Laurent phenomenon sequences
Author(s)Alman, Joshua; Cuenca, Cesar; Huang, Jiaoyang; Cuenca, Cesar A.
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In this paper, we undertake a systematic study of sequences generated by recurrences x[subscript m+n]x[subscript m]=P(x[subscript m+1],…,x[subscript m+n−1])xm+nxm=P(xm+1,…,xm+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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Journal of Algebraic Combinatorics
Alman, Joshua, Cesar Cuenca, and Jiaoyang Huang. “Laurent Phenomenon Sequences.” Journal of Algebraic Combinatorics 43, no. 3 (November 5, 2015): 589–633.
Author's final manuscript