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

%I A302250 #23 Aug 20 2023 10:49:16
%S A302250 2,3,10,347,79814832
%N A302250 The number of antichains in the lattice of set partitions of an n-element set.
%C A302250 Computing terms in this sequence is analogous to Dedekind's problem which asks for the number of antichains in the Boolean algebra.
%C A302250 This count includes the empty antichain consisting of no set partitions.
%H A302250 Dmitry I. Ignatov, <a href="https://doi.org/10.1007/978-3-031-40960-8_6">A Note on the Number of (Maximal) Antichains in the Lattice of Set Partitions</a>. In: Ojeda-Aciego, M., Sauerwald, K., Jäschke, R. (eds) Graph-Based Representation and Reasoning. ICCS 2023. Lecture Notes in Computer Science(). Springer, Cham.
%e A302250 For n = 3 the a(3) = 10 antichains are:
%e A302250   {}
%e A302250   {1/2/3}
%e A302250   {1/23}
%e A302250   {12/3}
%e A302250   {13/2}
%e A302250   {1/23, 12/3}
%e A302250   {1/23, 13/2}
%e A302250   {12/3, 13/2}
%e A302250   {1/23, 12/3, 13/2}
%e A302250   {123}.
%e A302250 Here we have used the usual shorthand notation for set partitions where 1/23 denotes {{1}, {2,3}}.
%o A302250 (Sage)
%o A302250 [Posets.SetPartitions(n).antichains().cardinality() for n in range(4)]
%Y A302250 Equals A302251 + 1, Cf. A000372, A007153, A003182, A014466.
%K A302250 nonn,hard,more
%O A302250 1,1
%A A302250 _John Machacek_, Apr 04 2018