Gaussian binomial coefficient


In mathematics, the Gaussian binomial coefficients are q-analogs of the binomial coefficients. The Gaussian binomial coefficient, written as or, 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 a finite field with q elements.

Definition

The Gaussian binomial coefficients are defined by
where m and r are non-negative integers. For 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. Note that the formula can be applied for, and gives 0 due to a factor in the numerator, in accordance with the second clause. All of the factors in numerator and denominator are divisible by, with as quotient a q number:
dividing out these factors gives the equivalent formula
which makes evident the fact that substituting into gives the ordinary binomial coefficient In terms of the q factorial, the formula can be stated as
a compact form, which however hides the presence of many common factors in numerator and denominator. This form does make obvious the symmetry for.
Unlike the ordinary binomial coefficient, the Gaussian binomial coefficient has finite values for :

Examples

Combinatorial description

Instead of these algebraic expressions, one can also give a combinatorial definition of Gaussian binomial coefficients. The ordinary binomial coefficient counts the -combinations chosen from an -element set. If one takes those elements to be the different character positions in a word of length, then each -combination corresponds to a word of length using an alphabet of two letters, say with copies of the letter 1 and letters 0.
The words using 0s and 1s would be 0011, 0101, 0110, 1001, 1010, 1100.
To obtain from this model the Gaussian binomial coefficient, it suffices to count each word with a factor, where is the number of "inversions" of the word: the number of pairs of positions for which the leftmost position of the pair holds a letter 1 and the rightmost position holds a letter 0 in the word. For example, there is one word with 0 inversions, 0011. There is 1 with only a single inversion, 0101. There are two words with 2 inversions, 0110, and 1001. There is one with 3, 1010, and finally one word with 4 inversions, 1100. This corresponds to the coefficients in. Noticing when q=1, the Gaussian binomial coefficient gives the same answer as the ordinary binomial coefficient does.
It can be shown that the polynomials so defined satisfy the Pascal identities given below, and therefore coincide with the polynomials given by the algebraic definitions. A visual way to view this definition is to associate to each word a path across a rectangular grid with sides of height and width, from the bottom left corner to the top right corner, taking a step right for each letter 0 and a step up for each letter 1. Then the number of inversions of the word equals the area of the part of the rectangle that is to the bottom-right of the path.

Balls into bins (urns)

Let be the number of ways of throwing balls into indistinguishable bins, where each bin can contain up to balls.
The Gaussian binomial coefficient can be used to characterize.
Indeed,
where denotes the coefficient of in polynomial .

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 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 become:
and
The first Pascal identity allows one to compute the Gaussian binomial coefficients recursively using the initial "boundary" values
and also incidentally shows that the Gaussian binomial coefficients are indeed polynomials. The second Pascal identity follows from the first using the substitution and the invariance of the Gaussian binomial coefficients under the reflection. Both Pascal identities together imply
which leads 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. 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 n. Furthermore, when q is 1, the Gaussian binomial coefficient yields the Euler characteristic of the corresponding complex Grassmannian.
The number of k-dimensional affine subspaces of Fqn is equal to
This allows another interpretation of the identity
as counting the -dimensional subspaces of -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 -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 and.

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: