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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.
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%I #16 Dec 15 2015 10:06:40

%S 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 749,337,65,1,255,2059,3969,3991,2505,919,129,1,511,6305,16129,20237,

%U 16201,8525,2569,257,1,1023,19171,65025,100087,97713,65911

%N 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 A(n,m) is the number of total difunctional (regular) binary relations between an n-element set and an m-element set.

%H Jasha Gurevich, <a href="/A265706/b265706.txt">Table of n, a(n) for n = 1..300</a>

%H 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_relation#Difunctional">Binary relation</a>

%F T(n, m) = Sum_{i=1..n} (Stirling2(m, i)* i! + Stirling2(m, i+1)*(i+1)!) *Stirling2(n, i).

%e Array A begins

%e 1 3 7 15 31 63 127 255 511

%e 1 5 19 65 211 665 2059 6305 19171

%e 1 9 49 225 961 3969 16129 65025 261121

%e 1 17 127 749 3991 20237 100087 489149 2379511

%e 1 33 337 2505 16201 97713 568177 3242265 18341401

%e 1 65 919 8525 65911 464645 3115519 20322605 130656871

%e 1 129 2569 29625 271561 2214009 16911049 124422105 896158921

%e 1 257 7327 105149 1137991 10657997 91989367 756570029 6046077511

%e 1 513 21217 380745 4857001 52034913 504717697 4611314745 40608430681

%p sum((Stirling2(m, i)*factorial(i)+Stirling2(m, i+1)*factorial(i+1))*Stirling2(n, i), i = 1 .. n);

%t 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 (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 Cf. A265417.

%K nonn,tabl

%O 1,2

%A _Jasha Gurevich_, Dec 14 2015