Counting subspaces of a finite vector space

In this expository article, we discuss the relation between the Gaussian binomial and multinomial coefficients and ordinary binomial and multinomial coefficients from a combinatorial viewpoint, based on expositions by Butler, Knuth and Stanley.Comment: 15 pages, 3 figures, corrected versio

Reduction theory for a rational function field

Let $G$ be a split reductive group over a finite field \Fq. Let F=\Fq(t) and let \A denote the ad\`eles of $F$. We show that every double coset in G(F)\bsl G(\A)/ K has a representative in a maximal split torus of $G$. Here $K$ is the set of integral ad\`elic points of $G$. When $G$ ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.Comment: 10 page