Gauss polynomials

Important

Gauss polynomials, also called $q$ -binomial coefficients, are polynomials that generalize the usual binomial coefficients. Instead of counting combinations, they count certain objects in combinatorics, but with a parameter $q$ that keeps track of extra information. The Gauss polynomial is written as $\displaystyle \binom{n}{k}_q$ and is defined by:

\binom{n}{k}_q = \frac{(1-q^n)(1-q^{n-1})\cdots(1-q^{n-k+1})}{(1-q^k)(1-q^{k-1})\cdots(1-q)}

When $q=1$ , this formula gives the usual binomial coefficient $\binom{n}{k}$ .

Important properties

Gauss polynomials are polynomials in $q$ with integer coefficients.
When $q=1$ , $\binom{n}{k}_q = \binom{n}{k}$ (the usual binomial coefficient).
They satisfy a recurrence relation similar to Pascal's triangle:

\binom{n}{k}_q = q^k \binom{n-1}{k}_q + \binom{n-1}{k-1}_q

They count the number of $k$ -dimensional subspaces of an $n$ -dimensional vector space over a finite field with $q$ elements.