A permutation matrix in compact (confusing) notation

Question

This article (free access), in equation 1.11 defines a permutation matrix as \begin{equation} P_{kl} = \begin{cases} \delta_{k,2l-1} \quad k\le n\\ \delta_{n+k,2l} \quad l \le n, \end{cases} \end{equation} where $\delta_{X,Y}$ is the Dirac delta function which is zero if $X\ne Y$ and has a value $1$ when $X=Y$. Now I am not sure how to construct this matrix, for a single element there are two possible outcomes. For example, if $n=2$, I can ask what is $P_{11}$ where note both $k = l = 1 <2$, and I have $\delta_{k,2l-1} = \delta_{1,1} = 1$ and $\delta_{n+k,2l} = \delta_{3,2} = 0$. Am I missing something here?

Answer 1

The matrix is $2n \times 2n$, it is worth saying it. I believe that there are typos in the conditions given by the authors, and it would be nice to write them. My impression is that the right formulas are $$P_{k,\ell} = \delta_{k,2\ell-1} \text{ if } 1 \le \ell \le n,$$
$$P_{k,\ell} = \delta_{k,2\ell-2n} \text{ if } n+1 \le \ell \ge 2n.$$ The corresponding permutation on $\{1,\ldots,2n\}$ sends $1,2\ldots,n$ on $1,3,\ldots,2n-1$ and sends $n+1,n+2\ldots,2n$ on $2,4,\ldots,2n$.