Say I have the following permutation
$$\sigma ={\begin{pmatrix}1&2&3&4&5&6&7&8\\1&2&3&8&4&5&6&7\end{pmatrix}}$$
which consists to let unchanged some first elements from $1$ to $k$ and to apply a circular shift on the elements $\{k+1, \ldots, n\}$.
What is the name for such transformation?
Thank you
$1, 2, 3$ each map to themselves. $1\to 1$. Period. $2\to 2.$ Period. $3\to 3.$ Period. We can represent this as $(1)(2)(3)$, but typically this is not customary.
So we can start with $4$:
$4 \to 8$, then
$8 \to 7$, then
$7\to 6$,
then $6\to 5$, and finally,
$5 \to 4$, completing the five cycle (bringing us back to the starting number for the cycle, $4$:
That gives us the permutation $\sigma = (48765)$. This can also be written as $$\sigma = (1)(2)(3)(48765) = (48765)$$
To get the cycle you are looking for, you'll need the following permutation:
$\alpha ={\begin{pmatrix}1&2&3&4&5&6&7&8\\1&2&3&5&6&7&8&4\end{pmatrix}} = (45678)$
The name for this transformation is merely "permutation" $\alpha$ on the set $S$ of $8$ numbers, specifically, both $\alpha, \sigma \in \mathbb S_8$, the group of permutations on $S$.
If we were talking about vertices of an octagon, numbered 1-8, then $(12345678)$ could be described as a rotation of the octagon.