Let $p(x) = a_{0} + a_{1}x + a_{2}x^{2} + \dots + a_{d}x^{d} + \dots + a_{2d+\alpha}x^{2d+\alpha} \in \mathbb{Z}[x]$ where $\alpha \in \{0,1\}$ with the constraints

$(1)$ $0 \leq a_{i},a_{2d+\alpha-i} \leq i + 1 + h$ for some constant $h$.

$(2)$ $p(a) = b$ for some fixed integers $a,b$ with $a \neq 1$.

$(3)$ $p(1) = s$ for some constant s.

How many such polynomials are possible?

Does it have similarity to intersection of $h+d$-bounded partition of $s$ into atmost $2d+\alpha+1$ sums and $h+d$-bounded $\{a^{i}\}$-weighted partition of $b$ into atmost $2d+\alpha+1$ sums?