Computing the Dimension of a Bipartition Matrix
Dawson Freeman, Ronald Umble
TL;DR
This work addresses the challenge of computing the dimension of a bipartition matrix (BPM), a quantity defined as the sum of the dimensions of its indecomposable factors. The authors introduce four interrelated routines—factoring BPMs into indecomposables, recovering bipartitions from indecomposable factorizations, factoring BPMs, and a transpose‑rotation based dimension computation—to support an overarching dimension algorithm. Central to the method are concepts such as embedding partitions, row/column equalizers, and the transverse decomposition, which together enable a constructive, algorithmic calculation of the BPM dimension. The implementation in Python, along with explicit examples and appendices, provides a practical tool for computing BPM dimensions and, by extension, the geometric dimensions of faces in related polytopes like bipermutahedra and biassociahedra, with potential extensions to generalized BPMs.
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).