A382350 Number of maximal antichains in the Bruhat order on B_n.

2, 5, 215, 24828398365

Offset

1,1

Comments

The number of maximal antichains in the Bruhat order of the Weyl group B_n (the hyperoctahedral group).

References

A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.

Links

Table of n, a(n) for n=1..4.

V. V. Deodhar, On Bruhat ordering and weight-lattice ordering for a Weyl group, Indagationes Mathematicae, vol. 81, 1 (1978), 423-435.

Example

For n=1 the elements are 1 (identity) and s1, the order contains pair (1, s1). The maximal antichains are {1} and {s1}.
For n=2 the line (Hasse) diagram is below.
      s2*s1*s2*s1
          /   \
    s2*s1*s2  s1*s2*s1
        |   X   |
      s2*s1   s1*s2
        |   X   |
        s2     s1
          \   /
            1
The set of maximal antichains is {{1}, {s2, s1}, {s2*s1, s1*s2}, {s2*s1*s2, s1*s2*s1}, {s2*s1*s2*s1}}.

Crossrefs

Cf. A382346 (antichains), A005900 (the number of join-irreducible elements), A378072 (the size of Dedekind-MacNeille completion)

Sequence in context: A096695 A084297 A041873 * A072118 A397277 A249180

Adjacent sequences: A382347 A382348 A382349 * A382351 A382352 A382353

Keyword

nonn,hard,more

Author

Dmitry I. Ignatov, May 30 2025

Extensions

a(4) from Dmitry I. Ignatov, Aug 15 2025

Status

approved