Gaussian binomial coefficient

Remarks

The coefficients satisfy the q-Pascal rule $\binom{n}{k}_q = \binom{n-1}{k-1}_q + q^k \binom{n-1}{k}_q$ for $n \ge 1$. By induction, $\binom{n}{k}_q$ is a polynomial in $q$ with nonnegative integer coefficients, of degree $k(n - k)$. At $q = 1$ it equals the binomial coefficient $\binom{n}{k}$. If $q$ is a prime power, it is the number of $k$-dimensional subspaces of $\F_q^n$.

Example: $\binom{4}{2}_q = \frac{(1 - q^4)(1 - q^3)}{(1 - q)(1 - q^2)} = (1 + q^2)(1 + q + q^2) = 1 + q + 2q^2 + q^3 + q^4$. At $q = 2$ this is $35$. Counting directly, $\F_2^4$ has $\frac{(2^4 - 1)(2^4 - 2)}{(2^2 - 1)(2^2 - 2)} = \frac{15 \cdot 14}{3 \cdot 2} = 35$ subspaces of dimension $2$.

Non-example: $q^{k(n-k)} \binom{n}{k}$ also equals $\binom{n}{k}$ at $q = 1$, but it is not the Gaussian binomial coefficient. For $n = 4$, $k = 2$ and $q = 2$ it gives $96$, not $35$.

The coefficient of $q^j$ in $\binom{n}{k}_q$ is the number of integer partitions of $j$ with at most $k$ parts, each at most $n - k$. Another common notation is $\left[ {n \atop k} \right]_q$.