I would like to know whether the notion of automorphism of the set of partitions of a positive integer $n$ has been considered so far or not. To make things clearer, I say that a partition of $n$ in $k$ summands $s\_1,s\_2,...s\_k$ sorted in decreasing order has signature $(a\_1,a\_2,...,a\_m)$ if and only if $\forall i\in\{1,\cdots,m\},\ \ a\_{i}>0$ and $\displaystyle{\sum\_{i=1}^{m}a\_{i}=k}$ and each $a\_i$ is the multiplicity of a summand: $a\_1$ is the number of times the largest summand appears, $a\_2$ the number of times the second largest summand appears, and so on. 


I thus call "automorphism of the set of partitions of $n$" any bijective map sending any partition of $n$ in $k$ summands of signature $(a\_1,a\_2,...a\_m)$ to a partition of $n$ in $k$ summands of signature $(a\_1, a\_2,...,a\_m)$.
For example, one can consider a map sending the partition $(6,5,1)$ of $12$ to the partition $(6,4,2)$.


If I'm not mistaken, the group of automorphisms of the set of partitions of $n$ is a subgroup $Aut\_{Part}(n)$ of $S\_{p(n)}$, where $p(n)$ is the number of partitions of $n$. What can be said about this group? Is it necessarily abelian? Solvable? Any interesting reference available on-line freely about related subjects?


Many thanks in advance.



Automorphisms of partitions
Let $P(n,\alpha)$ be the set of partitions of $n$ with signature $\alpha$. Such a set is invariant under your group, and any permutation of its elements is an automorphism. Your group is just $$G=\Pi\_\alpha S\_{|P(n,\alpha)|}.$$ 


