A056163 Number of ordered antichains on an unlabeled n-set; labeled T_1-hypergraphs with n hyperedges.

2, 3, 5, 11, 120, 191297

Offset: 0

Author

Vladeta Jovovic, Goran Kilibarda, Jul 31 2000

Keywords

• hard
• more
• nonn

Comments

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

Examples

			a(1)=1+2=3; a(2)=1+3+1=5; a(3)=1+4+4+2=11; a(4)=1+5+10+19+25+30+30=120; a(5)=1+6+20+90+454+2206+8340+20580+38640+60480+60480=191297.
There are 11 ordered antichains on an unlabeled 3-set: 0, (0), ({1}), ({1,2}), ({1,2,3}), ({1},{2}), ({1},{2,3}), ({2,3},{1}), ({1,2},{1,3}), ({1},{2},{3}), ({1,2},{1,3},{2,3}).
		

References

• 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.

Links

• K. S. Brown, Dedekind's problem
• Eric Weisstein's World of Mathematics, Antichain covers

Crossrefs

Cf. A000372 for (unordered) antichains on a labeled n-set, A056005, A056069-A056071, A056073, A056046-A056049, A056052, A056101, A056104, A051112-A051118.

Formula

a(n)=Sum_{k=0..C(n, floor(n/2))}b(k, n) where b(k, n) is the number of k-element ordered antichains on an unlabeled n-set.