Decomposable shuffles
João Dias, Bruno Dinis, Carlos Correia Ramos
TL;DR
The work addresses how to systematically encode total orders on $\\mathbb{N}$ that arise as shuffles—ordered concatenations of indexed sequences—by focusing on decomposable shuffles built from finite blocks and the ordinals $\\omega$ and $\\omega^*$. It develops a multi-layer representation via uniform sets and mixed-degree assemblies, a diagrammatic classification, and an involution that dualizes orders. It also introduces a composition operation that generalizes permutation composition and yields a group-like structure on a subclass, while clarifying its limits since not all order types (e.g., the rationals) are captureable by these shuffles. The framework connects the Šarkovskiĭ order with a broader combinatorial construction, enabling explicit recovery of total orders on $\\mathbb{N}$ and paving the way for algebraic manipulation of shuffles.
Abstract
We develop a combinatorial and order-theoretic framework for shuffles, understood as ordered concatenations of indexed families of sequences that induce total orders on the natural numbers. Motivated by the classical Šarkovskiĭ order, we introduce elementary building blocks that encode finite and infinite order patterns and focus on decomposable shuffles constructed from finite ordinals together with $ω$ and its dual $ω^*$. We define representations that allow individual elements to be located within a shuffle and show how suitable structural conditions yield total orders on $\mathbb{N}$