I'm reading this article: https://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732119-5/S0002-9947-1984-0732119-5.pdf , and I'm having trouble with lemma 4.3. More specifically I can't find a reason of why the action is transitive.
At the beginning of the article there is an explanation of the notation, but I'll write here something in that regard.
- $\pi(d)$ is the set of partitions of $d$, and by "partition of $d$" we mean a set of numbers that has the property that their sum equals $d$, so for example if $d=4$ then a partition of $4$ can be $[2,2]$ or $[3,1]$ or also $[4]$.
- To every partition of $d$ you can associate a permutation in the symmetric group $S_d$ which has the structure given by the partition, so we will write (improperly) "$\sigma\in [2,2]$" if $\sigma$ can be written as disjoint cycles of the form ( , )( , ) in $S_4$, or "$\sigma\in [3,1]$" if $\sigma$ has the form ( , , )( ) in $S_4$.
- At the beginning of the article there is a definition of the number $\nu (A)$ (where $A\in\pi(d)$), but basically if, for example $\nu (A)=d-t$, it means that every $\sigma\in A$ has exactly $t$ cycles (we count as cycles also the ones having only one element). For example $\nu([2,2])=4-2=2$, $\nu([3,1])=4-2=2$, $\nu([4])=4-1=3$.
- If $\sigma\in A$ (where $\sigma$ is a permutation and $A$ is a genuine partition) we write $\nu(\sigma):=\nu(A)$, and this is well defined.
Lemmas 4.1 and 4.2 can surely be useful, but I don't know how.