A382693 Number of dense symmetric binary relations on {1,...,n}.

1, 2, 4, 20, 234, 6308, 374586, 47740076, 12788143462, 7090729971308

Offset: 0

Author

Mark Bowron, Apr 03 2025

Keywords

• nonn
• more

Comments

A binary relation R is usually called dense if for all R-related x and y there exists z such that x and z and also z and y are R-related (see Wikipedia link), but here we define a symmetric binary relation R on {1,...,n} to be "dense" (see the second page of the paper in the links) iff for all x and y we have (zRx implies zRy for all z) implies (x <= y).

Equivalently, as mentioned in a SeqFan post (see link), for all x < y, some z satisfies (not zRx) AND zRy.

Note that terms were added for n = 7 and 8 in a SeqFan post (see link).

Links

• Francesco Ciraulo and Giovanni Sambin, The overlap algebra of regular opens, Journal of Pure and Applied Algebra, Volume 214, Issue 11, November 2010, Pages 1988-1995.
• Seqfan, number of dense symmetric binary relations on {1,...,n} (A382693).
• Wikipedia, Dense Order, Generalizations.

Crossrefs

Cf. A382839 (with symmetry added).

Extensions

a(9) from Christian Sievers, May 26 2025