K-promotion on m-packed labelings of posets

TL;DR

This paper explores the properties of a K-theoretic promotion operator on m-packed labelings of posets, revealing divisibility properties and orbit sizes, especially in rooted trees, expanding its application beyond Young tableaux.

Contribution

It extends the application of pro_K to general posets and rooted trees, providing new results on orbit sizes and divisibility properties.

Findings

Orbit sizes exhibit divisibility properties under certain conditions.

Complete determination of orbit sizes for specific rooted trees.

Application of pro_K yields interesting combinatorial structures in posets.

Abstract

Schutzenberger's promotion operator, pro, is a fundamental map in dynamical algebraic combinatorics. At first, its action was mainly considered on standard Young tableaux. But pro was subsequently shown to have interesting properties when applied to natural labelings of other posets. Pechenik defined a K-theoretic version of promotion, pro_K, on m-packed labelings of tableaux. The operator pro was then extended to increasing labelings of other posets. The purpose of the current work is to show that the original action of pro_K on m-packed labelings yields interesting results when applied to partially ordered sets in general, and to rooted trees in particular. We show that under certain conditions, the sizes of the orbits and order of pro_K exhibit nice divisibility properties. We also completely determine, for certain values of m, the orbit sizes for the action on various types of rooted trees such as extended stars, combs, zippers, and a type of three-leaved tree.