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

TL;DR

This paper investigates the structure of cyclic shift graphs in various plactic-like monoids, determining maximum diameters of connected components and summarizing known and new results in this area.

Contribution

It provides new bounds and exact values for the maximum diameter of cyclic shift graph components in several monoids, extending prior knowledge.

Findings

Hypoplactic monoid has maximum diameter n-1.

Sylvester and taiga monoids have maximum diameter between n-1 and n.

Stalactic monoid has maximum diameter 3 for ranks ≥3.

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. This paper examines the cyclic shift graphs of `plactic-like' monoids, whose elements can be viewed as combinatorial objects of some type: aside from the plactic monoid itself (the monoid of Young tableaux), examples include the hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary search trees with multiplicities), and the Baxter monoid (pairs of twin binary search trees). It was already known that for many of these monoids, connected components of the cyclic shift graph consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper focusses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank-$n$ case. For the hypoplactic monoid, this is $n-1$; for the sylvester and taiga monoids, at least $n-1$ and at most $n$; for the stalactic monoid, $3$ (except for ranks $1$ and $2$, when it is respectively $0$ and $1$); for the plactic monoid, at least $n-1$ and at most $2n-3$. The current state of knowledge, including new and previously-known results, is summarized in a table.