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.

15 pages

References in corpus (1)

• The Splitting Subspace Conjecture

Cited by in corpus (3)

• Subspace Profiles over Finite Fields and $q$-Whittaker Expansions of Symmetric Functions
• Enumeration of Anti-Invariant Subspaces and Touchard's Formula for the Entries of the $q$-Hermite Catalan Matrix
• Set partitions, tableaux, and subspace profiles under regular diagonal matrices