0,4

T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair of distinct nodes have a hyperedge containing one but not the other node.

V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)

V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

K. S. Brown, Dedekind's problem

a(n)=C(n + 7, 7) - 6*C(n + 5, 5) + 6*C(n + 4, 4) + 3*C(n + 3, 3) - 6*C(n + 2, 2) + 2*C(n + 1, 1)=n*(n - 2)*(n - 1)*(n + 1)*(n^3 + 30*n^2 + 131*n - 270)/5040.

G.f.: 1/(1-x)^8-6/(1-x)^6+6/(1-x)^5+3/(1-x)^4-6/(1-x)^3+2/(1-x)^2 = x^3*(2+3*x-6*x^2+2*x^3)/(1-x)^8.

Recurrence: a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8).

Generally, recurrence for the number of m - element ordered antichains on an unlabeled n - element set is a(m, n) = C(2^m, 1)*a(m, n - 1) - C(2^m, 2)*a(m, n - 2) + C(2^m, 3)*a(m, n - 3) + ... + ( - 1)^(k - 1)*C(2^m, k)*a(m, n - k) + ... - a(m, n - 2^m).

a(n)= A000580(n+7)-6*A000389(n+5)+6*A000332(n+4)+3*A000292(n+1)-6*A000217(n+1)+2*A000027(n+1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007

There are 19 3-element ordered antichains on an unlabeled 4-element set: ({4},{3},{2}), ({4},{3},{1,2}), ({4},{2,3},{1}), ({4},{2,3},{1,3}), ({3,4},{2},{1}), ({3,4},{2},{1,4}), ({3,4},{2,4},{2,3}), ({3,4},{2,4},{1}), ({3,4},{2,4},{1,4}), ({3,4},{2,4},{1,3}), ({3,4},{2,4},{1,2}), ({3,4},{2,4},{1,2,3}), ({3,4},{1,2},{2,4}), ({3,4},{1,2,4},{2,3}), ({3,4},{1,2,4},{1,2,3}), ({2,3,4},{1,4},{1,3}), ({2,3,4},{1,4},{1,2,3}), ({2,3,4},{1,3,4},{1,2}), ({2,3,4},{1,3,4},{1,2,4}).

Cf. A047707 for 3-element (unordered) antichains on a labeled n-element set.

Cf. A056069, A056070, A056071, A056073, A056163.

Sequence in context: A129446 A054570 A135436 this_sequence A034572 A041393 A107123

Adjacent sequences: A056002 A056003 A056004 this_sequence A056006 A056007 A056008

nonn

Vladeta Jovovic, Goran Kilibarda (vladeta(AT)eunet.rs), Jul 24 2000