# ALGEBRAIC COMBINATORICS

Counting Coxeter’s friezes over a finite field via moduli spaces
Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 225-240.

We count the number of Coxeter’s friezes over a finite field. Our method uses geometric realizations of the spaces of friezes in a certain completion of the classical moduli space ${ℳ}_{0,n}$ allowing repeated points in the configurations. Counting points in the completed moduli space over a finite field is related to the enumeration problem of counting partitions of cyclically ordered set of points into subsets containing no consecutive points. In the appendix we provide an elementary solution for this enumeration problem.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.140
Classification: 13F60,  11G25,  05A18
Keywords: Frieze, Moduli space, Finite field, Partitions, Stirling numbers, cluster variety
@article{ALCO_2021__4_2_225_0,
author = {Morier-Genoud, Sophie},
title = {Counting Coxeter{\textquoteright}s friezes over a finite field via moduli spaces},
journal = {Algebraic Combinatorics},
pages = {225--240},
publisher = {MathOA foundation},
volume = {4},
number = {2},
year = {2021},
doi = {10.5802/alco.140},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.140/}
}
Morier-Genoud, Sophie. Counting Coxeter’s friezes over a finite field via moduli spaces. Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 225-240. doi : 10.5802/alco.140. https://alco.centre-mersenne.org/articles/10.5802/alco.140/

[1] Baur, Karin; Marsh, Robert J. Frieze patterns for punctured discs, J. Algebraic Combin., Volume 30 (2009) no. 3, pp. 349-379 | Article | MR 2545501 | Zbl 1201.05103

[2] Bergeron, François; Reutenauer, Christophe $S{L}_{k}$-tilings of the plane, Illinois J. Math., Volume 54 (2010) no. 1, pp. 263-300 | Article | MR 2776996 | Zbl 1236.13018

[3] Boalch, Philip Wild character varieties, points on the Riemann sphere and Calabi’s examples, Representation theory, special functions and Painlevé equations—RIMS 2015 (Adv. Stud. Pure Math.), Volume 76, Math. Soc. Japan, Tokyo, 2018, pp. 67-94 | Article | MR 3837919

[4] Caldero, Philippe; Chapoton, Frédéric Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv., Volume 81 (2006) no. 3, pp. 595-616 | Article | MR 2250855 | Zbl 1119.16013

[5] Chapoton, Frédéric On the number of points over finite fields on varieties related to cluster algebras, Glasg. Math. J., Volume 53 (2011) no. 1, pp. 141-151 | Article | MR 2747140 | Zbl 1209.14021

[6] Chapoton, Frédéric On some varieties associated with trees, Michigan Math. J., Volume 64 (2015) no. 4, pp. 721-758 | Article | MR 3426614 | Zbl 1350.13022

[7] Conway, John H.; Coxeter, Harold S. M. Triangulated polygons and frieze patterns, Math. Gaz., Volume 57 (1973) no. 400, 401, p. 87-94, 175–183 | Article | MR 0461269 (57 #1254) | Zbl 0288.05021

[8] Coxeter, Harold S. M. Frieze patterns, Acta Arith., Volume 18 (1971), pp. 297-310 | Article | MR 286771 | Zbl 0217.18101

[9] Coxeter, Harold S. M. Regular complex polytopes, Cambridge University Press, Cambridge, 1991, xiv+210 pages | MR 1119304 | Zbl 0732.51002

[10] Cuntz, Michael On wild frieze patterns, Exp. Math., Volume 26 (2017) no. 3, pp. 342-348 | Article | MR 3642111 | Zbl 1367.05207

[11] Cuntz, Michael A combinatorial model for tame frieze patterns, Münster J. Math., Volume 12 (2019) no. 1, pp. 49-56 | Article | MR 3928082 | Zbl 1417.05025

[12] Cuntz, Michael; Holm, Thorsten Frieze patterns over integers and other subsets of the complex numbers, J. Comb. Algebra, Volume 3 (2019) no. 2, pp. 153-188 | Article | MR 3941490 | Zbl 1448.05222

[13] Fontaine, Bruce Non-zero integral friezes (2014) (https://arxiv.org/abs/1409.6026)

[14] Fontaine, Bruce; Plamondon, Pierre-Guy Counting friezes in type ${D}_{n}$, J. Algebraic Combin., Volume 44 (2016) no. 2, pp. 433-445 | Article | MR 3533561 | Zbl 1344.05152

[15] Galvin, David; Thanh, Do Trong Stirling numbers of forests and cycles, Electron. J. Combin., Volume 20 (2013) no. 1, Paper no. 73, 16 pages | MR 3040635 | Zbl 1266.05031

[16] Gelʼfand, Israel M.; MacPherson, Robert D. Geometry in Grassmannians and a generalization of the dilogarithm, Adv. in Math., Volume 44 (1982) no. 3, pp. 279-312 | Article | MR 658730 | Zbl 0504.57021

[17] Holm, Thorsten; Jørgensen, Peter A $p$-angulated generalisation of Conway and Coxeter’s theorem on frieze patterns, Int. Math. Res. Not. IMRN (2020) no. 1, pp. 71-90 | Article | MR 4050563

[18] Knuth, Donald E.; Lossers, O. P. Partitions of a circular set. Problem 11151, Volume 114, 2007 no. 3, p. 265-266

[19] Mabilat, Flavien Combinatorial description of the principal congruence subgroups $\Gamma \left(2\right)$ in $SL\left(2,ℤ\right)$ (2019) (https://arxiv.org/abs/1911.06717)

[20] Morier-Genoud, Sophie Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics, Bull. Lond. Math. Soc., Volume 47 (2015) no. 6, pp. 895-938 | Article | MR 3431573 | Zbl 1330.05035

[21] Morier-Genoud, Sophie; Ovsienko, Valentin; Schwartz, Richard E.; Tabachnikov, Serge Linear difference equations, frieze patterns, and the combinatorial Gale transform, Forum Math. Sigma, Volume 2 (2014), Paper no. e22, 45 pages | Article | MR 3264259 | Zbl 1297.39004

[22] Morier-Genoud, Sophie; Ovsienko, Valentin; Tabachnikov, Serge 2-frieze patterns and the cluster structure of the space of polygons, Ann. Inst. Fourier, Volume 62 (2012) no. 3, pp. 937-987 | Article | Numdam | MR 3013813 | Zbl 1290.13014

[23] OEIS, OEIS Foundation Inc The On-Line Encyclopedia of Integer Sequences, 2019 (http://oeis.org)

[24] Ovsienko, Valentin Partitions of unity in $SL\left(2,ℤ\right)$, negative continued fractions, and dissections of polygons, Res. Math. Sci., Volume 5 (2018) no. 2, Paper no. 21, 25 pages | Article | MR 3784137 | Zbl 1418.05047

[25] Sibuya, Yasutaka Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland Mathematics Studies, 18, North-Holland Publishing Co., Amsterdam-Oxford; Elsevier Publishing Co., Inc., New York, 1975, xv+290 pages | MR 0486867 | Zbl 0322.34006