Let $\lambda = (k_1^{m_1}\,k_2^{m_2})$ where $0<k_1<k_2$ be a partition of $n$ in the power notation.

Let $\mu = p_0^{r_0}\,p_1^{r_1}\,\cdots\, p_t^{r_t} \,(k_1^{m_1}\,k_2^{m_2})\,q_0^{s_0}\,q_1^{s_1}\,\cdots\, q_u^{s_u}$ where $0<p_0<p_1<\cdots<p_t<k_1<k_2<q_0<q_1<\cdots<q_u$ be a partition of some positive integer $m>n$.

Pictorially, I am adding new rows of length different from $r$ and $s$ on top and bottom of the Ferrer diagram of $\lambda$.

My question is, Given an $m > n$, how many such $\mu$'s are there which are also partition of $m$?

Also, what is the relation between $\lambda$ and $\mu$.

More precisely are they comparable in any natural partial order defined on partitions and set of all such $\mu$'a form the ideal generated by lambda?

Kindly share your thoughts.

Thank you.