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The number of maximal antichains in the lattice of set partitions of an n-element set.
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%I #22 Aug 20 2023 10:50:04

%S 1,2,3,32,14094

%N The number of maximal antichains in the lattice of set partitions of an n-element set.

%C Also similar to the number of maximal antichains in the Boolean lattice.

%H R. L. Graham, <a href="https://doi.org/10.1007/BF03023067">Maximum antichains in the partition lattice</a>, The Mathematical Intelligencer, 1 (1978), 84-86.

%H 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.

%H Dmitry I. Ignatov, <a href="https://github.com/dimachine/SetPartAnti/">Supporting iPython code and input files for counting (maximal) antichains of the set partition lattice up to n=5</a>, Github repository.

%e For n = 3 the a(3) = 3 maximal antichains are: {1|2|3}, {1|23, 12|3, 13|2}, and {123}. We use the typical shorthand notation for set partitions where 1|23 denotes {{1}, {2,3}}.

%Y Cf. A302250 (number of antichains in the lattice of set partitions).

%Y Cf. A326358 (number of maximal antichains in the Boolean lattice).

%K nonn,hard,more

%O 1,2

%A _Dmitry I. Ignatov_, Oct 29 2022