A147821 Number of consistent sets of 5 irreflexive binary order relationships over n objects.

108, 6180, 83952, 601944, 2991576, 11662056, 38167920, 109368864, 282174948, 668565612, 1475938464, 3069513720, 6065522736, 11466274512, 20850952608, 36639176832, 62447999580, 103567126068, 167581781136, 265177823064, 411169457160, 625796259000

Offset: 4

Author

R. H. Hardin, May 04 2009

Keywords

• nonn
• easy

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.

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.

Programs

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 *)

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)

Extensions

More terms from Vaclav Kotesovec, Apr 11 2020

Offset changed by Petros Hadjicostas, Apr 11 2020