Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, and related monoids

TL;DR

This paper investigates the structure of cyclic shift graphs in certain combinatorially significant monoids, revealing that the maximum diameter of connected components depends solely on the monoid's rank, with detailed proofs for the sylvester monoid.

Contribution

It provides new results on the diameters of connected components in cyclic shift graphs of finite-rank monoids, highlighting rank dependence.

Findings

Maximum diameter depends only on monoid rank

Connected components consist of elements with same evaluation

Results apply to plactic and sylvester monoids

Abstract

The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. For certain monoids connected with combinatorics, such as the plactic monoid (the monoid of Young tableaux) and the sylvester monoid (the monoid of binary search trees), connected components consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper discusses new results on the diameters of connected components of the cyclic shift graphs of the finite-rank analogues of these monoids, showing that the maximum diameter of a connected component is dependent only on the rank. The proof techniques are explained in the case of the sylvester monoid.