A384090 Number of ordered pairs in the Bruhat order on B_n.

3, 33, 847, 40249, 3089459, 350676009

Offset: 1

Author

Dmitry I. Ignatov, May 19 2025

Keywords

• nonn
• more

Comments

The number of ordered pairs in the Bruhat order of the Weyl group B_n (the hyperoctahedral group).

Examples

			For n=1 the elements are 1 (identity) and s1. The order relation consists of pairs (1, 1), (1, s1), and (s1, s1). So a(1) = 3.
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 order relation is formed by 8 reflexive pairs, 12 pairs shown as edges in the diagram, and 13 pairs taken by transitivity: (1, s2*s1), (1, s1*s2), (1, s2*s1*s2), (1, s1*s2*s1), (1, s2*s1*s2*s1), (s2, s2*s1*s2), (s2, s1*s2*s1), (s2, s2*s1*s2*s1), (s1, s2*s1*s2), (s1, s1*s2*s1), (s1, s2*s1*s2*s1), (s2*s1, s2*s1*s2*s1), (s1*s2, s2*s1*s2*s1). So a(2) = 8+12+13 = 33.
		

References

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

Links

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

Crossrefs

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