1,1

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

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

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

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.

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

Sequence in context: A371683 A124432 A234715 * A126466 A002112 A055549

Adjacent sequences: A384087 A384088 A384089 * A384091 A384092 A384093

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Dmitry I. Ignatov, May 19 2025

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