Computational Construction of a Magic Labeling for the Four-Dimensional Permutohedron

### Description This repository contains the Python script and the resulting integer labeling vector for the four-dimensional permutohedron (Pi_4), associated with the research paper titled: "Facet-Incidence Constraints and an Explicit Four-Dimensional Magic Permutohedron". The script provides the computational verification of the bijective integer labeling that attains the magic constants 1452 (for rank-1 and rank-4 facets) and 726 (for rank-2 and rank-3 facets). ### MethodologyThe problem is formulated as an Integer Linear Programming (ILP) model, utilizing an exact-cover orientation strategy. By pairing vertices $w_p^+ = (a,b,\tau)$ and $w_p^- = (b,a,\tau)$ and enforcing their labels to be complementary ($61-r$ and $60+r$), we effectively reduce the search space and solve the 93-dimensional affine solution space using the HiGHS solver integrated within scipy.optimize.milp. ### Requirements- Python 3.x- numpy- scipy ### How to useSimply run the script using a standard Python interpreter. The script will:1. Construct the 30x120 facet-vertex incidence matrix.2. Solve the orientation ILP model.3. Validate the solution against the full facet equations (M_5 * x = s).4. Output the bijective labeling vector for S_5 in lexicographic order. For any questions regarding the computational model, please refer to contact the author.