A358562 - OEIS

%I #18 Jan 12 2023 19:24:39

%S 2,3,8,83,28984,138832442543

%N The number of antichains in the Tamari lattice of order n.

%C Also the number of order ideals (down-sets) for the Tamari lattice of order n.

%D D. Tamari, The algebra of bracketings and their enumeration, Nieuw Archief voor Wiskunde, Series 3, 10 (1962), 131-146.

%H S. Huang and D. Tamari, <a href="https://doi.org/10.1016/0097-3165(72)90003-9">Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law</a>, J. of Comb. Theory, Series A, 13 (1972), 7-13.

%H Dmitry I. Ignatov, <a href="https://github.com/dimachine/TamariAnti">Supporting iPython code and input files for counting (maximal) antichains of the Tamari partition lattice up to n=6</a>, Github repository.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tamari_lattice">Tamari lattice</a>

%e 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))}.

%Y Cf. A000372 (number of antichains in the Boolean lattice).

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

%Y Cf. A358391 (number of antichains in the Kreweras lattice of non-crossing set partitions of an n-element set).

%Y Cf. A143673 (number of antichains in the lattice of Dyck paths).

%Y Cf. A027686.

%K nonn,hard,more

%O 1,1

%A _Dmitry I. Ignatov_, Nov 22 2022