0,2

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).

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.

Cf. A382839 (with symmetry added).

Sequence in context: A120388 A061348 A127103 * A059831 A064493 A145614

Adjacent sequences: A382690 A382691 A382692 * A382694 A382695 A382696

nonn,more

Mark Bowron, Apr 03 2025

a(9) from Christian Sievers, May 26 2025

approved