A302250 The number of antichains in the lattice of set partitions of an n-element set.

2, 3, 10, 347, 79814832

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 includes the empty antichain consisting of no set partitions.

Examples

			For n = 3 the a(3) = 10 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

• Dmitry I. Ignatov, A Note on the Number of (Maximal) Antichains in the Lattice of Set Partitions. In: Ojeda-Aciego, M., Sauerwald, K., Jäschke, R. (eds) Graph-Based Representation and Reasoning. ICCS 2023. Lecture Notes in Computer Science(). Springer, Cham.

Crossrefs

Equals A302251 + 1, Cf. A000372, A007153, A003182, A014466.

Programs

Sage

[Posets.SetPartitions(n).antichains().cardinality() for n in range(4)]