I am wondering if there is a standard notation and name for the following. Let $\lambda$ be a partition $\lambda\_1\geq \lambda\_2\geq\cdots\geq \lambda\_r\geq 1$ of $n$ into $r$ parts. Then we can form a partition $\mu$ of $r$ by keeping track of how many times each integer occurs in $\lambda$. For example the partition (3,3,2,2,2,1,1) of 14 into 7 parts has associated partition $\mu=(3,2,2)$ of $7$ because one integer appears 3 times and 2 integers appear twice. If $\mu$ is a partition of $r$, let $C\_{n,\mu}$ be the number of partitions $\lambda$ of $n$ into $r$ parts such the partition associated to $\lambda$ is $\mu$. 



> 
> 
> > 
> > **Question.** What is $C\_{n,\mu}$ called and what is the standard notation? Is the some formula or combinatorial rule to compute it?
> > 
> > 
> > 
> 
> 
> 



A question of terminology regarding integer partitions
Let $\mu=(\mu\_1,\dots,\mu\_m)$ and $\gamma\_i$ be the number of parts equal $i$ in $\mu$. Then
$$\sum\_{i=1}^r \gamma\_i = m\quad\text{and}\quad\sum\_{i=1}^r i\cdot\gamma\_i = r.$$


Then $C\_{n,\mu}$ equals $\frac{1}{\gamma\_1!\cdots \gamma\_r!}$ times the number of solutions to
$$(\star)\qquad \mu\_1\cdot y\_1 + \dots + \mu\_m\cdot y\_m = n$$
in pairwise distinct positive integers $y\_1,\dots, y\_n$.


Without of the distinctness requirement, the number of solutions to $(\star)$ has generating function:
$$F(\mu,x) = \frac{x^{\mu\_1+\dots+\mu\_m}}{(1-x^{\mu\_1})\cdots (1-x^{\mu\_m})}.$$
To enforce the distinctness, one can use inclusion-exclusion.


For any unordered partition $P=(P\_1,\dots,P\_k)$ of the set $[m]$, we define $\mu\_P$ as "contraction" of $\mu$, where parts of $\mu$ with indices from the same part of $P$ are summed up and represent a single element of $\mu\_P$.


By inclusion-exclusion, we have $C\_{n,\mu}$ equal the coefficient of $x^n$ in
$$\frac{1}{\gamma\_1!\cdots \gamma\_r!} \sum\_{P} (-1)^{m-k}\cdot (|P\_1|-1)!\cdots (|P\_k|-1)!\cdot F(\mu\_P,x).$$


**Example.** For $\mu=(3,2,2)$, we have $m=3$, $r=3+2+2=7$, $\gamma=(0,2,1,0,0,0,0,0)$. The have the following set partitions of $[3]=\{1,2,3\}$ and the corresponding contracted partitions $\mu\_P$:


$P=\{\{1,2,3\}\}\qquad \mu\_P=(7)$


$P=\{\{1,2\},\{3\}\}\qquad \mu\_P=(5,2)$


$P=\{\{1,3\},\{2\}\}\qquad \mu\_P=(5,2)$


$P=\{\{2,3\},\{1\}\}\qquad \mu\_P=(4,3)$


$P=\{\{1\},\{2\},\{3\}\}\qquad \mu\_P=(3,2,2)$


So, the generating function for $C\_{n,(3,2,2)}$ is
$$\frac{1}{2} (2\cdot F((7),x) - F((5,2),x) - F((5,2),x) - F((4,3),x) + F((3,2,2),x))$$
$$={\frac {{x}^{13} \left( 3\,{x}^{6}+6\,{x}^{5}+8\,{x}^{4}+8\,{x}^{3}+6\,{x}^{2}+3\,x+1 \right) (1-x)^2}{ (1-x^3) (1-x^4) (1-x^5) (1-x^7) }}$$
$$=x^{13} + x^{14} + 2\cdot x^{15} + x^{16} + 2\cdot x^{17} + 2\cdot x^{18} + 3\cdot x^{19} + 3\cdot x^{20} + 6\cdot x^{21} + 3\cdot x^{22} + \dots.$$


For example, the coefficient of $x^{21}$ enumerates the following six partitions of 21: $(7,7,2,2,1,1,1)$, $(6,6,3,3,1,1,1)$, $(5,5,4,4,1,1,1)$, $(5,5,3,3,3,1,1)$, $(4,4,3,3,3,2,2)$, and $(5,5,5,2,2,1,1)$.


