique relations and T(k,n) is the number of left unique relations: 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) = (k+1)^n - 1.
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EXAMPLE
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T(n,k) begins:
1, 2, 3, 4, 5, 6, 7, 8, ...
3, 8, 15, 24, 35, 48, 63, 80, ...
7, 26, 63, 124, 215, 342, 511, 728, ...
15, 80, 255, 624, 1295, 2400, 4095, 6560, ...
31, 242, 1023, 3124, 7775, 16806, 32767, 59048, ...
63, 728, 4095, 15624, 46655, 117648, 262143, 531440, ...
127, 2186, 16383, 78124, 279935, 823542, 2097151, 4782968, ...
255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720, ...
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MAPLE
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T:= (n, k)-> (k+1)^n-1:
seq(seq(T(1+d-k, k), k=1..d), d=1..12);
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MATHEMATICA
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T[n_, k_] := (k + 1)^n - 1; Table[T[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)
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PROG
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(MuPAD) T:=(n, k)->(k+1)^n-1:
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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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