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.
1, 3, 1, 7, 5, 1, 15, 19, 9, 1, 31, 65, 49, 17, 1, 63, 211, 225, 127, 33, 1, 127, 665, 961, 749, 337, 65, 1, 255, 2059, 3969, 3991, 2505, 919, 129, 1, 511, 6305, 16129, 20237, 16201, 8525, 2569, 257, 1, 1023, 19171, 65025, 100087, 97713, 65911
Offset: 1
Examples
Array A begins 1 3 7 15 31 63 127 255 511 1 5 19 65 211 665 2059 6305 19171 1 9 49 225 961 3969 16129 65025 261121 1 17 127 749 3991 20237 100087 489149 2379511 1 33 337 2505 16201 97713 568177 3242265 18341401 1 65 919 8525 65911 464645 3115519 20322605 130656871 1 129 2569 29625 271561 2214009 16911049 124422105 896158921 1 257 7327 105149 1137991 10657997 91989367 756570029 6046077511 1 513 21217 380745 4857001 52034913 504717697 4611314745 40608430681
Links
- Jasha Gurevich, Table of n, a(n) for n = 1..300
- Chris Brink, Wolfram Kahl, Gunther Schmidt, Relational Methods in Computer Science, Springer Science & Business Media, 1997, p. 200.
- J. Riguet, Relations binaires, fermetures, correspondances de Galois, Bulletin de la Société Mathématique de France (1948) Volume: 76, pp. 114-155.
- Wikipedia, Binary relation
Crossrefs
Cf. A265417.
Programs
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Maple
sum((Stirling2(m, i)*factorial(i)+Stirling2(m, i+1)*factorial(i+1))*Stirling2(n, i), i = 1 .. n);
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Mathematica
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 *) -
PARI
T(n, m) = sum(i=1, n, ( stirling(m, i, 2)*i! + stirling(m, i+1, 2)*(i+1)!)*stirling(n, i, 2));
Formula
T(n, m) = Sum_{i=1..n} (Stirling2(m, i)* i! + Stirling2(m, i+1)*(i+1)!) *Stirling2(n, i).
Comments