  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.

Combinatorics