The cyclic shift graph of a monoid is the graph whose vertices are the elements of the monoid and whose edges connect elements that differ by a cyclic shift. For several monoids arising from combinatorial objects like the plactic (Young tableaux) and the sylvester (binary search trees) two elements differ by a sequence of cyclic shifts if and only if they have the same number of each of its composing symbols (that is, the same evaluation). Therefore, the connected components of the cyclic graph for these monoids consist of the elements that have the same evaluation.

In this talk we will be interested in the rPS monoid, which arises from the combinatorial object that we obtain from a possible generalization of Patience Sorting to words, the rPS tableaux, and has also the evaluation property. In particular, we will discuss the diameter of a connected component of the rPS monoid for the finite rank case. Our goal is to explain how using the cloud-based calculation and computation platform CoCalc (previously known as SageMath), we were able to establish some conjectures and results regarding upper bounds for this diameter.