I'm solving a problem that a mathematician can probably help me with. I am not a mathematician myself, so please forgive the imprecise terminology.
Let us say that we denote a matrix $M$ as $M(n,k)$ if it satisfies the following conditions:
- it is a square matrix $n\times n$
- the matrix contains only zeros and ones
- the sum of the values in any column or row is equal to $k$
Thus, for example, the notation $M(5,2)$ can be given to this matrix:
\begin{bmatrix} 1&1&0&0&0\\ 1&0&1&0&0\\ 0&1&0&1&0\\ 0&0&0&1&1\\ 0&0&1&0&1 \end{bmatrix}
Let's say $V$ is any vector of length $m$ containing only zeros and ones. I am looking for a function that has the following properties:
- assigns exactly one matrix $M(n,k)$ to any vector $V$
- each matrix $M(n,k)$ is the image of at least one vector $V$
Is there any neat function that has these properties?
$m,k,n \in Z$; $k < n$; $m$ is a function of $k$ and $n$
(I know I can make a numbered list of all matrices $M(n,k)$ and binary encode their numbers, but I'm looking for a more elegant solution.)
