K-promotion on m-packed labelings of posets

TL;DR

This work extends dynamical algebraic combinatorics by studying the $K$-promotion action $\partial_K$ on $m$-packed labelings of finite posets, generalizing Pechenik’s promotion from tableaux to rooted trees and beyond. It establishes existence conditions for $m$-packed labelings, provides a toggle-based description of $\partial_K$, and proves a suite of orbit-divisibility results and exact orbit structures for specific rooted-tree families (extended stars, combs, zippers, and a three-leaved tree). A central theme is relating $\partial_K$ to rowmotion via equivariant bijections, enabling transfer of structural results and homomesy phenomena; the paper also discusses cyclic sieving prospects and outlines directions for future research on other posets and CSP. Together, these results broaden the toolkit for analyzing dynamic, combinatorial actions on labelings and highlight deep connections between $K$-promotion and classical poset dynamics in broader families.

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.

Figures (9)

• Figure 1: The promotion operator $\partial$
• Figure 2: The $K$-promotion operator $\partial_K$
• Figure 3: An example of the $K$-promotion toggle $s_2$
• Figure 4: The trunk of this poset $P$ is $T=\{u,v\}$
• Figure 5: $K$-promotion acts cyclically on the branch $B=\{2,5,6\}$

Theorems & Definitions (28)

• Proposition 1.1: DSV:ril
• proof
• Proposition 2.1
• proof
• Theorem 2.2: DSV:ril
• Proposition 2.3
• proof
• Theorem 2.4
• proof
• Corollary 2.5