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.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.