cp's OEIS Frontend

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.

%I A329940 #21 Apr 13 2020 01:41:52
%S A329940 1,3,2,7,8,3,15,26,15,4,31,80,63,24,5,63,242,255,124,35,6,127,728,
%T A329940 1023,624,215,48,7,255,2186,4095,3124,1295,342,63,8,511,6560,16383,
%U A329940 15624,7775,2400,511,80,9,1023,19682,65535,78124,46655,16806,4095,728,99,10
%N 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.
%C A329940 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.
%H A329940 Roy S. Freedman, <a href="https://arxiv.org/abs/1501.01914">Some New Results on Binary Relations</a>, arXiv:1501.01914 [cs.DM], 2015.
%F A329940 T(n,k) = (k+1)^n - 1.
%e A329940 T(n,k) begins:
%e A329940     1,    2,     3,      4,       5,       6,        7,        8, ...
%e A329940     3,    8,    15,     24,      35,      48,       63,       80, ...
%e A329940     7,   26,    63,    124,     215,     342,      511,      728, ...
%e A329940    15,   80,   255,    624,    1295,    2400,     4095,     6560, ...
%e A329940    31,  242,  1023,   3124,    7775,   16806,    32767,    59048, ...
%e A329940    63,  728,  4095,  15624,   46655,  117648,   262143,   531440, ...
%e A329940   127, 2186, 16383,  78124,  279935,  823542,  2097151,  4782968, ...
%e A329940   255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720, ...
%p A329940 T:= (n, k)-> (k+1)^n-1:
%p A329940 seq(seq(T(1+d-k, k), k=1..d), d=1..12);
%t A329940 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 *)
%o A329940 (MuPAD) T:=(n,k)->(k+1)^n-1:
%Y A329940 Cf. A037205 (main diagonal).
%Y A329940 Cf. A003992, A004248, A009998, A009999, A051128, A051129, A095884.
%K A329940 nonn,tabl,easy
%O A329940 1,2
%A A329940 _Roy S. Freedman_, Nov 24 2019