A note on quantum products of Schubert classes in a Grassmannian

Given two Schubert classes $\sigma_{\lambda}$ and $\sigma_{\mu}$ in the quantum cohomology of a Grassmannian, we construct a partition $\nu$, depending on $\lambda$ and $\mu$, such that $\sigma_{\nu}$ appears with coefficient 1 in the lowest (or highest) degree part of the quantum product $\sigma_{\lambda}\star\sigma_{\mu}$. To do this, we show that for any two partitions $\lambda$ and $\mu$, contained in a $k$-by-$(n-k)$ rectangle and such that the 180-degree rotation of one does not overlap the other, there is a third partition $\nu$, also contained in the rectangle, such that the Littlewood-Richardson number $c_{\lambda\mu}^{\nu}$ is 1.