The number of direct-sum decompositions of a finite vector space

TL;DR

This paper develops formulas for counting direct-sum decompositions of finite vector spaces over finite fields, extending classical combinatorial formulas like Stirling and Bell numbers to the q-analog setting.

Contribution

It introduces q-analog formulas for the number of direct-sum decompositions, paralleling set partition formulas such as Stirling and Bell numbers.

Findings

Formulas for the number of direct-sum decompositions with a given signature

q-analog of Stirling numbers of the second kind

q-analog of Bell numbers

Abstract

The theory of q-analogs develops many combinatorial formulas for finite vector spaces over a finite field with q elements--all in analogy with formulas for finite sets (which are the special case of q=1). A direct-sum decomposition of a finite vector space is the vector space analogue of a set partition. This paper develops the formulas for the number of direct-sum decompositions that are the q-analogs of the formulas for: (1) the number of set partitions with a given number partition signature; (2) the number of set partitions of an n-element set with m blocks (the Stirling numbers of the second kind); and (3) for the total number of set partitions of an n-element set (the Bell numbers).