Journal of Integer Sequences, Vol. 18 (2015), Article 15.7.5

Counting the Restricted Gaussian Partitions of a Finite Vector Space

Fusun Akman and Papa A. Sissokho
4520 Mathematics Department
Illinois State University
Normal, IL 61790-4520


A subspace partition Π of a finite vector space V = V(n,q) of dimension n over GF(q) is a collection of subspaces of V such that their union is V, and the intersection of any two subspaces in Π is the zero vector. The multiset TΠ of dimensions of subspaces in Π is called the type of Π, or, a Gaussian partition of V. Previously, we showed that subspace partitions of V and their types are natural, combinatorial q-analogues of the set partitions of {1,...,n} and integer partitions of n respectively. In this paper, we connect all four types of partitions through the concept of "basic" set, subspace, and Gaussian partitions, corresponding to the integer partitions of n. In particular, we combine Beutelspacher's classic construction of subspace partitions with some additional conditions to derive a special subset 𝓖 of Gaussian partitions of V. We then show that the cardinality of 𝓖 is a rational polynomial R(q) in q, with R(1) = p(n), where p is the integer partition function.

Full version:  pdf,    dvi,    ps,    latex    

Received September 5 2014; revised versions received December 16 2014; July 15 2015. Published in Journal of Integer Sequences, July 16 2015.

Return to Journal of Integer Sequences home page