A302251
The number of nonempty antichains in the lattice of set partitions.
Original entry on oeis.org
1, 2, 9, 346, 79814831
Offset: 1
For n = 3 the a(3) = 9 nonempty 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}}.
A358041
The number of maximal antichains in the lattice of set partitions of an n-element set.
Original entry on oeis.org
1, 2, 3, 32, 14094
Offset: 1
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}}.
- R. L. Graham, Maximum antichains in the partition lattice, The Mathematical Intelligencer, 1 (1978), 84-86.
- 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.
- Dmitry I. Ignatov, Supporting iPython code and input files for counting (maximal) antichains of the set partition lattice up to n=5, Github repository.
Cf.
A302250 (number of antichains in the lattice of set partitions).
Cf.
A326358 (number of maximal antichains in the Boolean lattice).
A358390
The number of maximal antichains in the Kreweras lattice of non-crossing set partitions of an n-element set.
Original entry on oeis.org
1, 2, 3, 25, 2117, 22581637702
Offset: 1
Cf.
A302250 (number of antichains in the lattice of set partitions).
Cf.
A326358 (number of maximal antichains in the Boolean lattice).
Cf.
A000108 (number of noncrossing partitions of a set of n elements)
A358391
The number of antichains in the Kreweras lattice of non-crossing set partitions of an n-element set.
Original entry on oeis.org
2, 3, 10, 234, 2342196
Offset: 1
Cf.
A000372 (number of antichains in the Boolean lattice).
Cf.
A143673 (number of antichains in the lattice of Dyck paths).
Cf.
A302250 (number of antichains in the lattice of set partitions).
A358562
The number of antichains in the Tamari lattice of order n.
Original entry on oeis.org
2, 3, 8, 83, 28984, 138832442543
Offset: 1
For n=3 the a(3)=8 antichains are {}, {((ab)c)d}, {(ab)(cd)}, {(a(bc))d}, {(ab)(cd), (a(bc))d}, {a((bc)d)}, {(ab)(cd), a((bc)d)}, {a(b(cd))}.
- D. Tamari, The algebra of bracketings and their enumeration, Nieuw Archief voor Wiskunde, Series 3, 10 (1962), 131-146.
Cf.
A000372 (number of antichains in the Boolean lattice).
Cf.
A302250 (number of antichains in the lattice of set partitions).
Cf.
A358391 (number of antichains in the Kreweras lattice of non-crossing set partitions of an n-element set).
Cf.
A143673 (number of antichains in the lattice of Dyck paths).
Showing 1-5 of 5 results.
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