# extending a partition of a number to get new partition

Let $$\lambda = (k_1^{m_1}\,k_2^{m_2})$$ where $$0 be a partition of $$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 $$0 be a partition of some positive integer $$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?