A395723 - OEIS

%I #33 Jul 14 2026 23:40:42

%S 1,4,34,526,13618,559654,35206834,3301531846,452274696418,

%T 89077635721894,24901395717678994,9775895465678639686,

%U 5341976869617159075778,4032518615672541333691174,4177918152562844767457824114,5907591811646289112930164831046,11344858065618251316427764256980898

%N Number of triples (P, I_1, I_2) where P is a partially ordered set on n labeled elements and I_1, I_2 are order ideals (down-sets) of P; equivalently, sum over labeled posets P on [n] of the square of the number of antichains of P.

%C a(n) = Sum_{P} d(P)^2 over the A001035(n) labeled posets P on [n], where d(P) is the number of antichains (equivalently order ideals / down-sets) of P. This is the k=2 member of the antichain-count moment family k=1..4 (k=1: A397542) used to compute A001035(19).

%C a(n) is the number of labeled posets on n+2 elements in which two designated elements are both maximal (with proof: attach to (P, I_1, I_2) a new maximal point above each of the two ideals; two maximal points are automatically incomparable, and the correspondence is a bijection).

%H Rafael Ayala, <a href="https://arxiv.org/abs/2606.31526">The number of labeled partial orders and topologies on 19 points</a>, arXiv:2606.31526 [math.CO], 2026.

%H Rafael Ayala, <a href="https://github.com/Rafael-Ayala/posets-and-topologies-19">Code and data for the antichain-count moments</a>

%Y Cf. A001035 (k=0, number of labeled posets), A397542 (k=1), A000798.

%K nonn,new

%O 0,2

%A _Rafael Ayala_, Jul 09 2026