  The $q$-commutation relations, formulated in the setting of the $q$-Fock
space of Bo\.zjeko and Speicher, interpolate between the classical commutation
relations (CCR) and the classical anti-commutation relations (CAR) defined on
the classical bosonic and fermionic Fock spaces, respectively. Interpreting the
$q$-Fock space as an algebra of "random variables" exhibiting a specific
commutativity structure, one can construct the so-called $q$-semicircular and
$q$-circular operators acting as $q$-deformations of the classical Gaussian and
complex Gaussian random variables, respectively. While the $q$-semicircular
operator is generally well understood, many basic properties of the
$q$-circular operator (in particular, a tractable expression for its norm)
remain elusive. Inspired by the combinatorial approach to free probability, we
revist the combinatorial formulations of these operators. We point out that a
finite alternating-sum expression for $2n$-norm of the $q$-semicircular is
available via generating functions of chord-crossing diagrams developed by
Touchard in the 1950s and distilled by Riordan in 1974. Extending these norms
as a function in $q$ onto the complex unit ball and taking the $n\to\infty$
limit, we recover the familiar expression for the norm of the $q$-semicircular
and show that the convergence is uniform on the compact subsets of the unit
ball. In contrast, the $2n$-norms of the $q$-circular are encoded by
chord-crossing diagrams that are parity-reversing, which have not yet been
characterized in the combinatorial literature. We derive certain combinatorial
properties of these objects, including closed-form expressions for the number
of such diagrams of any size with up to eleven crossings. These properties
enable us to conclude that the $2n$-norms of the $q$-circular operator are
significantly less well behaved than those of the $q$-semicircular operator.
