Editing Gaussian binomial coefficient (section)

==Applications==

Gauss originally used the Gaussian binomial coefficients in his determination of the sign of the [[quadratic Gauss sum]].<ref>{{Cite book |last=Gauß |first=Carl Friedrich |date=1808 |title=Summatio quarumdam serierum singularium |url=https://eudml.org/doc/203313 |language=LA|location=Göttingen|publisher=Dieterich}}</ref>

Gaussian binomial coefficients occur in the counting of [[symmetric polynomial]]s and in the theory of [[partition (number theory)|partitions]]. The coefficient of ''q''<sup>''r''</sup> in

:<math>{n+m \choose m}_q</math>

is the number of partitions of ''r'' with ''m'' or fewer parts each less than or equal to ''n''. Equivalently, it is also the number of partitions of ''r'' with ''n'' or fewer parts each less than or equal to ''m''.

Gaussian binomial coefficients also play an important role in the enumerative theory of [[projective space]]s defined over a finite field. In particular, for every [[finite field]] ''F''<sub>''q''</sub> with ''q'' elements, the Gaussian binomial coefficient

:<math>{n \choose k}_q</math>

counts the number of ''k''-dimensional vector subspaces of an ''n''-dimensional [[vector space]] over ''F''<sub>''q''</sub> (a [[Grassmannian]]).  When expanded as a polynomial in ''q'', it yields the well-known decomposition of the Grassmannian into Schubert cells.  For example, the Gaussian binomial coefficient

:<math>{n \choose 1}_q=1+q+q^2+\cdots+q^{n-1}</math>

is the number of one-dimensional subspaces in (''F''<sub>''q''</sub>)<sup>''n''</sup> (equivalently, the number of points in the associated [[projective space]]).  Furthermore, when ''q'' is 1 (respectively −1), the Gaussian binomial coefficient yields the [[Euler characteristic]] of the corresponding complex (respectively real) Grassmannian.

The number of ''k''-dimensional affine subspaces of ''F''<sub>''q''</sub><sup>''n''</sup> is equal to

:<math>q^{n-k} {n \choose k}_q</math>.

This allows another interpretation of the identity

:<math>{m \choose r}_q = {m-1 \choose r}_q + q^{m-r}{m-1 \choose r-1}_q</math>

as counting the (''r'' − 1)-dimensional subspaces of (''m'' − 1)-dimensional projective space by fixing a hyperplane, counting such subspaces contained in that hyperplane, and then counting the subspaces not contained in the hyperplane; these latter subspaces are in bijective correspondence with the (''r'' − 1)-dimensional affine subspaces of the space obtained by treating this fixed hyperplane as the hyperplane at infinity.

In the conventions common in applications to [[quantum groups]], a slightly different definition is used; the quantum binomial coefficient there is
:<math>q^{k^2 - n k}{n \choose k}_{q^2}</math>.
This version of the quantum binomial coefficient is symmetric under exchange of <math>q</math> and <math>q^{-1}</math>.