For a permutation $\pi\in S_n$ we define its bar permutation $\bar\pi$ as: $$\bar\pi=[\bar\pi(1),...\bar\pi(n)]\colon\; \bar\pi(i)=\#\{j\le i\colon \pi(j)\ge\pi(i)\}$$ If we define bar space as $B_n=[1]\times[2]\times ...[n-1]\times[n]$, the bar operation is a one-to-one map from $S_n$ to $B_n$ (see problem 3.16 in "Combinatorial Problems and Exercises" by Lovász). $B_n$ has the advantage that it decomposes the permutation to independent components (say if permutation is uniformly chosen at random its components in $B_n$ space are indpendent)
Moroever, for two permutations $\pi,\sigma\in S_n$ permutation composition is defined as: $$\sigma\circ\pi = [\sigma(\pi(1)),\sigma(\pi(2)),...,\sigma(\pi(n))]$$
Is it possible to express composition in the bar space, i.e., to compute $\overline{\sigma\circ\pi}$ as a function of $\bar\sigma$ and $\bar\pi$?