Geodesic
A combinatorial formula for rank 2 cluster variables
Journal of Algebraic Combinatorics, Tome 37 (2013) no. 1, pp. 67-85.

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

Summary: 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.
Keywords: Laurent polynomials, cluster algebras, lattice paths 
@article{JAC_2013__37_1_a4,
     author = {Lee, Kyungyong and Schiffler, Ralf},
     title = {A combinatorial formula for rank 2 cluster variables},
     journal = {Journal of Algebraic Combinatorics},
     pages = {67--85},
     publisher = {mathdoc},
     volume = {37},
     number = {1},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2013__37_1_a4/}
}
TY  - JOUR
AU  - Lee, Kyungyong
AU  - Schiffler, Ralf
TI  - A combinatorial formula for rank 2 cluster variables
JO  - Journal of Algebraic Combinatorics
PY  - 2013
SP  - 67
EP  - 85
VL  - 37
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JAC_2013__37_1_a4/
LA  - en
ID  - JAC_2013__37_1_a4
ER  - 
%0 Journal Article
%A Lee, Kyungyong
%A Schiffler, Ralf
%T A combinatorial formula for rank 2 cluster variables
%J Journal of Algebraic Combinatorics
%D 2013
%P 67-85
%V 37
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JAC_2013__37_1_a4/
%G en
%F JAC_2013__37_1_a4
Lee, Kyungyong; Schiffler, Ralf. A combinatorial formula for rank 2 cluster variables. Journal of Algebraic Combinatorics, Tome 37 (2013) no. 1, pp. 67-85. http://geodesic.mathdoc.fr/item/JAC_2013__37_1_a4/