Let $$p_{i,j} = \frac{\sum_{l=i}^{i+j-1} {l-1 \choose i-1} {m+n-l \choose m-i}}{{m+n \choose n}}$$ I am interested in approximating/upper bounding the sum $$ \sum_{i=1}^m \sum_{j=1}^n p_{i,j}(1-p_{i,j})$$ Putting it through Sympy's Concrete module showed no simplification. Does the summation have any reasonable closed form bounds?
I've been able to prove a few bounds for individual cases, say $m=1$ which was trivial but had a closed form solution.
For $m=1$, $i$ could only be $1$. $p_{i,j}$ simplifies to:
$$p_{1,j} = \frac{\sum_{l=1}^{j} 1}{{n+1 \choose n}}=\frac{j}{n+1}$$ Then $$\sum_{j=1}^n \frac{j}{n+1}(1-\frac{j}{n+1})$$ can be trivially evaluated and upper bounded using the equation for $\sum_i^n i$ and $\sum_i^n i^2 $