This is a follow-up question of my SO post I'll briefly mention it here.

So given a $n$ cycle say $(1,2,\ldots,n)$, what are the monotonic 2 -tuples, of the form $(a,b)(c,d)$, monotonicity in on the 2nd coordinate that is $b<d$ such that,

$$(1,2,\ldots,n)(a,b)(c,d)=\, \text{is still a $n$ cycle } $$ By default transposition $(a,b)$ imply $a<b$.

It was answered in the post that this construction can be extended to all $n$ cycles and the tuples are classified by the crossing chords etc.

I read in the post (genus zero permutation and noncrossing partition) about the bijection genus 0 permutation and non-crossing partitions.

Given a $n$ cycle $(1,2,\ldots,n)\in S_n$.

Let $H_g^{m}((1,2,\ldots, n);\mu)$ be the number of tuples $(\tau_1,\ldots,\tau_r)$ in symmetric group $S_n$ such that

$$(1,2,\ldots,n)\tau_1\ldots \tau_r =: \beta$$

has cycle type $\mu$, where $\tau_i$ are transposition written as $(a_i , b_i)$ with $a_i <b_i$, and $$ b_1\leq b_2\leq\ldots \leq b_r,$$ $$r=2g-1+\text{len}(\mu).$$

I guess the bijection between genus 0 permutation and non-crossing partition is a bijection between $H_0^{m}((n);\mu)$ where $\mu = 1^n$, not imposing the monotonicity conditions.

Now question is about bijection involving $H_g^{m}((1,2,\ldots,n);\mu)$ where $g\geq 1$ and $\mu$ can be of any cycle type and crossing and non-crossing partitions. Is there any bijection known to exist in this case? Say for example in the case of genus 1. So first thing that comes in mind is how the genus 1 is related to genus 0. So genus set of transposition differ from genus 0 by a pair of tuples $(a,b)(c,d)$ which keep the cycle $(1,2\ldots, n)$ fixed. These tuplese are given by crossing portion described in the post cited above.