cp's OEIS Frontend

A265706 Rectangular array A read by upward antidiagonals: A(n,m) is the number of total difunctional (regular) binary relations between an n-element set and an m-element set.

%I A265706 #16 Dec 15 2015 10:06:40
%S A265706 1,3,1,7,5,1,15,19,9,1,31,65,49,17,1,63,211,225,127,33,1,127,665,961,
%T A265706 749,337,65,1,255,2059,3969,3991,2505,919,129,1,511,6305,16129,20237,
%U A265706 16201,8525,2569,257,1,1023,19171,65025,100087,97713,65911
%N A265706 Rectangular array A read by upward antidiagonals: A(n,m) is the number of total difunctional (regular) binary relations between an n-element set and an m-element set.
%C A265706 A(n,m) is the number of total difunctional (regular) binary relations between an n-element set and an m-element set.
%H A265706 Jasha Gurevich, <a href="/A265706/b265706.txt">Table of n, a(n) for n = 1..300</a>
%H A265706 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 A265706 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 A265706 Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_relation#Difunctional">Binary relation</a>
%F A265706 T(n, m) = Sum_{i=1..n} (Stirling2(m, i)* i! + Stirling2(m, i+1)*(i+1)!) *Stirling2(n, i).
%e A265706 Array A begins
%e A265706 1   3     7     15      31       63       127        255         511
%e A265706 1   5    19     65     211      665      2059       6305       19171
%e A265706 1   9    49    225     961     3969     16129      65025      261121
%e A265706 1  17   127    749    3991    20237    100087     489149     2379511
%e A265706 1  33   337   2505   16201    97713    568177    3242265    18341401
%e A265706 1  65   919   8525   65911   464645   3115519   20322605   130656871
%e A265706 1 129  2569  29625  271561  2214009  16911049  124422105   896158921
%e A265706 1 257  7327 105149 1137991 10657997  91989367  756570029  6046077511
%e A265706 1 513 21217 380745 4857001 52034913 504717697 4611314745 40608430681
%p A265706 sum((Stirling2(m, i)*factorial(i)+Stirling2(m, i+1)*factorial(i+1))*Stirling2(n, i), i = 1 .. n);
%t A265706 Table[Sum[(StirlingS2[m, i] i! + StirlingS2[m, i + 1] (i + 1)!) StirlingS2[n - m + 1, i], {i, n - m + 1}], {n, 10}, {m, n, 1, -1}] // Flatten (* _Michael De Vlieger_, Dec 14 2015 *)
%o A265706 (PARI) T(n, m) = sum(i=1, n, ( stirling(m, i, 2)*i! + stirling(m, i+1, 2)*(i+1)!)*stirling(n, i, 2));
%Y A265706 Cf. A265417.
%K A265706 nonn,tabl
%O A265706 1,2
%A A265706 _Jasha Gurevich_, Dec 14 2015