Geodesic
Laurent phenomenon sequences
Journal of Algebraic Combinatorics, Tome 43 (2016) no. 3, pp. 589-633.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

In this paper, we undertake a systematic study of sequences generated by recurrences $x_{m+n}x_m = P(x_{m+1},\dots,x_{m+n-1})$ 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 T. Lam and P. Pylyavskyy ["Laurent phenomenon algebras", Camb. J. Math. 4, No. 1, 121--162 (2016; doi:10.4310/cjm.2016.v4.n1.a2)]. 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.
Classification : 13F60
Keywords: Laurent phenomenon, cluster algebra, LP algebra 
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     title = {Laurent phenomenon sequences},
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Alman, Joshua; Cuenca, Cesar; Huang, Jiaoyang. Laurent phenomenon sequences. Journal of Algebraic Combinatorics, Tome 43 (2016) no. 3, pp. 589-633. http://geodesic.mathdoc.fr/item/JAC_2016__43_3_a5/