A384062
Number of maximal antichains in the Bruhat order of type A_n.
Original entry on oeis.org
2, 4, 43, 183667
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.
s1*s2*s1
/ \
s2*s1 s1*s2
| X |
s2 s1
\ /
1
The set of maximal antichains is {{1}, {s2, s1}, {s2*s1, s1*s2}, {s1*s2*s1}}.
- A. Bjorner, 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.
A383875
Number of pairs in the Bruhat order of type A_n.
Original entry on oeis.org
1, 3, 19, 213, 3781, 98407, 3550919
Offset: 0
For n=0, the only element is 1 (identity) so a(0)=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.
s1*s2*s1
/ \
s2*s1 s1*s2
| X |
s2 s1
\ /
1
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.
- A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.
A384959
Number of chains in the Bruhat order of type A_n.
Original entry on oeis.org
4, 36, 4524, 15166380, 2010484649524, 14206021962108887860
Offset: 1
- A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.
Cf.
A061710 (maximal chains),
A000142 (the order size),
A005130 (the size of Dedekind-MacNeille completion),
A384061 (number of antichains).
Showing 1-3 of 3 results.
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