, 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
The Gaussian binomial coefficients are defined by:[1] where m and r are non-negative integers.
For r = 0, the value is 1 since both the numerator and denominator are empty products.
Although the formula at first appears to be a rational function, it actually is a polynomial, because the division is exact in Z[q] All of the factors in numerator and denominator are divisible by 1 − q, and the quotient is the q-number: Dividing out these factors gives the equivalent formula In terms of the q factorial
The Gaussian binomial coefficient has finite values as
: One combinatorial description of Gaussian binomial coefficients involves inversions.
counts the r-combinations chosen from an m-element set.
If one takes those m elements to be the different character positions in a word of length m, then each r-combination corresponds to a word of length m using an alphabet of two letters, say {0,1}, with r copies of the letter 1 (indicating the positions in the chosen combination) and m − r letters 0 (for the remaining positions).
, each word is associated with a factor qd, where d is the number of inversions of the word, where, in this case, an inversion is a pair of positions where the left of the pair holds the letter 1 and the right position holds the letter 0.
This is also the number of left-shifts of the 1s from the initial position.
Another way to see this is to associate each word with a path across a rectangular grid with height r and width m − r, going from the bottom left corner to the top right corner.
An inversion switches the directions of a step (right+up becomes up+right and vice versa), hence the number of inversions equals the area under the path.
: In particular, The evaluation of a Gaussian binomial coefficient at q = 1 is i.e. the sum of the coefficients gives the corresponding binomial value.
The analogs of Pascal's identity for the Gaussian binomial coefficients are:[2] and When
The first Pascal analog allows computation of the Gaussian binomial coefficients recursively (with respect to m ) using the initial values and also shows that the Gaussian binomial coefficients are indeed polynomials (in q).
and the invariance of the Gaussian binomial coefficients under the reflection
These identities have natural interpretations in terms of linear algebra.
is r-dimensional, and we must also keep track of the linear function
The second identity has a similar interpretation, taking
There is an analog of the binomial theorem for q-binomial coefficients, known as the Cauchy binomial theorem: Like the usual binomial theorem, this formula has numerous generalizations and extensions; one such, corresponding to Newton's generalized binomial theorem for negative powers, is In the limit
gives the generating functions for distinct and any parts respectively.
With the ordinary binomial coefficients, we have: With q-binomial coefficients, the analog is: Gauss originally used the Gaussian binomial coefficients in his determination of the sign of the quadratic Gauss sum.
[3] Gaussian binomial coefficients occur in the counting of symmetric polynomials and in the theory of partitions.
The coefficient of qr in 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 spaces defined over a finite field.
In particular, for every finite field Fq with q elements, the Gaussian binomial coefficient counts the number of k-dimensional vector subspaces of an n-dimensional vector space over Fq (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 is the number of one-dimensional subspaces in (Fq)n (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 Fqn is equal to This allows another interpretation of the identity 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 This version of the quantum binomial coefficient is symmetric under exchange of