Assume I have $n \in \mathbb{N}$ and $p_0, ..., p_{n-1}$ such that $0 \le p_k \le 1$ for all $k = 0,...,n-1$.
Now consider the following recurrence relation:
$\begin{align}P_{0,0}&=1\\P_{a,0}&=1-\sum_{k=0}^{a-1}\binom{a}{k}P_{k,a-k}\;\mathrm{ when } \; a\ge1\\P_{a,b}&=P_{a,0} \prod_{k=0}^{a}(1-p_k)^{b\binom{a}{k}} \;\mathrm{ when } \; a\ge1, b\ge1 \end{align}$
I am mainly insterested in the values $P_{a,n-a}$ for $a = 0,...,n$. I have the following conjecture which I have verified analytically using Mathematica up to $n=5$:
$P_{a,n-a} \le \left(\frac{a}{n}\right)^{a} \left(\frac{n-a}{n}\right)^{n-a} $
Note that obviously always $P_{a,b}<1$. So far I have figured, that the equality can be reached if we set $p_0 = \frac{a}{n}$ and $p_k=0$ for $k=1,...,n-1$.
My question is how would you proceed to prove this kind of bound? I thought maybe I could $log$ the whole thing and then try to apply Jensen's inequality to simplify it, but did not succeed. My attempt to prove by contradiction did not go very far either.