%I #19 Oct 05 2024 09:41:02

%S 1,1,2,6,23,102,497,2586,14127,80146,468688,2810163,17206549,

%T 107261051,679096359,4358360362,28309516828,185862601727,

%U 1232042778231,8238155634738,55521191613041,376888928783870,2575334987109807,17704834935517727,122401523831513816

%N The number of 1+1+1-free ordered posets of [n].

%C A partial order R on [n] is ordered if xRy implies x < y; i.e., the natural order (<) is a linear extension of R. 1+1+1-free posets are those with width (longest antichain) at most 2.

%F Conjectured g.f.: 2 - 2*x/(B(x)-1+x), where B(x) is the o.g.f. for A001181. - _Michael D. Weiner_, Oct 04 2024

%e The six 1+1+1-free ordered posets of [3] are those with covering relations {(1,2)}, {(1,3)}, {(2,3)}, {(1,2), (1,3)}, {(1,2), (2,3)} and {(1,3), (2,3)}.

%Y See A006455 for the number of all ordered posets on [n], and A135922 for the number of ordered posets on [n] with height at most two.

%Y Cf. A001181.

%K nonn

%O 0,3

%A _David Bevan_, Jul 27 2022