For context:
This problem arose when attempting to implement the Lanczos algorithm for putting a matrix into tri-diagonal form in order to find the ground state energy and first excited state energy for a 1D spin chain.
The problem:
I will preface this question by saying that I'm am aware that given a permutation $\pi$, that we can find a permutation matrix $P_{\pi}$ with elements defined by
\begin{equation} p_{ij}= \begin{cases} 1 & \pi(i)=j\\ 0 & \text{otherwise} \end{cases}. \end{equation}
Now, according to my notes we can write the matrix $P_{ij}$ as,
\begin{equation} P_{ij}=2\vec{S_i}\cdot\vec{S_j}+\frac{1}{2}I= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \end{equation}
where $\vec{S_{i,j}}$ are $2\times 2$ spin matrices. According to my professor (and here is where my doubts begin),
\begin{equation} P_{01}\begin{pmatrix}x_0 \\ x_1 \\ x_2 \\ x_3 \end{pmatrix}=\begin{pmatrix}x_0 \\ x_2 \\ x_1 \\ x_3 \end{pmatrix}. \end{equation}
That is, we permute elements $1$ and $2$. My question is, should we not be permuting elements $0$ and $1$, since the matrix is $P_{01}$, and not $P_{12}$.
My professor then goes onto state that the Hamiltonian for a 1D spin chain can be written as $$H=\sum_{i=0}^{L-1}\vec{S_i}\cdot S_{i+1}\frac{1}{2}\bigg(\sum_{i=1}^{L-1}P_{ij}-\frac{L}{2}I\bigg).$$ I suppose my trouble with understanding this, is that I have no idea what $i,j$ are supposed to represent. If anyone can explain what $P_{ij}$ actually is, that would be appreciated.