  Jean-Louis Loday introduced a class of symmetric operads generated by one
bilinear operation subject to one relation making each left-normed product of
three elements equal to a linear combination of right-normed products:
  \[ (a_1a_2)a_3=\sum_{\sigma\in S_3}x_\sigma\,
a_{\sigma(1)}(a_{\sigma(2)}a_{\sigma(3)})\ ;
  \] such an operad is called a parametrized one-relation operad. For a
particular choice of parameters $\{x_\sigma\}$, this operad is said to be
regular if each of its components is the regular representation of the
symmetric group; equivalently, the corresponding free algebra on a vector space
$V$ is, as a graded vector space, isomorphic to the tensor algebra of $V$. We
classify, over an algebraically closed field of characteristic zero, all
regular parametrized one-relation operads. In fact, we prove that each such
operad is isomorphic to one of the following five operads: the left-nilpotent
operad defined by the identity $((a_1a_2)a_3)=0$, the associative operad, the
Leibniz operad, the dual Leibniz (Zinbiel) operad, and the Poisson operad. Our
computational methods combine linear algebra over polynomial rings,
representation theory of the symmetric group, and Gr\"obner bases for
determinantal ideals and their radicals.
