Enumeration of Anti-Invariant Subspaces and Touchard's Formula for the Entries of the $q$-Hermite Catalan Matrix

Abstract

We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the $q$-Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries.

15 pages, final version

References in corpus (2)

• The Splitting Subspace Conjecture
• Splitting subspaces and a finite field interpretation of the Touchard-Riordan Formula

Cited by in corpus (1)

• Subspace Profiles over Finite Fields and $q$-Whittaker Expansions of Symmetric Functions