Nonsymmetric versions of binary quadratic operads

In this paper, we study the white Manin product of the associative operad $\text{As}$ with a binary quadratic operad $\text{Var}$. We introduce the notion of a nonsymmetric version of $\text{Var}$ and provide a criterion for determining when the operad $\text{As}\circ \text{Var}$ has this property. We illustrate the construction with several examples and counterexamples. Finally, for some operads admitting nonsymmetric versions, we describe their combinatorial properties.