I am playing with some questions concerning connections between 
certain poset partitions and their linear extensions. This is not
my usual playground, I just happened to stumble upon something.

When I was playing with these things, I came up with a very
simple construction that I cannot find anywhere.I suspect that this construction is a minor particular case of something well known
and much more general. It reminds of an equivariant map, but the group
is not fixed here. 


Let $n>2$, let $X$ be a finite set of cardinality $n$. Let $\oplus_n$ be an action
of $Z_n$ on $X$ such that the action of $1$ is a cycle of length $n$.
Pick $x\in X$ and let $\pi$ be a partition of the set $X$ such that
$\{x,x\oplus_n 1\}$ is the only nonsingleton block of $\pi$.

In a very simple way, this gives rise to an action $\oplus_{n-1}$ of $Z_{n-1}$ on $\pi$
by ,,squeezing the action'' at $x$:

Put

 * $\{x,x\oplus_n 1\}\oplus_{n-1} 1=\{x\oplus_n 2\}$
 * $\{x\oplus_n(n-1)\}\oplus_{n-1} 1=\{x,x\oplus_n 1\}$
 * $\{y\\}\oplus_{n-1} 1=\{y\oplus_n 1\}$ for any $y\neq x\oplus_n(n-1)$.

It can be visualized in a simple way by a digraph construction: if we
identify the action of $1$ on $X$ with an oriented cycle, this construction
corresponds to a contraction of an edge. 


Has anyone seen this construction before? 
Is there any name for it?