Computing the Dimension of a Bipartition Matrix

TL;DR

This paper introduces methods to compute the dimension of bipartition matrices by decomposing them into indecomposable factors and applying four key routines for factorization and reconstruction.

Contribution

It presents a systematic approach with four routines to efficiently compute the dimension of bipartition matrices and their indecomposable components.

Findings

Four routines for bipartition matrix analysis

Efficient decomposition into indecomposables

Methods for reconstructing bipartitions from factors

Abstract

The dimension of a bipartition matrix (BPM) is the sum of the dimensions of its indecomposable factors. The dimension of an indecomposable BPM is the sum of its row, column, and entry dimensions. To compute these dimensions, we apply four routines of independent interest: (1) Factor a bipartition as a product of indecomposables; (2) recover a bipartition from its indecomposable factorization; (3) factor a BPM as a product of indecomposables; and (4) compute the "transpose-rotation" (the column dimension of a BPM is the row dimension of its transpose-rotation).