A382346 - OEIS

%I #11 May 29 2025 15:46:01

%S 3,12,2247

%N Number of antichains in the Bruhat order on B_n.

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

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

%H V. V. Deodhar, <a href="https://doi.org/10.1016/1385-7258(78)90059-8">On Bruhat ordering and weight-lattice ordering for a Weyl group</a>, Indagationes Mathematicae, vol. 81, 1 (1978), 423-435.

%e For n=1 the elements are 1 (identity) and s1, the order contains pair (1, s1). The antichains are {}, {1}, and {s1}.

%e For n=2 the line (Hasse) diagram is below.

%e       s2*s1*s2*s1

%e           /   \

%e     s2*s1*s2  s1*s2*s1

%e         |   X   |

%e       s2*s1   s1*s2

%e         |   X   |

%e         s2     s1

%e           \   /

%e             1

%e The set of antichains is  {{}, {1}, {s2}, {s2, s1}, {s1}, {s2*s1}, {s2*s1, s1*s2}, {s1*s2}, {s2*s1*s2}, {s2*s1*s2, s1*s2*s1}, {s1*s2*s1}, {s2*s1*s2*s1}}.

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

%K nonn,hard,more,bref

%O 1,1

%A _Dmitry I. Ignatov_, May 18 2025