  The associative spectrum of a groupoid (i.e., a set with a binary operation)
measures its nonassociativity while the associative-commutative spectrum
measures both nonassociativity and noncommutativity of the groupoid. The two
spectra are also the coefficients of the Hilbert series of certain operads. We
establish upper bounds for the two spectra of various varieties of groupoids
defined by different sets of identities and provide examples (often groupoids
with three elements) for which the upper bounds are achieved. Our results have
connections to many interesting combinatorial objects and integer sequences and
naturally lead to some questions for future studies.
