A397542 - OEIS

%I #6 Jul 07 2026 18:57:14

%S 1,2,10,98,1678,46922,2049550,135499898,13243258318,1878894285002,

%T 381362574101710,109430505055180058,43954986389427881998,

%U 24508712037068391623402,18835419731708228516982670,19828793128398628087131589178,28441643117705315333254490986318

%N Number of pairs (P, I) where P is a partially ordered set on n labeled elements and I is an order ideal (down-set) of P; equivalently, sum over labeled posets P on [n] of the number of antichains of P.

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

%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), A000798.

%K nonn,new

%O 0,2

%A _Rafael Ayala_, Jun 30 2026