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Counting the Restricted Gaussian Partitions of a Finite Vector Space
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Fusun Akman and Papa A. Sissokho

4520 Mathematics Department

Illinois State University

Normal, IL 61790-4520

USA

**Abstract:**

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 *T*_{Π} 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.

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

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