A397764 Number of quadruples (P, I_1, I_2, I_3) where P is a partially ordered set on n labeled elements and I_1, I_2, I_3 are order ideals (down-sets) of P; equivalently, sum over labeled posets P on [n] of the cube of the number of antichains of P.

1, 8, 118, 2942, 117286, 7195958, 660822358, 88931884742, 17250697648246, 4759215340095638, 1847052091786573078, 999208523732653011302, 747640657240669159797046, 768599412047968397655319478, 1079396516088637321967914136278, 2060461398693819222679181880377222, 5322963351172775869497071016032650486

Offset

0,2

Comments

a(n) = Sum_{P} d(P)^3 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=3 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+3 elements in which three designated elements are all maximal (with proof: attach a new maximal point above each of the three ideals; maximal points are automatically pairwise incomparable, and the correspondence is a bijection).

a(16) is the frontier moment from which A001035(19) was computed; see the linked paper.

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), A395723 (k=2), A000798.

Sequence in context: A385320 A267752 A386365 * A318009 A368911 A228752

Adjacent sequences: A397761 A397762 A397763 * A397765 A397768 A397770

Keyword

nonn,new

Author

Rafael Ayala, Jul 09 2026

Status

approved