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{\bf Gregg Musiker and James Propp}
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{\bf Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type}
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Fomin and Zelevinsky show that a certain two-parameter family of
rational recurrence relations, here called the $(b,c)$ family,
possesses the Laurentness property: for all $b,c$, each term of the
$(b,c)$ sequence can be expressed as a Laurent polynomial in the two
initial terms. In the case where the positive integers $b,c$ satisfy
$bc<4$, the recurrence is related to the root systems of
finite-dimensional rank $2$ Lie algebras; when $bc>4$, the recurrence
is related to Kac-Moody rank $2$ Lie algebras of general type. Here
we investigate the borderline cases $bc=4$, corresponding to Kac-Moody
Lie algebras of affine type. In these cases, we show that the Laurent
polynomials arising from the recurence can be viewed as generating
functions that enumerate the perfect matchings of certain graphs. By
providing combinatorial interpretations of the individual coefficients
of these Laurent polynomials, we establish their positivity.
\bye