Mappings of Subspaces into Subsets

Abstract Let Vn(q) denote the n-dimensional vector space over the finite field with q elements, and Ln(q) be the lattice of subspaces of Vn(q). Two rank- and order-preserving maps from Ln(q) onto the lattice of subsets of an n-set are constructed. Three equivalent formulations of these maps are given: an inductive procedure based on an elementary combinatorial interpretation of a well-known pair of difference equations satisfied by the Gaussian coefficients [ n k ], a direct set-theoretical definition, and, a direct definition involving a certain pair of modular chains in Ln(q). The direct set-theoretical definition of one of these maps has already been given by Knuth. Knuth's map, however, may be systematically discovered by means of the inductive procedure and the direct lattice-theoretic definition shows how it can be generalized. As a further application of the pair of difference equations satisfied by [ n k ], a direct-combinatorial proof of an identity of Carlitz that expands Gaussian coefficients in terms of binomial coefficients has been formulated.