Let $V=F_q^n$ be a vector space of dimension $n$ over the $q$-element field $F_q$ ($q$ is a prime number). Given a $i$-dimensional subspace of $V$. What is the number of the $j$-dimensional subspaces of $V$ which contain the given $i$-dimensional subspace ($j>i$). In the case $n=4$, $i=1$, $j=3$, the number is ${3 \brack 1}_q = \frac{q^3-1}{q-1}$. What is this number in general? Thank you very much.


Hint (lattice correspondence): if $W\le V$ then

$$\{A:W\le A\le V\}\cong\{B:0\le B\le V/W\}.$$

Make this correspondence more precise by partitioning each set according to dimension.

This reduces the problem to counting fixed dimension subspaces of a given finite vector space.


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