A combinatorial formula for rank 2 cluster variables |
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| Authors: | Kyungyong Lee Ralf Schiffler |
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| Institution: | 1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA
2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA
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| Abstract: | Let r be any positive integer, and let x 1,x 2 be indeterminates. We consider the sequence {x n } defined by the recursive relation $$x_{n+1} =\bigl(x_n^r +1\bigr)/{x_{n-1}}$$ for any integer n. Finding a combinatorial expression for x n as a rational function of x 1 and x 2 has been an open problem since 2001. We give a direct elementary formula for x n in terms of subpaths of a specific lattice path in the plane. The formula is manifestly positive, providing a new proof of a result by Nakajima and Qin. |
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