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A341633 a(n) is the cardinality of the central rank of the free distributive lattice on n generators.

%I A341633 #67 Jul 29 2025 15:37:06
%S A341633 1,2,4,24,621,492288,81203064840
%N A341633 a(n) is the cardinality of the central rank of the free distributive lattice on n generators.
%C A341633 Sequence for 2 <= n <= 5 is given in Church (1940); n = 1 obtained trivially from {} - {{}} - {{}, {1}}; n = 6 and n = 7 obtained from the triangle A269699.
%C A341633 a(n) is also provably the number of downward closed subsets of the powerset of {1,2,3,...,n} which have the cardinality 2^(n-1).
%C A341633 If FD(n) (the free distributive lattice on n generators) is rank unimodal for all n, then a(n) is the largest cardinality of any rank of FD(n).
%C A341633 If FD(n) is rank unimodal and Sperner for all n, then a(n) is the width of FD(n). (Consequences provable, antecedents are open questions - e.g., Stanley (1991))
%C A341633 This sequence is related (at least methodologically) to the n-th Dedekind number (A000372), which is obtained from the cardinality of FD(n).
%C A341633 a(n) is also the number of balanced monotone Boolean functions. - _Aniruddha Biswas_, Nov 22 2024
%H A341633 Bruno L. O. Andreotti, <a href="/A341633/a341633.py.txt">Python program for n = 1 to 6</a>
%H A341633 Aniruddha Biswas and Palash Sarkar, <a href="https://arxiv.org/abs/2304.14069">Counting unate and balanced monotone Boolean functions</a>, arXiv:2304.14069 [math.CO], 2023.
%H A341633 Aniruddha Biswas and Palash Sarkar, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Biswas/biswas6.html">Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy</a>, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 3, 11-12.
%H A341633 Randolph Church, <a href="https://doi.org/10.1215/S0012-7094-40-00655-X">Numerical analysis of certain free distributive structures</a>, Duke Math. J. 6 (1940). 732--734.
%H A341633 Randolph Church, <a href="/A000372/a000372_3.pdf">Numerical analysis of certain free distributive structures</a>, Duke Math. J. 6 (1940). 732--734.
%H A341633 Richard P. Stanley, <a href="https://doi.org/10.1016/0166-218X(91)90089-F">Some application of algebra to combinatorics</a>, Discrete Applied Mathematics, 34 (1991), 241-277.
%F A341633 a(n) = A269699(n, 2^(n-1)).
%e A341633 a(4)=24 is obtained from the 24 downsets on the 8th and central rank of FD(4), each containing 8 members (enumeration is arbitrary):
%e A341633    1  {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%e A341633    2  {{},{1},{2},{4},{1,2},{1,4},{2,4},{1,2,4}}
%e A341633    3  {{},{1},{3},{4},{1,3},{1,4},{3,4},{1,3,4}}
%e A341633    4  {{},{2},{3},{4},{2,3},{2,4},{3,4},{2,3,4}}
%e A341633    5  {{},{1},{2},{3},{4},{1,2},{1,3},{2,3}}
%e A341633    6  {{},{1},{2},{3},{4},{1,2},{1,4},{2,4}}
%e A341633    7  {{},{1},{2},{3},{4},{1,3},{1,4},{3,4}}
%e A341633    8  {{},{1},{2},{3},{4},{2,3},{2,4},{3,4}}
%e A341633    9  {{},{1},{2},{3},{4},{1,2},{1,3},{1,4}}
%e A341633   10  {{},{1},{2},{3},{4},{1,2},{2,3},{2,4}}
%e A341633   11  {{},{1},{2},{3},{4},{1,3},{2,3},{3,4}}
%e A341633   12  {{},{1},{2},{3},{4},{1,4},{2,4},{3,4}}
%e A341633   13  {{},{1},{2},{3},{4},{1,2},{2,3},{3,4}}
%e A341633   14  {{},{1},{2},{3},{4},{1,2},{1,4},{3,4}}
%e A341633   15  {{},{1},{2},{3},{4},{1,2},{1,4},{2,3}}
%e A341633   16  {{},{1},{2},{3},{4},{1,2},{1,3},{3,4}}
%e A341633   17  {{},{1},{2},{3},{4},{1,2},{2,4},{3,4}}
%e A341633   18  {{},{1},{2},{3},{4},{1,2},{1,3},{2,4}}
%e A341633   19  {{},{1},{2},{3},{4},{1,3},{1,4},{2,3}}
%e A341633   20  {{},{1},{2},{3},{4},{1,3},{1,4},{2,4}}
%e A341633   21  {{},{1},{2},{3},{4},{1,3},{2,4},{3,4}}
%e A341633   22  {{},{1},{2},{3},{4},{1,3},{2,3},{2,4}}
%e A341633   23  {{},{1},{2},{3},{4},{1,4},{2,3},{3,4}}
%e A341633   24  {{},{1},{2},{3},{4},{1,4},{2,3},{2,4}}
%o A341633 (Python) # See Andreotti link.
%Y A341633 Cf. A000372, A269699, A371722.
%K A341633 nonn,hard,more
%O A341633 1,2
%A A341633 _Bruno L. O. Andreotti_, Feb 16 2021