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Séminaire Lotharingien de Combinatoire, 82B.81 (2019), 12 pp.

# Rachel Bailey and Emily Gunawan

# Cluster algebras and binary words

**Abstract.**
We establish a connection between binary subwords and perfect matchings of a snake graph, an important tool in the theory of cluster algebras. Every binary expansion *w* can be associated to a piecewise-linear poset *P* and a snake graph *G*. We describe bijections from the subwords of *w* to the antichains of *P* and to the perfect matchings of *G*. We also construct a tree structure called the antichain trie which is isomorphic to the trie of subwords introduced by Leroy, Rigo, and Stipulanti.

Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.

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