My question is the following: An integer partition $\lambda$ can be represented as an integer sequence $(f_1,f_2,f_3, \cdots)$ where $f_i$ is the number of parts used in $\lambda$. For instance, $4 + 2 + 2 + 1 + 1 +1$ can be represented as $(3 , 2 , 0 , 1, 0 , 0, \cdots)$. Let $C$ be the set of all sequences $(f_1,f_2,f_3, \cdots)$ where
Each $f_i$ is a nonnegative integer
Only finitely many $f_i$s are nonzero.
Then, $\le$ gives a partial order on $C$ where $f = (f_1,f_2,f_3, \cdots) \le g = (g_1,g_2,g_3 , \cdots)$ if and only if $f_i \le g_i$ for all $i$. Is there any other partial order defined on $C$? If yes, can you share the partial order relation as well as the area of the mathematics it appears and why it is a natural partial order to discuss etc.
Motivation: There is a concept of partition ideal which is defined as follows: Let $I$ be a subset of $C$ such that for any $\pi \in I$ and $\lambda \in C$, if $\lambda \le \pi$, then $\lambda \in I$. We have some conjectures related to generating function of certain partition ideals. Thus, I am wondering what happens to the generating functions if we change the partial order relation all together.
Thanks in advance.
