Let $p_0=0$ and $p_i=a_1+\cdots+a_i$. We need a permutation $p$ of $1,\ldots,n$ with $-1\le p_i-p_{i-1}\le b_i$.

Whenever the permutation decreases, it must decrease by exactly one: a larger drop is forbidden, and equal values would repeat a prefix sum.

Suppose its first value is $r$. It must begin $r,r-1,\ldots,1$. If it increases before reaching $1$, it can never return to the remaining smaller values: moving down to them would require passing through a value already used.

The same argument applies to the unused values. Thus every valid permutation is obtained by splitting $1,2,\ldots,n$ into consecutive blocks and reversing each block. For example, blocks $[1,2]$, $[3]$, $[4,5,6]$ give $2,1,3,6,5,4$.

Inside a block, $a_i=-1$, which always satisfies its bound. Only the jumps at block starts need to be checked.