q-binomial coefficient

Tags: #definition

q-binomial

The $q$ -binomial coefficient, also called Gaussian coefficients is given by

{[\binom{n}{k}]}_{q} = \frac{[n]_{q}!}{[k]_{q}! [n - k]_{q}}

These satisfies the q-Pascal recurrence relation:

{[\binom{n}{k}]}_{q} = {[\binom{n - 1}{k}]}_{q} + q^{k} {[\binom{n - 1}{k}]}_{q}

which creates a $q$ -Pascal triangle
Pasted image 20260307174846.png|500

Properties

this is always a polynomial
positive integer coefficients
symmetric
The above 3 can be proven with this theorem.
unimodal

Example:

{[\binom{4}{2}]}_{q} = \frac{[4]_{q} [3]_{q}}{[2]_{q}} = \frac{(1 + q + q^{2}) (1 + q + q^{2} + q^{3})}{1 + q} = 1 + q + 2 q^{2} + q^{3} + q^{4}