  A systematic study of holomorphic gauge invariant operators in general
$\mathcal{N}=1$ quiver gauge theories, with unitary gauge groups and
bifundamental matter fields, was recently presented in [1]. For large ranks a
simple counting formula in terms of an infinite product was given. We extend
this study to quiver gauge theories with fundamental matter fields, deriving an
infinite product form for the refined counting in these cases. The infinite
products are found to be obtained from substitutions in a simple building block
expressed in terms of the weighted adjacency matrix of the quiver. In the case
without fundamentals, it is a determinant which itself is found to have a
counting interpretation in terms of words formed from partially commuting
letters associated with simple closed loops in the quiver. This is a new
relation between counting problems in gauge theory and the Cartier-Foata
monoid. For finite ranks of the unitary gauge groups, the refined counting is
given in terms of expressions involving Littlewood-Richardson coefficients.
