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

1, 2, 4, 24, 621, 492288, 81203064840

Offset: 1

Author

Bruno L. O. Andreotti, Feb 16 2021

Keywords

• nonn
• hard
• more

Comments

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.

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).

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).

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))

This sequence is related (at least methodologically) to the n-th Dedekind number (A000372), which is obtained from the cardinality of FD(n).

a(n) is also the number of balanced monotone Boolean functions. - Aniruddha Biswas, Nov 22 2024

Examples

			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):
   1  {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
   2  {{},{1},{2},{4},{1,2},{1,4},{2,4},{1,2,4}}
   3  {{},{1},{3},{4},{1,3},{1,4},{3,4},{1,3,4}}
   4  {{},{2},{3},{4},{2,3},{2,4},{3,4},{2,3,4}}
   5  {{},{1},{2},{3},{4},{1,2},{1,3},{2,3}}
   6  {{},{1},{2},{3},{4},{1,2},{1,4},{2,4}}
   7  {{},{1},{2},{3},{4},{1,3},{1,4},{3,4}}
   8  {{},{1},{2},{3},{4},{2,3},{2,4},{3,4}}
   9  {{},{1},{2},{3},{4},{1,2},{1,3},{1,4}}
  10  {{},{1},{2},{3},{4},{1,2},{2,3},{2,4}}
  11  {{},{1},{2},{3},{4},{1,3},{2,3},{3,4}}
  12  {{},{1},{2},{3},{4},{1,4},{2,4},{3,4}}
  13  {{},{1},{2},{3},{4},{1,2},{2,3},{3,4}}
  14  {{},{1},{2},{3},{4},{1,2},{1,4},{3,4}}
  15  {{},{1},{2},{3},{4},{1,2},{1,4},{2,3}}
  16  {{},{1},{2},{3},{4},{1,2},{1,3},{3,4}}
  17  {{},{1},{2},{3},{4},{1,2},{2,4},{3,4}}
  18  {{},{1},{2},{3},{4},{1,2},{1,3},{2,4}}
  19  {{},{1},{2},{3},{4},{1,3},{1,4},{2,3}}
  20  {{},{1},{2},{3},{4},{1,3},{1,4},{2,4}}
  21  {{},{1},{2},{3},{4},{1,3},{2,4},{3,4}}
  22  {{},{1},{2},{3},{4},{1,3},{2,3},{2,4}}
  23  {{},{1},{2},{3},{4},{1,4},{2,3},{3,4}}
  24  {{},{1},{2},{3},{4},{1,4},{2,3},{2,4}}
		

Links

• Bruno L. O. Andreotti, Python program for n = 1 to 6
• Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
• Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 3, 11-12.
• Randolph Church, Numerical analysis of certain free distributive structures, Duke Math. J. 6 (1940). 732--734.
• Randolph Church, Numerical analysis of certain free distributive structures, Duke Math. J. 6 (1940). 732--734.
• Richard P. Stanley, Some application of algebra to combinatorics, Discrete Applied Mathematics, 34 (1991), 241-277.

Crossrefs

Cf. A000372, A269699, A371722.

Programs

Python

# See Andreotti link.

Formula

a(n) = A269699(n, 2^(n-1)).