A382346
Number of antichains in the Bruhat order on B_n.
Original entry on oeis.org
For n=1 the elements are 1 (identity) and s1, the order contains pair (1, s1). The antichains are {}, {1}, and {s1}.
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 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}}.
- A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.
Cf.
A005900 (the number of join-irreducible elements),
A378072 (the size of Dedekind-MacNeille completion).
A382350
Number of maximal antichains in the Bruhat order on B_n.
Original entry on oeis.org
2, 5, 215, 24828398365
Offset: 1
For n=1 the elements are 1 (identity) and s1, the order contains pair (1, s1). The maximal antichains are {1} and {s1}.
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 set of maximal antichains is {{1}, {s2, s1}, {s2*s1, s1*s2}, {s2*s1*s2, s1*s2*s1}, {s2*s1*s2*s1}}.
- A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.
Cf.
A382346 (antichains),
A005900 (the number of join-irreducible elements),
A378072 (the size of Dedekind-MacNeille completion)
A384090
Number of ordered pairs in the Bruhat order on B_n.
Original entry on oeis.org
3, 33, 847, 40249, 3089459, 350676009
Offset: 1
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.
- A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.
Cf.
A005900 (the number of join-irreducible elements),
A378072 (the size of Dedekind-MacNeille completion).
A384687
Number of elements in the Dedekind-MacNeille completion of the Bruhat order on D_n.
Original entry on oeis.org
4, 42, 1292, 114976, 29735760
Offset: 2
For n=2 the Bruhat order on D_2 consists of four elements, 1 (identity), s1, s2, and s2*s1. Its completion forms the diamond lattice and coincides with the order.
s2*s1
/ \
s1 s2
\ /
1
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