Let $\nu$ is a partition on $n$. Given a $n$ cycle $(1,2,\ldots,n)\in S_n$.
Let define $H_g^{m}((n);\mu)$ count the number of tuples $(\tau_1,\ldots,\tau_r)$ in symmetric group $S_n$.Let $\beta$ denote the cycle type.
$$ (1,2,\ldots,n)\tau_1\ldots \tau_r =\beta$$
$\tau_i$ are transposition written as $(a_i , b_i)$ and $a_i <b_i$ such that $$ b_1\leq b_2\leq\ldots \leq b_r. $$ $$r=2g-1+\text{len}(\beta) $$ A result in the paper https://arxiv.org/pdf/1005.0151.pdf state that when $\beta= 1^n$ then $$H_g^m((12..n), \beta )= Cat_{n-1}T(n+g-1,n-1).....(**) $$
I have the following expression the numbers which are called central factorial numbers $$T(a,b)= 2 \sum_{j={0}}^{b}(-1)^{b-j} \frac{j^{2a}}{(b-j)!(b+j)!}$$ where $Cat_n$ dentoe the catalan number. Even the paper gives a combinatorial description of $T(a,b)$.
My question is if there exists a combinatorial bijection? That is given a the seq in LHS of (**) I would construct an element belong to $Cat_{n-1}$ and $T(n-1+g,n-1)$ ?
I am also placing an example in the case $n=3, g=1$
The LHS set is given as follows $$[[(2,3),(2,3),(2,3),(1,3)],[(2,3),(1,3),(2,3),(2,3)],[(2,3),(1,3),(1,3),(1,3)],[(1,3),(2,3),(1,3),(2, 3)],[(1,3),(1,3),(2,3),(1,3)],[(1,2),(2,3),(2,3),(2,3)],[(1,2),(2,3),(1,3),(1,3)],[(1,2),(1,3),(1,3), (2,3)],[(1,2),(1,2),(2,3),(1,3)],[(1,2),(1,2),(1,2),(2,3)]] $$ The above gives a set of 10 tuple of length 4 each. it monotonic
Now $$Cat_2 =2$$ and $T(3,2)=5$ The elements of $Cat_2$ can be given as follows 2 cyles $$[[(2,3),(1,3)],[(1,2),(2,3)]]$$ I have no idea how to construct the set $T(3,2)$ to make an combinatorial bijection?