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Séminaire Lotharingien de Combinatoire, 80B.55 (2018), 12 pp.

# Melissa Sherman-Bennett

# Combinatorics of **X**-variables in Finite Type Cluster Algebras

**Abstract.**
We compute the number of **X**-variables (also called
coefficients) of a cluster algebra of finite type when the underlying
semifield is the universal semifield. For non-exceptional types, these
numbers arise from a bijection between coefficients and quadrilaterals
(with a choice of diagonal) appearing in triangulations of certain
marked surfaces. We conjecture that similar results hold for cluster
algebras from arbitrary marked surfaces, and obtain corollaries
regarding the structure of finite type cluster algebras of geometric
type.

Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.

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