The number of slim rectangular lattices

Slim rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in 2009 . They are finite semimodular lattices L such that the ordered set Ji L of join-irreducible elements of L is the cardinal sum of two nontrivial chains. After describing these lattices of a given length n by permutations, we determine their number, |SRectL( n )|. Besides giving recursive formulas, which are effective up to about n = 1000, we also prove that |SRectL( n )| is asymptotically ( n - 2)! · e 2 / 2 . Similar results for patch lattices, which are special rectangular lattices introduced by G. Czédli and E. T. Schmidt in 2013, and for slim rectangular lattice diagrams are also given.