Decomposable shuffles

TL;DR

This paper introduces a combinatorial framework for understanding shuffles as ordered concatenations, focusing on decomposable shuffles built from finite and infinite order patterns, with applications to total orders on natural numbers.

Contribution

It develops a novel order-theoretic approach to shuffles, including elementary building blocks and representations for analyzing total orders on natural numbers.

Findings

Defined elementary order patterns for finite and infinite sequences

Constructed decomposable shuffles from finite ordinals and infinite orders

Established conditions for total orders on natural numbers

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 \v{S}arkovski\u{i} order, we introduce elementary building blocks that encode finite and infinite order patterns and focus on decomposable shuffles constructed from finite ordinals together with $\omega$ and its dual $\omega^*$. We define representations that allow individual elements to be located within a shuffle and show how suitable structural conditions yield total orders on $\mathbb{N}$