From stylic monoid to Catalan monoid

@misc{stein2026from,
  title = {{From stylic monoid to Catalan monoid}},
  author = {Itamar Stein},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07983},
  eprint = {2610.07983},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07983},
}

TY  - UNPB
TI  - From stylic monoid to Catalan monoid
AU  - Itamar Stein
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.07983
UR  - https://arxiv.org/abs/2610.07983
AB  - The stylic monoid Styl_n, introduced by Abram and Reutenauer, is the quotient of the plactic monoid by the relations x^2=x, and its elements are represented by N-tableaux. Volkov showed that the Catalan monoid Cat_n of order-preserving, order-decreasing self-maps of {0,1,…,n} is a quotient of Styl_n. However, the quotient map is defined on generators, and it is not apparent how to see, from an N-tableau, the map in Cat_n it corresponds to. In this paper we give a simple visual way to read off this map, and some of its main properties, from the N-tableau. The new ingredient is that we do not insist on drawing an N-tableau as a classical Young tableau: we allow the entries of each row to be shifted relative to the row below, as long as each entry stays above a smaller one. We call this a positioning, and prove that the column word read from any positioning is plactically equivalent to the usual column word; so every positioning can be used to compute the quotient map. We work with the tight positioning, in which each entry is pushed as far right as possible, and define the full core of an N-tableau: the part of each column that climbs by consecutive values from the bottom row. We call an N-tableau full if it equals its full core. We prove that passing to the full core does not change the image in Cat_n, that full N-tableaux are in bijection with Cat_n, and we show how to read the corresponding map directly off a full N-tableau.
ER  -