Geodesic
Combinatorics of X-variables in Finite Type Cluster Algebras
Séminaire lotharingien de combinatoire, 80B (2018)

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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. 

@article{SLC_2018_80B_a54,
     author = {Melissa Sherman-Bennett},
     title = {Combinatorics of {X-variables} in {Finite} {Type} {Cluster} {Algebras}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {80B},
     year = {2018},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a54/}
}
TY  - JOUR
AU  - Melissa Sherman-Bennett
TI  - Combinatorics of X-variables in Finite Type Cluster Algebras
JO  - Séminaire lotharingien de combinatoire
PY  - 2018
VL  - 80B
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SLC_2018_80B_a54/
ID  - SLC_2018_80B_a54
ER  - 
%0 Journal Article
%A Melissa Sherman-Bennett
%T Combinatorics of X-variables in Finite Type Cluster Algebras
%J Séminaire lotharingien de combinatoire
%D 2018
%V 80B
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SLC_2018_80B_a54/
%F SLC_2018_80B_a54
Melissa Sherman-Bennett. Combinatorics of X-variables in Finite Type Cluster Algebras. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a54/