A302251 The number of nonempty antichains in the lattice of set partitions.

1, 2, 9, 346, 79814831

Offset

1,2

Comments

Computing terms in this sequence is analogous to Dedekind's problem which asks for the number of antichains in the Boolean algebra.

This count excludes the empty antichain consisting of no set partitions.

Links

Table of n, a(n) for n=1..5.

Sebastian Bozlee, Bob Kuo, and Adrian Neff, A classification of modular compactifications of the space of pointed elliptic curves by Gorenstein curves, arXiv:2105.10582 [math.AG], 2021.

Example

For n = 3 the a(3) = 9 nonempty antichains are:
{1/2/3}
{1/23}
{12/3}
{13/2}
{1/23, 12/3}
{1/23, 13/2}
{12/3, 13/2}
{1/23, 12/3, 13/2}
{123}
Here we have used the usual shorthand notation for set partitions where 1/23 denotes {{1}, {2,3}}.

Prog

(Sage)
[Posets.SetPartitions(n).antichains().cardinality() - 1 for n in range(4)]
# minus removes the empty antichain

Crossrefs

Equals A302250 - 1, Cf. A000372, A007153, A003182, A014466.

Sequence in context: A005271 A258668 A012938 * A013093 A013169 A012991

Adjacent sequences: A302248 A302249 A302250 * A302252 A302253 A302254

Keyword

nonn,hard,more

Author

John Machacek, Apr 04 2018

Status

approved