Abstract
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.
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