>**Question:**  

>What are, provided their existence, examples of functions $f$ with the following properties:  
<br>
>\begin{align}f:& \ \mathbb{N}\times\mathbb{N}\ni(i,j)&\mapsto\  \quad\quad\quad\quad\quad\ \beta\in\lbrace 0,1\rbrace\\
&f(i,i)&=\quad\quad\quad\quad\quad\quad\quad\quad\quad\ 0\\
&f(i,j)&=\quad\quad\quad\quad\quad\quad\quad\ f(j,i)\\
&\sum_{i=1}^{\infty}{f(i,j)}&=\quad k\in\mathbb{N}\quad \forall n\ge n_0\ge k\\
&\sum_{j=1}^{\infty}{f(i,j)}&=\quad k\in\mathbb{N}\quad \forall n\ge n_0\ge k\\
&F\in\lbrace0,1\rbrace^{n\times n},\ 1\le i,j \le n,\ F_{ij}=f(i,j)&\implies\quad\quad\quad\quad\quad\quad \ |F|\ne 0\end{align}  
The calculation of $f(i,j)$ must not depend on $n$ but would ideally be parameterized by $k\in\mathbb{N}$   

>In reply to @PuckRombach's comment:
>Another restriction is that, given $k$ the $f(i,j)$ must yield matrices with the described properties for all $n\ge n_0\ge k$