A Short Proof of the Rank Formula for Inclusion Matrices using the Representation Theory of the Symmetric Group

TL;DR

This paper offers a new proof for the rank formula of inclusion matrices by leveraging the representation theory of the symmetric group, providing a novel algebraic approach to a classical combinatorial result.

Contribution

It introduces a module construction approach that simplifies the proof of the rank formula using symmetric group representations.

Findings

Constructed a $k ext{S}_n$-module from the inclusion matrix columns

Calculated the module's dimension to determine the matrix rank

Provided a new algebraic proof of the classical rank formula

Abstract

We present a new proof of the well known formula for the rank of the inclusion matrix by constructing a $k\mathcal{S}_n$-module spanned by the columns of this matrix and calculating its dimension.