Subvarieties of Schubert varieties over finite fields

Question

Let $\mathbb{F}_q$ be a finite fields and suppose $\Omega_{\alpha} (\ell, m)$ denote the Schubert variety given by $$\Omega_{\alpha} (\ell, m)= \{ [P] \in G(\ell, m) : \dim (P \cap A_i) \ge i\}$$ where $A_1 \subset \dots \subset A_{\ell}$ is a fixed flag with dimension sequence $\alpha = (\alpha_1, \dots, \alpha_{\ell})$ inside an $m$-dimensional vector space $V$ over $\mathbb{F}_q$. I am interested in the following question: If $\beta = (\beta_1, \dots, \beta_{\ell})$ be an $\ell$-tuple that is smaller than $\alpha$ in Bruhat order, then how many "copies" of $\Omega_{\beta} (\ell, m)$ are contained in $\Omega_{\alpha}(\ell, m)$? I believe that this question is rather simple to answer when $\alpha = (m-\ell + 1, \dots, m)$ and $\beta = (m- \ell, \dots, m-1)$. I suppose in this case, it is enough to look for how many copies of $G(\ell, m - 1)$ are there inside a fixed $G(\ell, m)$, which is the same as counting the number of $m-1$ dimensional subspaces of $V$.