We study and classify faithfully balanced modules for the algebra of triangular n by n matrices and more generally for Nakayama algebras. The theory extends known results about tilting modules, which are classified by binary trees, and counted with the Catalan numbers. The number of faithfully balanced modules is a 2-factorial number. Among them are n! modules with n indecomposable summands, which can be classified by interleaved binary trees or by increasing binary trees.

中文翻译：

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忠实平衡模块的组合

我们研究和分类三角n × n矩阵的代数的忠实平衡模块，以及中山代数的更一般的分类。该理论扩展了有关倾斜模块的已知结果，倾斜模块通过二叉树进行分类，并使用加泰罗尼亚数进行计数。忠实平衡的模块数是2阶乘数。其中有n个！具有n个不可分解求和数的模块，可以按交错二叉树或按递增二叉树分类。