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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.

%I A329655 #26 Nov 25 2019 16:20:21
%S A329655 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,
%T A329655 42,7,8,56,228,500,500,228,56,8,9,72,357,1044,1545,1044,357,72,9,10,
%U A329655 90,528,1960,4050,4050,1960,528,90,10,11,110,747,3392,9275,13326,9275,3392,747,110,11
%N 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.
%C A329655 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.
%H A329655 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 A329655 T(n,k) = Sum_{j=1..k} binomial(n,j)*binomial(k,j)*j!.
%F A329655 T(n,k) = A088699(n,k)-1.
%e A329655 The symmetric array T(n,k) begins:
%e A329655   1,   2,    3,    4,     5,      6,       7,       8,        9, ...
%e A329655   2,   6,   12,   20,    30,     42,      56,      72,       90, ...
%e A329655   3,  12,   33,   72,   135,    228,     357,     528,      747, ...
%e A329655   4,  20,   72,  208,   500,   1044,    1960,    3392,     5508, ...
%e A329655   5,  30,  135,  500,  1545,   4050,    9275,   19080,    36045, ...
%e A329655   6,  42,  228, 1044,  4050,  13326,   37632,   93288,   207774, ...
%e A329655   7,  56,  357, 1960,  9275,  37632,  130921,  394352,  1047375, ...
%e A329655   8,  72,  528, 3392, 19080,  93288,  394352, 1441728,  4596552, ...
%e A329655   9,  90,  747, 5508, 36045, 207774, 1047375, 4596552, 17572113, ...
%p A329655 T:= (n,k)-> value(Sum(binomial(n,j)*binomial(k, j)*j!, j=1..k)):
%p A329655 seq(seq(T(n, 1+d-n), n=1..d), d=1..12);
%t A329655 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 *)
%o A329655 (MuPAD) T:=(n,k)->_plus (binomial(n,j)*binomial(k, j)* j! $ j=1..k):
%Y A329655 The diagonal T(n,n) is A097662.  T(1,k)=A000027; T(2,k)=A002378; T(3,k)=A054602.
%K A329655 nonn,tabl,easy
%O A329655 1,2
%A A329655 _Roy S. Freedman_, Nov 18 2019