# Shuffles of trees

## TL;DR

This paper introduces a new concept of shuffles for trees, extending classical shuffles, with algebraic, combinatorial, and topological characterizations, motivated by operad theory and dendroidal sets.

## Contribution

It defines and explores a novel notion of tree shuffles, providing multiple descriptions and properties, independent of existing operad and dendroidal set frameworks.

## Key findings

- Provides equivalent descriptions of tree shuffles
- Establishes algebraic and combinatorial properties
- Characterizes shuffles via topological open sets

## Abstract

We discuss a notion of shuffle for trees which extends the usual notion of a shuffle for two natural numbers. We give several equivalent descriptions, and prove some algebraic and combinatorial properties. In addition, we characterize shuffles in terms of open sets in a topological space associated to a pair of trees. Our notion of shuffle is motivated by the theory of operads and occurs in the theory of dendroidal sets, but our presentation is independent and entirely self-contained.