Decomposable 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 $\mathbb{N}$