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

1,2

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).

Links

Table of n, a(n) for n=1..7.

Bruno L. O. Andreotti, Python program for n = 1 to 6

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.

R. P. Stanley, Some application of algebra to combinatorics, Discrete Applied Mathematics, 34 (1991), 241-277.

Formula

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

Example

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}}

Prog

(Python) # See Andreotti link.

Crossrefs

Cf. A000372, A269699.

Sequence in context: A009273 A332539 A296787 * A128299 A143672 A001510

Adjacent sequences: A341630 A341631 A341632 * A341634 A341635 A341636

Keyword

nonn,hard,more

Author

Bruno L. O. Andreotti, Feb 16 2021

Status

approved