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A147821
Number of consistent sets of 5 irreflexive binary order relationships over n objects.
8
108, 6180, 83952, 601944, 2991576, 11662056, 38167920, 109368864, 282174948, 668565612, 1475938464, 3069513720, 6065522736, 11466274512, 20850952608, 36639176832, 62447999580, 103567126068, 167581781136, 265177823064, 411169457160, 625796259000
OFFSET
4,1
COMMENTS
It seems that a(n) = A081064(n,5) is the number of labeled acyclic directed graphs with n nodes and k = 5 arcs (see Rodionov (1992)). The reason is that we may label the graphs with the n objects and draw an arc from X towards Y if and only if X < Y. The 5 arcs of the directed graph correspond to the 3-set of binary order relationships and they are consistent because the directed graph is acyclic. - Petros Hadjicostas, Apr 10 2020
LINKS
V. I. Rodionov, On the number of labeled acyclic digraphs, Discr. Math. 105 (1-3) (1992), 319-321.
FORMULA
a(n) = (n-3)*(n-2)*(n-1)*n*(n^6 + n^5 - 15*n^4 - 45*n^3 - 4*n^2 + 326*n + 900)/120. - Vaclav Kotesovec, Apr 11 2020
Conjectures from Colin Barker, Apr 11 2020: (Start)
G.f.: 12*x^4*(9 + 416*x + 1826*x^2 + 46*x^3 + 291*x^4 - 78*x^5 + 10*x^6) / (1 - x)^11.
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n>12.
(End)
MAPLE
a := n -> (1/120)*(n-3)*(n-2)*(n-1)*n*(n*(n*(n*(n*(n^2+n-15)-45)-4)+326)+900):
seq(a(n), n=4..25); # Peter Luschny, Apr 11 2020
MATHEMATICA
Table[(n - 3)*(n - 2)*(n - 1)*n*(n^6 + n^5 - 15*n^4 - 45*n^3 - 4*n^2 + 326*n + 900)/120, {n, 4, 25}] (* Wesley Ivan Hurt, Apr 11 2020 *)
CROSSREFS
Related sequences for the number of consistent sets of k irreflexive binary order relationships over n objects: A147796 (k = 3), A147817 (k = 4), A147860 (k = 6), A147872 (k = 7), A147881 (k = 8), A147883 (k = 9), A147964 (k = 10).
Column k = 5 of A081064.
Sequence in context: A138784 A035812 A054624 * A269148 A143403 A269210
KEYWORD
nonn,easy
AUTHOR
R. H. Hardin, May 04 2009
EXTENSIONS
More terms from Vaclav Kotesovec, Apr 11 2020
Offset changed by Petros Hadjicostas, Apr 11 2020
STATUS
approved