# Combinatorics of faithfully balanced modules

## TL;DR

This paper classifies faithfully balanced modules over lower triangular matrix algebras, extending tilting module theory, and reveals their enumeration relates to factorial numbers and binary tree structures.

## Contribution

It introduces a classification of faithfully balanced modules, generalizing tilting modules, and connects their enumeration to factorial numbers and binary trees.

## Key findings

- Number of faithfully balanced modules is a 2-factorial number.
- Includes classification of modules with n! indecomposable summands.
- Establishes connections to binary trees and Catalan numbers.

## Abstract

We study and classify faithfully balanced modules for the algebra of lower triangular $n$ by $n$ matrices. 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.