Counting subspaces of a finite vector space

TL;DR

This paper explores the combinatorial relationships between Gaussian binomial coefficients, multinomial coefficients, and ordinary binomial coefficients, providing an expository overview based on classical expositions.

Contribution

It offers a clear combinatorial explanation of the connections between these coefficients, synthesizing prior expositions by Butler, Knuth, and Stanley.

Findings

Clarifies the combinatorial interpretation of Gaussian binomial coefficients

Establishes links between different types of binomial coefficients

Provides insights into counting subspaces in finite vector spaces

Abstract

In this expository article, we discuss the relation between the Gaussian binomial and multinomial coefficients and ordinary binomial and multinomial coefficients from a combinatorial viewpoint, based on expositions by Butler, Knuth and Stanley.