A384061 Number of antichains in the Bruhat order of type A_n.

3, 9, 250, 67595432

Offset: 1

Author

Dmitry I. Ignatov, May 18 2025

Keywords

• nonn
• hard
• more

Comments

The number of antichains in the Bruhat order of the Weyl group A_n (isomorphic to the symmetric group S_{n+1}).

Examples

			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.
       s1*s2*s1
        /   \
      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}, {s1*s2*s1}}.
		

References

• 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.

Crossrefs

Cf. A000142 (the order size), A005130 (the size of Dedekind-MacNeille completion), A384062.