Gaussian Binomial Coefficient - Applications

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 F_q with q elements, the Gaussian binomial coefficient

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 coefficient

is the number of different lines in F_qn (a projective space).

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 .