This question is inspired by this coding puzzle. I've tried to figure a pattern but couldn't really, but ideally I'd like to see a proof even if I did see it (if it's possible).
Let $\{ \sigma_1,...\sigma_{n-1} \} $ be a family of permutations on $S_n$ defined as follows:
\begin{equation} \sigma_j(i)= \begin{cases} i, & \text{if}\ i\in[1,...,j-1] \\ n, & \text{if}\ i=j \\ i-1, & \text{if}\ i\in[j+1,...,n] \end{cases} \end{equation}
Concretely, $$\sigma_1 = (1\ n\ (n-1)\ ...\ 3\ 2)$$ $$\sigma_2 = (2\ n\ (n-1)\ ...\ 4\ 3)(1)$$ $$\sigma_3 = (3\ n\ (n-1)\ ...\ 5\ 4)(1)(2)$$ $$\sigma_{n-1} = (n\ (n-1))(1)(2)...(n-2) $$
In words, these can be visualized as follows: if the number $123...n$ represents $S_n$, and 'moving to the left' sends the left most digit to the right end, then
$\sigma_1$ moves $123...n$ to the left. $\sigma_2$ fixes $1$ and moves $23...n$ to the left. $\sigma_3$ fixes $12$ and moves $34...n$ to the left. And so on.
For example, in $S_5$, $\sigma_1$ transforms $12345$ to $23451$. $\sigma_2$ transforms $12345$ to $13452$. Etc.
My question is: What does $\sigma := \sigma_n\sigma_{n-1}...\sigma_1$ actually look like?
The reason I ask is because if we look at the "visualization" I described, we get the following:
$$n=1:\ \ \sigma (1) = 1$$ $$n=2:\ \ \sigma(12) =21$$ $$n=3:\ \ \sigma(123) =213$$ $$n=4:\ \ \sigma(1234) =2431$$ $$n=5:\ \ \sigma(12345) =24153$$ $$n=6:\ \ \sigma(123456) =246315$$ $$n=7:\ \ \sigma(1234567) =2461537$$ $$n=8:\ \ \sigma(12345678) =24683751$$ $$n=9:\ \ \sigma(123456789) =246815973$$
All I wanted, really, was to predict how $\sigma$ transforms $123...n$ in its respective $S_n$. I can see the pattern of all even numbers neatly distributed to the left, but can't really figure out anything for the remainder of the odd numbers to the right. Is this just a conincidence? Because looking at the cycles doesn't actually seem to show a pattern at all...:
$$n=1:\ \ \sigma = (1)$$ $$n=2:\ \ \sigma = (12)$$ $$n=3:\ \ \sigma = (12)$$ $$n=4:\ \ \sigma =(142)$$ $$n=5:\ \ \sigma =(13542)$$ $$n=6:\ \ \sigma =(156342)$$ $$n=7:\ \ \sigma =(142)(36)$$ $$n=8:\ \ \sigma =(1842)(3576)$$ $$n=9:\ \ \sigma =(156397842)$$