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A265707 Rectangular array A read by upward antidiagonals: A(n,m) is the number of surjective difunctional (regular) binary relations between an n-element set and an m-element set.

%I A265707 #13 Dec 15 2015 10:06:51
%S A265707 1,1,3,1,5,7,1,9,19,15,1,17,49,65,31,1,33,127,225,211,63,1,65,337,749,
%T A265707 961,665,127,1,129,919,2505,3991,3969,2059,255,1,257,2569,8525,16201,
%U A265707 20237,16129,6305,511,1,513,7327,29625,65911,97713
%N A265707 Rectangular array A read by upward antidiagonals: A(n,m) is the number of surjective difunctional (regular) binary relations between an n-element set and an m-element set.
%C A265707 A(n,m) is the number of surjective difunctional (regular) binary relations between an n-element set and an m-element set.
%H A265707 Jasha Gurevich, <a href="/A265707/b265707.txt">Table of n, a(n) for n = 1..300</a>
%H A265707 Chris Brink, Wolfram Kahl, Gunther Schmidt, <a href="http://dx.doi.org/10.1007/978-3-7091-6510-2">Relational Methods in Computer Science</a>, Springer Science & Business Media, 1997, p. 200.
%H A265707 J. Riguet, <a href="http://www.numdam.org/item?id=BSMF_1948__76__114_0">Relations binaires, fermetures, correspondances de Galois</a>, Bulletin de la Société Mathématique de France (1948) Volume: 76, pp. 114-155.
%H A265707 Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_relation#Difunctional">Binary relation</a>
%F A265707 T(n, m) = Sum_{i=1..n} (Stirling2(m, i-1)* i! + Stirling2(m, i)* i! ) * Stirling2(n, i).
%e A265707 Array A begins
%e A265707     1     1      1       1        1         1         1          1           1
%e A265707     3     5      9      17       33        65       129        257         513
%e A265707     7    19     49     127      337       919      2569       7327       21217
%e A265707    15    65    225     749     2505      8525     29625     105149      380745
%e A265707    31   211    961    3991    16201     65911    271561    1137991     4857001
%e A265707    63   665   3969   20237    97713    464645   2214009   10657997    52034913
%e A265707   127  2059  16129  100087   568177   3115519  16911049   91989367   504717697
%e A265707   255  6305  65025  489149  3242265  20322605 124422105  756570029  4611314745
%e A265707   511 19171 261121 2379511 18341401 130656871 896158921 6046077511 40608430681
%p A265707 sum((Stirling2(m, i-1)*factorial(i)+Stirling2(m,i)*factorial(i))*Stirling2(n, i), i = 1 .. n);
%o A265707 (PARI) T(n, m) = sum(i=1, n, (stirling(m, i-1, 2)*i! + stirling(m, i, 2)*i!)*stirling(n, i, 2));
%Y A265707 Cf. A265417, A265706.
%K A265707 nonn,tabl
%O A265707 1,3
%A A265707 _Jasha Gurevich_, Dec 14 2015