Given positive integers $k$ and $n$, we define the probability distribution $p_{n,k}$ on $S_n$ as:
$$ p_{n,k}(\sigma):=\frac{\#\{(\tau_1,\dots,\tau_s)\mid \sigma=\tau_1\dots\tau_s, \text{ each }\tau_i\in S_n \text{ is a transposition and } s \in \{0,\dots,k\}\}}{\sum_{s=0}^k\binom{n}{2}^s}\quad (\sigma\in S_n). $$ This assigns the same mass to elements of each conjugacy class, but is in general different from the uniform distribution on $S_n$ - which we denote by $u_n$. For instance, its support is not the whole $S_n$ if $k<n-1$.
I am interested to see what can be said about the total variation distance (denoted by $\delta$) between $p_{n,k}$ and $u_n$, at least asymptotically. More precisely:
- Is there any upper bound for $\delta(p_{n,k},u_n)$ in terms of $k$ and $n$? With $n$ fixed, for what value of $k$ (if any) this distance is minmized? What is the asymptotic behavior if $k$ is a function of $n$ e.g. $k=n-1$?
