Journal of Algebraic Combinatorics

, Volume 43, Issue 3, pp 589–633

Laurent phenomenon sequences


DOI: 10.1007/s10801-015-0647-5

Cite this article as:
Alman, J., Cuenca, C. & Huang, J. J Algebr Comb (2016) 43: 589. doi:10.1007/s10801-015-0647-5


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.


Laurent phenomenon Cluster algebra LP algebra 

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA

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