Counting the restricted Gaussian partitions of a finite vector space

Akman, Fusun ; Sissokho, Papa A.

Journal of integer sequences, Tome 18 (2015) no. 7

Résumé

A subspace partition $\Pi $ 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 $\Pi $ is the zero vector. The multiset $T_{\Pi }$ of dimensions of subspaces in $\Pi $ is called the type of $\Pi , or,$ 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 ${1,\dots ,n}$ 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 ? of Gaussian partitions of $V$. We then show that the cardinality of ? is a rational polynomial $R(q)$ in $q$, with $R(1) = p(n)$, where $p$ is the integer partition function.

Détail

Keywords: subspace partition, vector space partition, Gaussian partition, integer partition, q-analogue