In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients. The Gaussian binomial coefficient
Contents
Definition
The Gaussian binomial coefficients are defined by
where m and r are non-negative integers. For r = 0 the value is 1 since numerator and denominator are both empty products. Although the formula in the first clause appears to involve a rational function, it actually designates a polynomial, because the division is exact in Z[q]. Note that the formula can be applied for r = m + 1, and gives 0 due to a factor 1 − q0 = 0 in the numerator, in accordance with the second clause (for even larger r the factor 0 remains present in the numerator, but its further factors would involve negative powers of q, whence explicitly stating the second clause is preferable). All of the factors in numerator and denominator are divisible by 1 − q, with as quotient a q number:
dividing out these factors gives the equivalent formula
which makes evident the fact that substituting q = 1 into
a compact form (often given as only definition), which however hides the presence of many common factors in numerator and denominator. This form does make obvious the symmetry
Instead of these algebraic expressions, one can also give a combinatorial definition of Gaussian binomial coefficients. The ordinary binomial coefficient
Unlike the ordinary binomial coefficient, the Gaussian binomial coefficient has finite values for
Examples
Properties
Like the ordinary binomial coefficients, the Gaussian binomial coefficients are center-symmetric, i.e., invariant under the reflection
In particular,
The name Gaussian binomial coefficient stems from the fact that their evaluation at q = 1 is
for all m and r.
The analogs of Pascal identities for the Gaussian binomial coefficients are
and
There are analogs of the binomial formula, and of Newton's generalized version of it for negative integer exponents, although for the former the Gaussian binomial coefficients themselves do not appear as coefficients:
and
which, for
and
The first Pascal identity allows one to compute the Gaussian binomial coefficients recursively (with respect to m ) using the initial "boundary" values
and also incidentally shows that the Gaussian binomial coefficients are indeed polynomials (in q). The second Pascal identity follows from the first using the substitution
which leads (when applied iteratively for m, m − 1, m − 2,....) to an expression for the Gaussian binomial coefficient as given in the definition above.
Applications
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
Triangles
The Gaussian binomial coefficients can be arranged in a triangle for each q, which is Pascal's triangle for q=1.
Read line by line these triangles form the following sequences in the OEIS: