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 A383875 #51 Jul 02 2025 15:50:14
%S A383875 1,3,19,213,3781,98407,3550919
%N A383875 Number of pairs in the Bruhat order of type A_n.
%C A383875 The number of ordered pairs in the Bruhat poset of the Weyl group A_n (isomorphic to the symmetric group S_{n+1}).
%D A383875 A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.
%H A383875 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 A383875 For n=0, the only element is 1 (identity) so a(0)=1.
%e A383875 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 A383875 For n=2 the line (Hasse) diagram is below.
%e A383875 s1*s2*s1
%e A383875 / \
%e A383875 s2*s1 s1*s2
%e A383875 | X |
%e A383875 s2 s1
%e A383875 \ /
%e A383875 1
%e A383875 The order relation consists of the six reflexive pairs, the eight pairs shown in the diagram as edges, and the five pairs (1, s2*s1), (1, s1*s2), (1, s1*s2*s1), (s1, s1*s2*s1), and (s2, s1*s2*s1). So a(2) = 6+8+5 = 19.
%Y A383875 Cf. A000142 (the order size), A002538 (edges in the cover relation), A005130 (the size of Dedekind-MacNeille completion), A384061 (antichains), A384062 (maximal antichains).
%K A383875 nonn,more
%O A383875 0,2
%A A383875 _Dmitry I. Ignatov_, May 18 2025
%E A383875 a(0)=1 prepended by _Sara Billey_, Jul 02 2025