The Stylic Monoid

TL;DR

The paper introduces the stylic monoid, a finite monoid derived from the free monoid on a totally ordered alphabet, exploring its structure, combinatorial properties, and connections to tableaux and the longest decreasing subword function.

Contribution

It defines the stylic monoid as a quotient of the plactic monoid, provides its presentation, and establishes its combinatorial and algebraic properties, including a bijection with N-tableaux and its role as a syntactic monoid.

Findings

Cardinality equals the number of set partitions on |A|+1 elements.

Presented as generated by A with plactic and idempotent relations.

Identified as the syntactic monoid for the longest decreasing subword function.

Abstract

The free monoid $A^*$ on a finite totally ordered alphabet $A$ acts at the left on columns, by Schensted left insertion. This defines a finite monoid, denoted $Styl(A)$ and called the stylic monoid. It is canonically a quotient of the plactic monoid. Main results are: the cardinality of $Styl(A)$ is equal to the number of partitions of a set on $|A|+1$ elements. We give a bijection with so-called $N$-tableaux, similar to Schensted's algorithm, explaining this fact. Presentation of $Styl(A)$: it is generated by $A$ subject to the plactic (Knuth) relations and the idempotent relations $a^2=a$, $a\in A$. The canonical involutive anti-automorphism on $A^*$, which reverses the order on $A$, induces an involution of $Styl(A)$, which similarly to the corresponding involution of the plactic monoid, may be computed by an evacuation-like operation (Sch\"utzenberger involution on tableaux) on so-called standard immaculate tableaux (which are in bijection with partitions). The monoid $Styl(A)$ is $J$-trivial, and the $J$-order of $Styl(A)$ is graded: the co-rank is given by the number of elements in the $N$-tableau. The monoid $Styl(A)$ is the syntactic monoid for the the function which associates to each word $w\in A^*$ the length of its longest strictly decreasing subword.