Associative-commutative spectra for some varieties of groupoids

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.