Frieze patterns, as introduced by Coxeter in the 1970s, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper, we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing classic results for frieze patterns. As a consequence, we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated with classic Conway–Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (nonzero) frieze patterns with coefficients with entries in a subset of the complex numbers without an accumulation point.

中文翻译：

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带系数的楣纹图案

Coxeter 在 1970 年代介绍的 Frieze 模式与没有系数的簇代数密切相关。与带系数的簇代数相关联的带状图案的适当推广仅在 Propp 未发表的手稿中短暂出现过。在本文中，我们系统地研究了这些带系数的带状图案，并证明了各种基本结果，概括了带状图案的经典结果。因此，我们看到如何通过从与经典 Conway-Coxeter 饰带图案相关的三角多边形中切出子多边形，从经典饰带图案中获得带系数的饰带图案。我们解决了以这种方式可以获得哪些带系数的带状图案的问题，并完全解决了三角形的这个问题。最后，