For $k\ge2$ and $M\ge k+2$ two integers, a permutation code matrix $C$ is a $\binom Mk\times M$ matrix which columns contain all distinct permutations of $M-k$ zeroes and $k$ ones.
Page 44 of his work, Dahlin writes that for different rows $x$ and $y$ of $C$ there are $\binom{M-2}{k-1}$ coordinates in which $x$ has a one and $y$ has a zero, and vice versa.
How can one prove this? Is there a paper on this?
There are $\binom Mk$ permutations of $k$ ones and $M-k$ zeroes. This is clear because this corresponds to the number of ways one can chose $k$ elements in a set of $M$, these elements being the positions of the ones.
Now consider the rows $x$ and $y$. They contain the value of $x^{\text{th}}$ and $y^{\text{th}}$ elements in the different permutations. If the $x^{\text{th}}$ element of a permutation is $1$ and the $y^{\text{th}}$ is $0$ or vice-versa, we have $M-2$ places left and $k-1$ ones left, so there are $\binom{M-2}{k-1}$ permutations verifying these conditions.