This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384090 #11 May 24 2025 00:03:26 %S A384090 3,33,847,40249,3089459,350676009 %N A384090 Number of ordered pairs in the Bruhat order on B_n. %C A384090 The number of ordered pairs in the Bruhat order of the Weyl group B_n (the hyperoctahedral group). %D A384090 A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64. %H A384090 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. %H A384090 Wikipedia, <a href="http://en.wikipedia.org/wiki/Bruhat_order">Bruhat order</a> %e A384090 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. %e A384090 For n=2 the line (Hasse) diagram is below. %e A384090 s2*s1*s2*s1 %e A384090 / \ %e A384090 s2*s1*s2 s1*s2*s1 %e A384090 | X | %e A384090 s2*s1 s1*s2 %e A384090 | X | %e A384090 s2 s1 %e A384090 \ / %e A384090 1 %e A384090 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. %Y A384090 Cf. A005900 (the number of join-irreducible elements), A378072 (the size of Dedekind-MacNeille completion). %K A384090 nonn,more %O A384090 1,1 %A A384090 _Dmitry I. Ignatov_, May 19 2025