Let $S_k$ be the set of all permutations of $k+1$ elements $0,1,...,k$. introduce boundary maps $d_i : S_k \rightarrow S_{k-1}$ by deleting from permutation $\eta$ element $\eta(i)$ and monotone reordering and degeneracy
$s_i :S_k \rightarrow S_{k+1} $ by adding 1 to all elements with $\eta(j)>\eta(i)$ and incerting into the result a new element $\eta(i)+1$ right after $\eta(i)$ on $i+1$ place. It is a simplicial set, contractible and classifies reorderings of simplicial sets.

Is it known? May be in higher symmetric something?