Let $\nu$ is a partition on $n$. Given a $n$ cycle $(1,2,\ldots,n)\in S_n$.
Let define $H^{m}((n);\mu)$ count the number of tuples $(\alpha,\tau_1,\ldots,\tau_r ,\beta)$ in symmetric group $S_n$ where $\alpha$ is fixed cycle of type $(n)$ and $\beta$ cycle of type $\mu$ and
1) r = len($\mu$)-1 where len() imply length of partition.
2) $$ (1,2,\ldots,n)\tau_1\ldots \tau_r =\beta$$
3) $\tau_i$ are transposition written as $(a_i , b_i)$ and $a_i <b_i$ such that $$ b_1\leq b_2\leq\ldots \leq b_r. $$
For example in $S_5$ elements $H^m(5,(2^2 , 1))$ are collection of monotonic transposition of length 2. They are given by
$(1,2)(2,4)$ , $(12)(35)$ , $(13)(34)$ this when multiply by $(1,2,3,4,5)$ gives a $(2^2 ,1)$ cycle. Is there is combinatorial characterisation of such sequence? What would be a characteization of such a sequence ?