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

1, 2, 9, 346, 79814831

Offset: 1

Author

John Machacek, Apr 04 2018

Keywords

• nonn
• hard
• more

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.

Examples

			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}}.
		

Links

• 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.

Crossrefs

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

Programs

Sage

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