Partitions index the irreducible POLYNOMIAL representations of $GL(V)$. But the dual of a polynomial representation is not going to be polynomial. Recall that, if a representation $W$ is given by the map $\rho: GL_n \to GL_N$, then $W^{\ast}$ is given by $(\rho^T)^{-1}$. The presence of that inverse means that it is unlikely to be a polynomial map.
We can broaden the class of polynomal representation to that of algebraic representations, meaning that the map $\rho$ must be made of rational functions whose denominators are powers of the determinant. Irreducible rational representations of $GL_n$ are indexed by $n$-tuples of integers $(\lambda_1, \lambda_2, \ldots, \lambda_n)$ with $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n$; the polynomial representations correspond to the subset where $\lambda_1 \geq 0$. Then we can state that taking the dual replaces $(\lambda_1, \lambda_2, \ldots, \lambda_n)$ by $(-\lambda_n, -\lambda_{n-1}, \cdots, - \lambda_1)$.
Since I don't know how you are most comfortable thinking of the classification of representations of $GL_n$, I won't try to justify this, but it is pretty easy from most perspectives.