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Laurent phenomenon sequences

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Abstract

In this paper, we undertake a systematic study of sequences generated by recurrences \(x_{m+n}x_m = P(x_{m+1}, \ldots , x_{m+n-1})\) which exhibit the Laurent phenomenon. Some of the most famous among these are the Somos and the Gale-Robinson sequences. Our approach is based on finding period 1 seeds of Laurent phenomenon algebras of Lam–Pylyavskyy. We completely classify polynomials P that generate period 1 seeds in the cases of \(n=2,3\) and of mutual binomial seeds. We also find several other interesting families of polynomials P whose generated sequences exhibit the Laurent phenomenon. Our classification for binomial seeds is a direct generalization of a result by Fordy and Marsh, that employs a new combinatorial gadget we call a double quiver.

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Acknowledgments

This research was conducted at the 2013 summer REU (Research Experience for Undergraduates) program at the University of Minnesota, Twin Cities, and was supported by NSF grants DMS-1067183 and DMS-1148634. We would like to thank Professors Dennis Stanton, Gregg Musiker, Joel Lewis, and especially our mentor Pavlo Pylyavskyy, who directed the program, for their advice and support throughout this project. We would also like to thank Al Garver, Andrew Hone and other REU participants for their help in editing this paper and the anonymous reviewers for careful reading and valuable comments.

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Correspondence to Joshua Alman.

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Alman, J., Cuenca, C. & Huang, J. Laurent phenomenon sequences. J Algebr Comb 43, 589–633 (2016). https://doi.org/10.1007/s10801-015-0647-5

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