In [8] we studied Koszulity of a family ${t𝒜\mathrm{𝑠𝑠}}_d^n$ of operads depending on a natural number $n\in \mathbb{N}$ and on the degree $d\in \mathbb{Z}$ of the generating operation. While we proved that, for $n\le 7$, the operad ${t𝒜\mathrm{𝑠𝑠}}_d^n$ is Koszul if and only if $d$ is even, and while it follows from [4] that ${t𝒜\mathrm{𝑠𝑠}}_d^n$ is Koszul for $d$ even and arbitrary $n$, the (non)Koszulity of ${t𝒜\mathrm{𝑠𝑠}}_d^n$ for $d$ odd and $n\ge 8$ remains an open problem. In this note we describe some related numerical experiments, and formulate a conjecture suggested by the results of these computations.

This is an extended version of a talk presented by the second author on the Third Mile High Conference on Nonassociative Mathematics (August 2013, Denver, CO). The purpose of this paper is twofold. First, we would like to review the technique developed in a series of papers for various classes of di-algebras and show how the same ideas work for tri-algebras. Second, we present a general approach to the definition of pre- and post-algebras which turns out to be equivalent to the construction of dendriform...