A302251 - OEIS

%I #17 Aug 30 2021 21:02:24

%S 1,2,9,346,79814831

%N The number of nonempty antichains in the lattice of set partitions.

%C Computing terms in this sequence is analogous to Dedekind's problem which asks for the number of antichains in the Boolean algebra.

%C This count excludes the empty antichain consisting of no set partitions.

%H Sebastian Bozlee, Bob Kuo, and Adrian Neff, <a href="https://arxiv.org/abs/2105.10582">A classification of modular compactifications of the space of pointed elliptic curves by Gorenstein curves</a>, arXiv:2105.10582 [math.AG], 2021.

%e For n = 3 the a(3) = 9 nonempty antichains are:

%e {1/2/3}

%e {1/23}

%e {12/3}

%e {13/2}

%e {1/23, 12/3}

%e {1/23, 13/2}

%e {12/3, 13/2}

%e {1/23, 12/3, 13/2}

%e {123}

%e Here we have used the usual shorthand notation for set partitions where 1/23 denotes {{1}, {2,3}}.

%o (Sage)

%o [Posets.SetPartitions(n).antichains().cardinality() - 1 for n in range(4)]

%o # minus removes the empty antichain

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

%K nonn,hard,more

%O 1,2

%A _John Machacek_, Apr 04 2018