Splitting subspaces and a finite field interpretation of the Touchard-Riordan Formula

TL;DR

This paper explores the enumeration of T-splitting subspaces over finite fields and provides a new proof of the Touchard-Riordan formula by connecting subspace counts with chord diagram crossings.

Contribution

It introduces a novel enumeration of T-splitting subspaces and links finite field operator theory with combinatorial formulas for chord diagrams.

Findings

Enumerates T-splitting subspaces for arbitrary operators

Provides a new proof of the Touchard-Riordan formula

Connects finite field operator theory with combinatorial enumeration

Abstract

We enumerate the number of $T$-splitting subspaces of dimension $m$ for an arbitrary operator $T$ on a $2m$-dimensional vector space over a finite field. When $T$ is regular split semisimple, comparison with an alternate method of enumeration leads to a new proof of the Touchard-Riordan formula for enumerating chord diagrams by their number of crossings.