Topological connectivity of random permutation complexes

Let Sₙ denote the symmetric group on n=\1, , n\ with the uniform probability measure. For a permutation Sₙ let X_ denote the simplicial complex on the vertex set n whose simplices are all \i₀, , iₘ\ n such that i₀<<iₘ and (i₀) < < (iₘ). For r 0 let pᵣ (n) denote the probability that X_ is not topologically r-connected for Sₙ. It is shown that for fixed r 0 there exist constants 0<Cᵣ, Cᵣ' < such that \ Cᵣ (n) ʳn pᵣ (n) Cᵣ' (n) ^2rn. \