Let $e=[1,1,\ldots,1]\in\mathbb{Z}^n$. I am looking for a way to find a vector $a\in\mathbb{Z}^n$ such that:

- $\langle a,e\rangle=0$ and
- for every nonnegative $v\in\mathbb{Z}^n$ such that $\langle e,v\rangle=n$ and $v\neq e$ we have $\langle a,v\rangle\neq 0$

Here the vector $v$ is nonnegative if it has nonnegative entries. The $\langle a,v\rangle$ denotes the standard scalar product. I am looking for the $a$'s whose lengths would polynomially depend on $n$ (actually my cost is the maximum of the absolute values $|\langle a,v\rangle|$ over all $v$'s as above).

The question is motivated by my attempts to find a convenient way to cancel out, in calculations, the inputs of nontrivial partitions, leaving only the one coming from $1+1+\ldots+1=n$.