varieties and Schubert classes by partitions. Let $\Lambda(r,n-r)$ denote the set of partitions with $r$-parts, whose largest part is at most $n-r$, i.e. $$\Lambda(r,n-r) = 
\{ \lambda = \mathbf{0} \leq \lambda_1 \leq \cdots \leq \lambda_r 
\leq \mathbf{n-r} \}.$$ There is a simple correspondence between these partitions and $01$-strings: given a partition $\lambda \in \Lambda(r,n-r)$, one can construct a string as follows.

> 0\. Begin with the empty string and $l=1$.\
> 1. Append $\lambda_l - \lambda_{l-1}$ [[‘$0$’]{}s ]{}.\
> 2. If $l=n-r$ stop.\
> 3. Append a [[‘$1$’]{} ]{}.\
> 4. Increment $l$ and repeat steps 1-4.\

$\lambda = \mathbf{0} \leq 0 \leq 1 \leq 3 \leq 3 \leq \mathbf{5}$, corresponds to the string $101001100$.

In the case of Grassmannians, we will sometimes find it more convenient to use $\Lambda(r,n-r)$ to index our Schubert classes and Schubert varieties. We therefore denote these $S^\lambda$ and $\bar{\Omega}_\lambda$ respectively. It is worth noting at this time that $$\dim \Omega_\lambda = \sum_{l=1}^{n-r} \lambda_l =: |\lambda|.$$

### Induced flags

Whenever we have a full flag $F$ on a vector space $W$, and $V \subset W$ is a subspace, we get induced flags $F_V$ and $F_{W/V}$ on $V$ and $W/V$ respectively. These can be thought of 