×

Cluster algebras and binary subwords. (English) Zbl 1520.13032

Summary: This paper establishes a connection between binary subwords and perfect matchings of a snake graph, an important tool in the theory of cluster algebras. Every binary expansion \(w\) can be associated to a piecewise-linear poset \(P\) and a snake graph \(G\). We construct a tree structure called the antichain trie which is isomorphic to the trie of subwords introduced by Leroy, Rigo, and Stipulanti. We then present bijections from the subwords of \(w\) to the antichains of \(P\) and to the perfect matchings of \(G\).

MSC:

13F60 Cluster algebras
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
06A07 Combinatorics of partially ordered sets
68R15 Combinatorics on words
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Broline, D.; Crowe, DW; Isaacs, IM, The geometry of frieze patterns, Geom. Dedicata., 3, 171-176 (1974) · Zbl 0292.05009
[2] Caldero, P.; Chapoton, F., Cluster algebras as Hall algebras of quiver representations, Comment. Math Helv., 81, 3, 595-616 (2006) · Zbl 1119.16013
[3] Claussen, A.: Expansion posets for polygon cluster algebras. arXiv:2005.02083 (2020)
[4] Çanakçı, İ.; Schiffler, R., Snake graph calculus and cluster algebras from surfaces, J. Algebra, 382, 240-281 (2013) · Zbl 1319.13012
[5] Çanakçı, İ.; Schiffler, R., Cluster algebras and continued fractions, Compos. Math., 154, 3, 565-593 (2018) · Zbl 1437.13033
[6] Çanakçı, İ.; Schroll, S., Lattice bijections for string modules, snake graphs and the weak Bruhat order, Adv. Appl. Math., 126, 102094 (2021) · Zbl 1484.16022
[7] Felsner, S., Lattice structures from planar graphs., Electron. J. Combin. 11(1):Research Paper, 15, 24 (2004) · Zbl 1056.05039
[8] Fomin, S.; Zelevinsky, A., Cluster algebras, I. Foundations. J. Amer. Math. Soc., 15, 2, 497-529 (2002) · Zbl 1021.16017
[9] Gunawan, E.; Musiker, G.; Vogel, H., Cluster algebraic interpretation of infinite friezes, European J. Combin., 81, 22-57 (2019) · Zbl 1420.05033
[10] Knauer, K.; Martínez-Sandoval, L.; Ramírez Alfonsín, JL, On lattice path matroid polytopes: Integer points and Ehrhart polynomial, Discrete Comput. Geom., 60, 3, 698-719 (2018) · Zbl 1494.52014
[11] Lee, K.; Li, L.; Nguyen, B., New combinatorial formulas for cluster monomials of type A quivers, Electron. J Combin., 24(2):Paper, 41, 2.42 (2017) · Zbl 1401.13068
[12] Leroy, J.; Rigo, M.; Stipulanti, M., Counting the number of non-zero coefficients in rows of generalized Pascal triangles, Discrete Math., 340, 5, 862-881 (2017) · Zbl 1357.05004
[13] Lee, K., Schiffler, R.: Cluster algebras and Jones polynomials. Selecta. Math. (N.S.), 25(4):Paper No. 58 (2019) · Zbl 1442.13076
[14] Morier-Genoud, S., Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics, Bull. Lond. Math. Soc., 47, 6, 895-938 (2015) · Zbl 1330.05035
[15] McConville, T., Sagan, B.E., Smyth, C.: On a rank-unimodality conjecture of Morier-Genoud and Ovsienko. arXiv:2008.13232 (2020) · Zbl 1515.06003
[16] Musiker, G.; Schiffler, R.; Williams, L., Positivity for cluster algebras from surfaces, Adv. Math., 227, 6, 2241-2308 (2011) · Zbl 1331.13017
[17] Musiker, G.; Schiffler, R.; Williams, L., Bases for cluster algebras from surfaces, Compos. Math, 149, 2, 217-263 (2013) · Zbl 1263.13024
[18] Nagai, W.; Terashima, Y., Cluster variables, ancestral triangles and Alexander polynomials, Adv. Math., 363, 106965, 37 (2020) · Zbl 1434.57010
[19] Propp, J.: Lattice structure for orientations of graphs. arXiv:math/0209005 (2002)
[20] Propp, J.: The combinatorics of frieze patterns and Markoff numbers. Integers 20(A12) (2020) · Zbl 1435.05018
[21] Schiffler, R., A cluster expansion formula (An case), Electron. J. Combin., 15(1):Research paper, 64, 9 (2008) · Zbl 1184.13064
[22] Schiffler, R., Quiver representations. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC (2014), Cham: Springer, Cham · Zbl 1310.16015
[23] Schiffler, R.; Thomas, H., On cluster algebras arising from unpunctured surfaces, Int. Math. Res. Not., 2009, 17, 3160-3189 (2009) · Zbl 1171.30019
[24] Yacavone, M.: Cluster Algebras and the HOMFLY Polynomial. arXiv:1910.10267 (2019)
[25] Yurikusa, T., Combinatorial cluster expansion formulas from triangulated surfaces, Electron. J. Combin., 26(2):Paper, 2.33, 39 (2019) · Zbl 1436.13054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.