Let $c_i$ be a given non-negative integer for all $i\in\{1,\ldots,n\}$. I would like to find the non-negative integers $a_i$ and $b_i$ for all $i\in\{1,\ldots,n\}$ such that:
\begin{align} \begin{cases} c_i = a_i+b_i, & \text{ for all } i\in\{1,\ldots,n\}\\ a_i < a_{i+1}, & \text{ for all } i\in\{1,\ldots,n-1\}\\ b_i > b_{i+1}, & \text{ for all } i\in\{1,\ldots,n-1\}\\ b_i\ge 0 \text{ and integer }, & \text{ for all } i\in\{1,\ldots,n\}\\ a_i\ge 0 \text{ and integer }, & \text{ for all } i\in\{1,\ldots,n\} \end{cases} \end{align}
How to solve this system? Is there a single solution?
What if $c_i=0$ $\forall i$? Then for the first condition you'd need $a_i=b_i=0$ $\forall i$. But then the 2nd and 3rd conditions cannot be satisfied.