Gaussian binomial coefficient

Short description: Family of polynomials

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, written as [math]\displaystyle{ \binom nk_q }[/math] or [math]\displaystyle{ \begin{bmatrix}n\\ k\end{bmatrix}_q }[/math], 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 [math]\displaystyle{ \mathbb{F}_q }[/math], a finite field with q elements; i.e. it is the number of points in the finite Grassmannian [math]\displaystyle{ \mathrm{Gr}(k, \mathbb{F}_q^n) }[/math].

Definition

The Gaussian binomial coefficients are defined by:

[math]\displaystyle{ {m \choose r}_q = \frac{(1-q^m)(1-q^{m-1})\cdots(1-q^{m-r+1})} {(1-q)(1-q^2)\cdots(1-q^r)} }[/math]

where m and r are non-negative integers. If r > m, this evaluates to 0. 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:

[math]\displaystyle{ [k]_q=\sum_{0\leq i\lt k}q^i=1+q+q^2+\cdots+q^{k-1}= \begin{cases} \frac{1-q^k}{1-q} & \text{for} & q \neq 1 \\ k & \text{for} & q = 1 \end{cases}, }[/math]

Dividing out these factors gives the equivalent formula

[math]\displaystyle{ {m \choose r}_q=\frac{[m]_q[m-1]_q\cdots[m-r+1]_q}{[1]_q[2]_q\cdots[r]_q}\quad(r\leq m). }[/math]

In terms of the q factorial [math]\displaystyle{ [n]_q!=[1]_q[2]_q\cdots[n]_q }[/math], the formula can be stated as

[math]\displaystyle{ {m \choose r}_q=\frac{[m]_q!}{[r]_q!\,[m-r]_q!}\quad(r\leq m). }[/math]

Substituting q = 1 into [math]\displaystyle{ \tbinom mr_q }[/math] gives the ordinary binomial coefficient [math]\displaystyle{ \tbinom mr }[/math].

The Gaussian binomial coefficient has finite values as [math]\displaystyle{ m\rightarrow \infty }[/math]:

[math]\displaystyle{ {\infty \choose r}_q = \lim_{m\rightarrow \infty} {m \choose r}_q = \frac{1} {(1-q)(1-q^2)\cdots(1-q^r)} = \frac{1}{[r]_q!\,(1-q)^r} }[/math]

Examples

[math]\displaystyle{ {0 \choose 0}_q = {1 \choose 0}_q = 1 }[/math]

[math]\displaystyle{ {1 \choose 1}_q = \frac{1-q}{1-q}=1 }[/math]

[math]\displaystyle{ {2 \choose 1}_q = \frac{1-q^2}{1-q}=1+q }[/math]

[math]\displaystyle{ {3 \choose 1}_q = \frac{1-q^3}{1-q}=1+q+q^2 }[/math]

[math]\displaystyle{ {3 \choose 2}_q = \frac{(1-q^3)(1-q^2)}{(1-q)(1-q^2)}=1+q+q^2 }[/math]

[math]\displaystyle{ {4 \choose 2}_q = \frac{(1-q^4)(1-q^3)}{(1-q)(1-q^2)}=(1+q^2)(1+q+q^2)=1+q+2q^2+q^3+q^4 }[/math]

[math]\displaystyle{ {6 \choose 3}_q = \frac{(1-q^6)(1-q^5)(1-q^4)}{(1-q)(1-q^2)(1-q^3)}=(1+q^2)(1+q^3)(1+q+q^2+q^3+q^4)=1 + q + 2 q^2 + 3 q^3 + 3 q^4 + 3 q^5 + 3 q^6 + 2 q^7 + q^8 + q^9 }[/math]

Combinatorial descriptions

Inversions

One combinatorial description of Gaussian binomial coefficients involves inversions.

The ordinary binomial coefficient [math]\displaystyle{ \tbinom mr }[/math] 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).

So, for example, the [math]\displaystyle{ {4 \choose 2} = 6 }[/math] words using 0s and 1s are [math]\displaystyle{ 0011, 0101, 0110, 1001, 1010, 1100 }[/math].

To obtain the Gaussian binomial coefficient [math]\displaystyle{ \tbinom mr_q }[/math], each word is associated with a factor q^d, 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.

With the example above, there is one word with 0 inversions, [math]\displaystyle{ 0011 }[/math], one word with 1 inversion, [math]\displaystyle{ 0101 }[/math], two words with 2 inversions, [math]\displaystyle{ 0110 }[/math], [math]\displaystyle{ 1001 }[/math], one word with 3 inversions, [math]\displaystyle{ 1010 }[/math], and one word with 4 inversions, [math]\displaystyle{ 1100 }[/math]. This is also the number of left-shifts of the 1s from the initial position.

These correspond to the coefficients in [math]\displaystyle{ {4 \choose 2}_q = 1+q+2q^2+q^3+q^4 }[/math].

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. The path takes a step right for each 0 and a step up for each 1. 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.

Balls into bins

Let [math]\displaystyle{ B(n,m,r) }[/math] be the number of ways of throwing [math]\displaystyle{ r }[/math] indistinguishable balls into [math]\displaystyle{ m }[/math] indistinguishable bins, where each bin can contain up to [math]\displaystyle{ n }[/math] balls. The Gaussian binomial coefficient can be used to characterize [math]\displaystyle{ B(n,m,r) }[/math]. Indeed,

[math]\displaystyle{ B(n,m,r)= [q^r] {n+m \choose m}_q. }[/math]

where [math]\displaystyle{ [q^r]P }[/math] denotes the coefficient of [math]\displaystyle{ q^r }[/math] in polynomial [math]\displaystyle{ P }[/math] (see also Applications section below).

Properties

Reflection

Like the ordinary binomial coefficients, the Gaussian binomial coefficients are center-symmetric, i.e., invariant under the reflection [math]\displaystyle{ r \rightarrow m-r }[/math]:

[math]\displaystyle{ {m \choose r}_q = {m \choose m-r}_q. }[/math]

In particular,

[math]\displaystyle{ {m \choose 0}_q ={m \choose m}_q=1 \, , }[/math]

[math]\displaystyle{ {m \choose 1}_q ={m \choose m-1}_q=\frac{1-q^m}{1-q}=1+q+ \cdots + q^{m-1} \quad m \ge 1 \, . }[/math]

Limit at q = 1

The evaluation of a Gaussian binomial coefficient at q = 1 is

[math]\displaystyle{ \lim_{q \to 1} \binom{m}{r}_q = \binom{m}{r} }[/math]

i.e. the sum of the coefficients gives the corresponding binomial value.

Degree of polynomial

The degree of [math]\displaystyle{ \binom{m}{r}_q }[/math] is [math]\displaystyle{ \binom{m+1}{2}-\binom{r+1}{2}-\binom{(m-r)+1}{2} = r(m-r) }[/math].

q identities

Analogs of Pascal's identity

The analogs of Pascal's identity for the Gaussian binomial coefficients are:

[math]\displaystyle{ {m \choose r}_q = q^r {m-1 \choose r}_q + {m-1 \choose r-1}_q }[/math]

and

[math]\displaystyle{ {m \choose r}_q = {m-1 \choose r}_q + q^{m-r}{m-1 \choose r-1}_q. }[/math]

When [math]\displaystyle{ q=1 }[/math], these both give the usual binomial identity. We can see that as [math]\displaystyle{ m\to\infty }[/math], both equations remain valid.

The first Pascal analog allows computation of the Gaussian binomial coefficients recursively (with respect to m ) using the initial values

[math]\displaystyle{ {m \choose m}_q ={m \choose 0}_q=1 }[/math]

and also shows that the Gaussian binomial coefficients are indeed polynomials (in q).

The second Pascal analog follows from the first using the substitution [math]\displaystyle{ r \rightarrow m-r }[/math] and the invariance of the Gaussian binomial coefficients under the reflection [math]\displaystyle{ r \rightarrow m-r }[/math].

These identities have natural interpretations in terms of linear algebra. Recall that [math]\displaystyle{ \tbinom{m}{r}_q }[/math] counts r-dimensional subspaces [math]\displaystyle{ V\subset \mathbb{F}_q^m }[/math], and let [math]\displaystyle{ \pi:\mathbb{F}_q^m \to \mathbb{F}_q^{m-1} }[/math] be a projection with one-dimensional nullspace [math]\displaystyle{ E_1 }[/math]. The first identity comes from the bijection which takes [math]\displaystyle{ V\subset \mathbb{F}_q^m }[/math] to the subspace [math]\displaystyle{ V' = \pi(V)\subset \mathbb{F}_q^{m-1} }[/math]; in case [math]\displaystyle{ E_1\not\subset V }[/math], the space [math]\displaystyle{ V' }[/math] is r-dimensional, and we must also keep track of the linear function [math]\displaystyle{ \phi:V'\to E_1 }[/math] whose graph is [math]\displaystyle{ V }[/math]; but in case [math]\displaystyle{ E_1\subset V }[/math], the space [math]\displaystyle{ V' }[/math] is (r−1)-dimensional, and we can reconstruct [math]\displaystyle{ V=\pi^{-1}(V') }[/math] without any extra information. The second identity has a similar interpretation, taking [math]\displaystyle{ V }[/math] to [math]\displaystyle{ V' = V\cap E_{n-1} }[/math] for an (m−1)-dimensional space [math]\displaystyle{ E_{m-1} }[/math], again splitting into two cases.

Proofs of the analogs

Both analogs can be proved by first noting that from the definition of [math]\displaystyle{ \tbinom{m}{r}_q }[/math], we have:

[math]\displaystyle{ \binom{m}{r}_q = \frac{1-q^m}{1-q^{m-r}} \binom{m-1}{r}_q }[/math]

(1)

[math]\displaystyle{ \binom{m}{r}_q = \frac{1-q^m}{1-q^r} \binom{m-1}{r-1}_q }[/math]

(2)

[math]\displaystyle{ \frac{1-q^r}{1-q^{m-r}}\binom{m-1}{r}_q = \binom{m-1}{r-1}_q }[/math]

(3)

As

[math]\displaystyle{ \frac{1-q^m}{1-q^{m-r}}=\frac{1-q^r+q^r-q^m}{1-q^{m-r}}=q^r+\frac{1-q^r}{1-q^{m-r}} }[/math]

Equation (1) becomes:

[math]\displaystyle{ \binom{m}{r}_q = q^r\binom{m-1}{r}_q + \frac{1-q^r}{1-q^{m-r}}\binom{m-1}{r}_q }[/math]

and substituting equation (3) gives the first analog.

A similar process, using

[math]\displaystyle{ \frac{1-q^m}{1-q^r}=q^{m-r}+\frac{1-q^{m-r}}{1-q^r} }[/math]

instead, gives the second analog.

q-binomial theorem

There is an analog of the binomial theorem for q-binomial coefficients, known as the Cauchy binomial theorem:

[math]\displaystyle{ \prod_{k=0}^{n-1} (1+q^kt)=\sum_{k=0}^n q^{k(k-1)/2} {n \choose k}_q t^k . }[/math]

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

[math]\displaystyle{ \prod_{k=0}^{n-1} \frac{1}{1-q^kt}=\sum_{k=0}^\infty {n+k-1 \choose k}_q t^k. }[/math]

In the limit [math]\displaystyle{ n\rightarrow\infty }[/math], these formulas yield

[math]\displaystyle{ \prod_{k=0}^{\infty} (1+q^kt)=\sum_{k=0}^\infty \frac{q^{k(k-1)/2}t^k}{[k]_q!\,(1-q)^k} }[/math]

and

[math]\displaystyle{ \prod_{k=0}^\infty \frac{1}{1-q^kt}=\sum_{k=0}^\infty \frac{t^k}{[k]_q!\,(1-q)^k} }[/math].

Setting [math]\displaystyle{ t=q }[/math] gives the generating functions for distinct and any parts respectively. (See also Basic hypergeometric series.)

Central q-binomial identity

With the ordinary binomial coefficients, we have:

[math]\displaystyle{ \sum_{k=0}^n \binom{n}{k}^2 = \binom{2n}{n} }[/math]

With q-binomial coefficients, the analog is:

[math]\displaystyle{ \sum_{k=0}^n q^{k^2}\binom{n}{k}_q^2 = \binom{2n}{n}_q }[/math]

Applications

Gauss originally used the Gaussian binomial coefficients in his determination of the sign of the quadratic Gauss sum.

Gaussian binomial coefficients occur in the counting of symmetric polynomials and in the theory of partitions. The coefficient of q^r in

[math]\displaystyle{ {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 spaces defined over a finite field. In particular, for every finite field F_q with q elements, the Gaussian binomial coefficient

[math]\displaystyle{ {n \choose k}_q }[/math]

counts the number of k-dimensional vector subspaces of an n-dimensional vector space over F_q (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]\displaystyle{ {n \choose 1}_q=1+q+q^2+\cdots+q^{n-1} }[/math]

is the number of one-dimensional subspaces in (F_q)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 F_qⁿ is equal to

[math]\displaystyle{ q^{n-k} {n \choose k}_q }[/math].

This allows another interpretation of the identity

[math]\displaystyle{ {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]\displaystyle{ q^{k^2 - n k}{n \choose k}_{q^2} }[/math].

This version of the quantum binomial coefficient is symmetric under exchange of [math]\displaystyle{ q }[/math] and [math]\displaystyle{ q^{-1} }[/math].

See also

• List of q-analogs

References

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