A395723 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.

1, 4, 34, 526, 13618, 559654, 35206834, 3301531846, 452274696418, 89077635721894, 24901395717678994, 9775895465678639686, 5341976869617159075778, 4032518615672541333691174, 4177918152562844767457824114, 5907591811646289112930164831046, 11344858065618251316427764256980898

Offset

0,2

Comments

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).

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).

Links

Table of n, a(n) for n=0..16.

Rafael Ayala, The number of labeled partial orders and topologies on 19 points, arXiv:2606.31526 [math.CO], 2026.

Rafael Ayala, Code and data for the antichain-count moments

Crossrefs

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

Sequence in context: A198908 A207864 A222077 * A081972 A158961 A134354

Adjacent sequences: A395717 A395720 A395721 * A395725 A395726 A395727

Keyword

nonn,new

Author

Rafael Ayala, Jul 09 2026

Status

approved