🤖 AI Summary
This study addresses the problem of maximizing the number of triangular faces in simple pseudoline arrangements formed by an odd number of pseudolines. By employing a depth-first search that branches only on generators at even positions, combined with geometric pruning and reduced word enumeration, the approach systematically handles structural biases arising when $n \equiv 1 \pmod{6}$. Arrangements are hierarchically classified according to swap, Euclidean, and projective equivalence relations. The work achieves, for the first time, a complete enumeration of maximal-triangle arrangements together with a multi-level classification into equivalence classes, fully recovering the symmetry groups of each projective class and the orbit–stabilizer structures of their Euclidean subclasses. The enumeration yields 85,562,064 wiring diagrams for $n=27$, grouped into 56,646 projective classes, and provides initial results extending up to $n=93$.
📝 Abstract
We describe algorithms for the exhaustive enumeration and classification of simple arrangements of $n$ pseudolines ($n$ odd) maximizing the number of triangular faces. The depth-first search enumerates reduced words for the longest permutation $w_0$ by branching only on the even-indexed generators, using pruning constraints imposed by the geometry of optimal arrangements. The approach handles both perfect arrangements with a regular triangular pattern and unavoidable deviations from it for $n \equiv 1 \pmod 6$.
The output is classified into a hierarchy of equivalence classes: by commutation, by Euclidean transformations, and by projective transformations. For each projective class we recover its full symmetry group $G \subseteq S_{n+1}$ together with the orbit-stabilizer profile of its Euclidean subclasses. Completeness of the search and classification is proved: every wiring diagram is reached.
We report full enumerations; e.g. for $n=27$, 85,562,064 wiring diagrams partitioned into 56,646 projective classes. For larger $n$ (up to $n=93$), where exhaustive enumeration is out of reach, we report partial (first-hit) results.