Gaussian binomial coefficient

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are <a href="/facts/Q-analog/7jXz12AP">q-analogs</a> of the <a href="/facts/Binomial_coefficients/Tsx7eY5h">binomial coefficients</a>. The Gaussian binomial coefficient, written as 
 
 
 
 
 
 
 
 (
 
 
 n
 k
 
 
 )
 
 
 
 
 q
 
 
 
 
 {\displaystyle {\binom {n}{k}}_{q}}
 
 or 
 
 
 
 
 
 
 [
 
 
 
 n
 
 
 
 
 k
 
 
 
 ]
 
 
 
 q
 
 
 
 
 {\displaystyle {\begin{bmatrix}n\\k\end{bmatrix}}_{q}}
 
, is a polynomial in q with integer coefficients, whose value when q is set to a prime power counts the number of subspaces of dimension k in a vector space of dimension n over 
 
 
 
 
 
 F
 
 
 q
 
 
 
 
 {\displaystyle \mathbb {F} _{q}}
 
, a <a href="/facts/Finite_field/UkMGJo3m">finite field</a> with q elements; i.e. it is the number of points in the finite <a href="/facts/Grassmannian/QcmgLRq8">Grassmannian</a> 
 
 
 
 
 G
 r
 
 (
 k
 ,
 
 
 F
 
 
 q
 
 
 n
 
 
 )
 
 
 {\displaystyle \mathrm {Gr} (k,\mathbb {F} _{q}^{n})}
 
.

Gaussian binomial coefficient open-in-new

Gaussian binomial coefficient