Billey, Konvalinka, Petersen, Solfstra, and Tenner recently presented a method for counting parabolic double cosets in Coxeter groups, used it to compute $p_n$, the number of $S_n$, $n\leq13$. In this paper, we derive new formula $p_n$ an efficient polynomial time algorithm evaluating formula. We use these results $n\leq5000$ prove asymptotic that was conjectured by Billey et al.

This paper studies the algebraic structure of transposition hypergroups with idempotent identity. Their subhypergroups and their properties are examined. Right, left and double cosets are defined through symmetric subhypergroups and their properties are studied. Further- more, this paper examines the homomorphisms, the behaviour of attrac- tive and non-attractive elements through them, as well ...

Let G be a connected reductive group over an algebraically closed field k of characteristic not 2; let θ ∈ Aut(G) be an involution and K = Gθ ⊆ G the fixed point group of θ and let P ⊆ G be a parabolic subgroup. The set K\G/P of (K , P)-double cosets in G plays an important role in the study of Harish Chandra modules. In [BH00] we gave a description of the orbits of symmetric subgroups in a fla...

We explore classical (relative) difference sets intersected with the cosets of a subgroup of small index. The intersection sizes are governed by quadratic Diophantine equations. Developing the intersections in the subgroup yields an interesting class of group divisible designs. From this and the Bose-Shrikhande-Parker construction, we obtain some new sets of mutually orthogonal latin squares. W...