It is well-known, that any element $\rho$ of the symmetric group $S_n$ with $n-p$ cycles admits a unique presentation as a product of a sequence of transpositions $\{(a_i\,b_i)\}_{i = 1}^p$ with $a_i < b_i$ and $b_i < b_{i+1}$ for any $i$. Another way to say it is that the sum $\sum_{j = 1}^n e_j(J_2,\ldots,J_n)$ of elementary symmetric polynomials evaluated on Jucys-Murphy elements is a central element of the group algebra $\mathbb C[S_n]$, that can be presented as $\sum_{\tau \in S_n} \tau$. We refer to a sequence of transpositions $\{(a_i\,b_i)\}_{i = 1}^p$ with $a_i < b_i$ and $b_i < b_{i+1}$ for any $i$ to as a strictly monotone factorization.

Thus, there is a natural action of $S_n$ on the set of strictly monotone factorizations, isomorphic to the action of $S_n$ on itself by conjugations. This action can be completely described in the terms of strictly monotone factorizations only.

We may also consider the set $M_p$ of weakly monotone factorizations, i.e. the set of sequences $\{(a_i\,b_i)\}_{i = 1}^p$ with $a_i < b_i$ and $b_i \le b_{i+1}$. This set is naturally related to the complete homogeneous symmetric polynomial $h_p$ evaluated on Jucys-Murphys. The set $M_p$ admits a natural map $\pi\colon M_p \to S_n$ -- the product of the elements of the sequence in the prescribed order.

Is there a natural structure of $S_n$-space on $M_p$, such that $\pi$ is a morphism of $S_n$-spaces, where $S_n$ acts on itself by conjugations?