I am really trying to understand descent set permutations, but before that there are some things I want to make clear from this paper I am reading. That is:
http://emis.ams.org/journals/EJC/Volume_16/PDF/v16i1r32.pdf
Defintion 1.1: For $\sigma \in \zeta_n,$ let $\phi_{j,k}(\sigma) =\tau_1\cdots\tau_{j-1}k\tau_j\cdots\tau_n$ where: $$\tau_i = \begin{cases} \sigma_i, & \text{if $\sigma_i< k$} \\ \sigma_i+1, & \text{if $\sigma_i\ge k$} \end{cases}$$
Similarly, $\psi_j(\sigma)=\tau_1\cdots\tau_{j-1}\tau_{j+1}\cdots\tau_n$ where: $$\tau_i = \begin{cases} \sigma_i, & \text{if $\sigma_i< \sigma_j$} \\ \sigma_i-1, & \text{if $\sigma_i> \sigma_j$} \end{cases}$$
On page 3, Definition 1.1, could some one please explain how this definition works, because I do not get the explanation of what this $\phi_{j,k}(\sigma)$ and $\psi_j(\sigma)$ are supposed to do. A numerical example will be an excellent way for me to understand.
I would appreciate the help.
Seems straightforward from their description below. "Thus, $φ_{j,k}$ inserts the element k at position j, increasing elements larger than k by one and shifting elements to the right of position j one step further to the right. The map $ψ_j$ removes the element at position j, decreasing larger elements by one and shifting those to its right one step left.
Take a permutation of length 10 ($\zeta_n$). Now, if you want to create a permutation of size 11, you can introduce a number from 1-11 somewhere. Say you want to permute 4 to 6 ($\phi_{4,6}$), you introduce 6 at the position of 4, and increase all the previous assignments greater than equal to 6 by 1. I.e. if 7 permuted to 6 earlier, it would now permute to 7.
Removal is similar.