Once established a certain partition of n, I am interested in finding out which is the exact number of partitions with same amount of parts $l(\lambda)$ but that are dominated by the first one.
for a detailed description on dominance order recall this wikipedia entry:
Dominance ordering on partitions of n.
So find a way to calculate $f_\lambda$ that:
$$f_\lambda = \#{\{\mu\in P ; l(\mu)=l(\lambda); \mu<\lambda \}}$$
Of course I expect to be related to $\lambda$'s components $\{\lambda_i\}$. For instance a multivariate generating function (as depends on a general partition definition) or a recursive formula would represent a feasible answer to this question.
Partial answers for certain partition length would also be appreciated.
I have already found assuming base partition is $\lambda=\{\lambda_1\ge\lambda_2...\}$:
If $l(\lambda)=1$ then $f_\lambda = |\lambda|=\lambda_1$
If $l(\lambda)=2$ then $f_\lambda = \sum_{k=\lambda_2}^{\lfloor(\lambda_1-\lambda_2)/2\rfloor}{k}+\sum_{k=1}^{\lfloor(\lambda_1-\lambda_2)/2\rfloor}{k}=(2(\lfloor(\lambda_1-\lambda_2)/2\rfloor)^2-\lambda_2^2+(\lfloor(\lambda_1-\lambda_2)/2\rfloor))/2$
No further advance seems easy.
Thanks in advance.