The theory of how packs of cards may be shuffled together to form a single pack has already been remarkably well-studied in combinatorics, group theory, statistics, and other areas of mathematics. This talk considers natural extensions where 1/ We may have infinitely many cards in a deck, 2/ We may take the result of previous shuffles as one of our decks of cards (i.e. shuffles are hierarchical), and 3/ There may be arbitrarily many decks of cards.

Such hierarchical shuffles are naturally studied using operads, and lead directly to infinitary analogues of (standard or otherwise) Young tableaux. Based on these, we consider a special sub-operad of shuffles that is isomorphic to structures commonly used to label facets of associahedra -- geometric figures based on associativity laws that occur in a remarkably wide range of fields.

The advantage of labeling facets (vertices, edges, faces, etc.) of associahedra with elements of this operad is that we have a natural in-built notion of mapping between them. Further, such mappings are simple arithmetic operations. This allows us to give elementary arithmetic interpretations of deep structures from several branches of mathematics related to associahedra.

We illustrate this by example, and demonstrate how well-known structures from topology, group theory, logic, and coherence arise in this way, giving a description of complex structures in elementary terms. We also aim to give novel structures that related different examples in a natural way. We finish by showing how, even though we may give interpretations in terms of elementary arithmetic, classic unsolved conjectures arise naturally at even the simplest level.