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

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.