A302251 -id:A302251 - cp's OEIS frontend

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

Original entry on oeis.org

2, 3, 10, 347, 79814832

Offset: 1

John Machacek, Apr 04 2018

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.

			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}}.

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.

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

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

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