Roy S. Freedman has authored 3 sequences.
A329943
Square array read by antidiagonals: T(n,k) is the number of right total relations between set A with n elements and set B with k elements.
Original entry on oeis.org
1, 3, 1, 7, 9, 1, 15, 49, 27, 1, 31, 225, 343, 81, 1, 63, 961, 3375, 2401, 243, 1, 127, 3969, 29791, 50625, 16807, 729, 1, 255, 16129, 250047, 923521, 759375, 117649, 2187, 1, 511, 65025, 2048383, 15752961, 28629151, 11390625, 823543, 6561, 1
Offset: 1
T(n,k) begins, for 1 <= n,k <= 9:
1, 1, 1, 1, 1, 1, 1
3, 9, 27, 81, 243, 729, 2187
7, 49, 343, 2401, 16807, 117649, 823543
15, 225, 3375, 50625, 759375, 11390625, 170859375
31, 961, 29791, 923521, 28629151, 887503681, 27512614111
63, 3969, 250047, 15752961, 992436543, 62523502209, 3938980639167
127, 16129, 2048383, 260144641, 33038369407, 4195872914689, 532875860165503
A092477 = T(k,n) is the number of left total relations between A and B.
A053440 is the number of relations that are both right unique (see
A329940) and right total.
A089072 is the number of functions from A to B: relations between A and B that are both right unique and left total.
A019538 is the number of surjections between A and B: relations that are right unique, right total, and left total.
A008279 is the number of injections: relations that are right unique, left total, and left unique.
A000142 is the number of bijections: relations that are right unique, left total, right total, and left unique.
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rt:=(n,k)->(2^n-1)^k:
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T[n_, k_] := (2^n - 1)^k; Table[T[n - k + 1, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)
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rt:=(n,k)->(2^n-1)^k:
A329940
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.
Original entry on oeis.org
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
Offset: 1
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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T:= (n, k)-> (k+1)^n-1:
seq(seq(T(1+d-k, k), k=1..d), d=1..12);
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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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T:=(n,k)->(k+1)^n-1:
A329655
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.
Original entry on oeis.org
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
Offset: 1
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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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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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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T:=(n,k)->_plus (binomial(n,j)*binomial(k, j)* j! $ j=1..k):
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