Conjugacy in Patience Sorting monoids

Abstract

The cyclic shift graph of a monoid is the graph whose vertices are the elements of the monoid and whose edges connect elements that are cyclic shift related. The Patience Sorting algorithm admits two generalizations to words, from which two kinds of monoids arise, the $\mathrm{rps}$ monoid and the $\mathrm{lps}$ (also known as Bell) monoid. Like other monoids arising from combinatorial objects such as the plactic and the sylvester, the connected components of the cyclic shift graph of the $\mathrm{rps}$ monoid consists of elements that have the same number of each of its composing symbols. In this paper, with the aid of the computational tool SageMath, we study the diameter of the connected components from the cyclic shift graph of the $\mathrm{rps}$ monoid. Within the theory of monoids, the cyclic shift relation, among other relations, generalizes the relation of conjugacy for groups. We examine several of these relations for both the $\mathrm{rps}$ and the $\mathrm{lps}$ monoids.