# Word operads and admissible orderings

## TL;DR

This paper explains the origins of shuffle tree sequences using operad theory, introduces new monomial orders, and demonstrates a quadratic Gr"obner basis for the Poisson operad.

## Contribution

It provides a conceptual framework linking combinatorial operads to shuffle tree sequences and introduces novel monomial orders with algebraic applications.

## Key findings

- Connected shuffle tree sequences to operad theory
- Defined new monomial orders for shuffle trees
- Established a quadratic Gr"obner basis for the Poisson operad

## Abstract

We use Giraudo's construction of combinatorial operads from monoids to offer a conceptual explanation of the origins of Hoffbeck's path sequences of shuffle trees, and use it to define new monomial orders of shuffle trees. One such order is utilised to exhibit a quadratic Gr\"obner basis of the Poisson operad.