Gaussian binomial

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In mathematics, the Gaussian binomials (sometimes called the Gaussian coefficients, or the q-binomial coefficients) are the q-analogs of the binomial coefficients.

[edit] Definition

The Gaussian binomials are defined by

${m \choose r}_q = \frac{(1-q^m)(1-q^{m-1})\cdots(1-q^{m-r+1})} {(1-q)(1-q^2)\cdots(1-q^r)}.$

One can prove that

$\lim_{q\rightarrow 1^-} {m \choose r}_q = {m \choose r}.$

The Pascal identities for the Gaussian binomials are

${m \choose r}_q = q^r {m-1 \choose r}_q + {m-1 \choose r-1}_q$

and

${m \choose r}_q = {m-1 \choose r}_q + q^{m-r}{m-1 \choose r-1}_q.$

The Newton binomial formulas are

$\prod_{k=0}^{n-1} (1+q^kt)=\sum_{k=0}^n q^{k(k-1)/2} {n \choose k}_q t^k$

and

$\prod_{k=0}^{n-1} \frac{1}{(1-q^kt)}=\sum_{k=0}^\infty {n+k-1 \choose k}_q t^k.$

Like the ordinary binomial coefficients, the Gaussian binomials are center-symmetric i.e. invariant under the reflection $r \rightarrow m-r$ :

${m \choose r}_q = {m \choose m-r}_q.$

The first Pascal identity allows one to compute the Gaussian binomials recursively (with respect to m ) using the initial "boundary" values

${m \choose m}_q ={m \choose 0}_q=1$

and also incidentally shows that the Gaussian binomials are indeed polynomials (in q). The second Pascal identity follows from the first using the substitution $r \rightarrow m-r$ and the invariance of the Gaussian binomials under the reflection $r \rightarrow m-r$ . Both Pascal identities together imply

${m \choose r}_q = {{1-q^{m}}\over {1-q^{m-r}}} {m-1 \choose r}_q$

which leads (when applied iteratively for m, m − 1, m − 2,....) to an expression for the Gaussian binomial as given in the definition above.

[edit] Applications

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

${n+m \choose m}_q$

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

Gaussian binomials also play an important role in the enumerative theory of symmetric spaces defined over a finite field. In particular, for every finite field F_q with q elements, the Gaussian binomial

${n \choose k}_q$

counts the number v_n,k;q of different k-dimensional vector subspaces of an n-dimensional vector space over F_q (a Grassmannian). For example, the Gaussian binomial