A358041 -id:A358041 - cp's OEIS frontend

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A358563 The number of maximal antichains in the Tamari lattice of order n.

Original entry on oeis.org

1, 2, 4, 26, 1979, 161117453

Offset: 1

Dmitry I. Ignatov, Nov 22 2022

Also the number of maximal order ideals in the Tamari lattice of order n.

Maximal antichains are those which cannot be extended without violating the antichain condition.

			The line (Hasse) diagram of the Tamari lattice for n=3 is
     ((ab)c)d
      /     \
 (a(bc))d (ab)(cd)
     |       /
  a((bc)d)  /
      \    /
     a(b(cd))
with the a(3)=4 maximal antichains {((ab)c)d}, {(ab)(cd), (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.

S. Huang and D. Tamari, Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law, J. of Comb. Theory, Series A, 13 (1972), 7-13.
Dmitry I. Ignatov, Supporting iPython code and input files for counting (maximal) antichains of the Tamari partition lattice up to n=5, Github repository.
Wikipedia, Tamari lattice

Cf. A358562 (number of antichains in the Tamari lattice).
Cf. A326358 (number of maximal antichains in the Boolean lattice).
Cf. A358041 (number of maximal antichains in the lattice of set partitions of an n-element set).
Cf. A358390 (number of maximal antichains in the Kreweras lattice of non-crossing set partitions).
Cf. A143674 (number of maximal antichains in the lattice of Dyck paths).

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