Combinatorial generation via permutation languages. VII. Supersolvable hyperplane arrangements

TL;DR

The paper establishes that supersolvable hyperplane arrangements admit Hamiltonian cycles in their graph of regions $G( H)$ and that any lattice quotient of the region lattice $P( H,R_0)$ inherits a Hamiltonian path on its cover graph. It develops a unifying zigzag framework to generate a wide range of combinatorial objects via signed and unsigned permutations, polygons, and acyclic orientations, using the suspension- and quotient-based structure of supersolvable arrangements. The results recover classical Gray codes (e.g., binary, permutahedral, and associahedral traversals) and provide new cyclic Gray codes for type $B$ analogues, including the $B$-associahedron. A notable contribution is the systematic treatment of signed permutation patterns, culminating in explicit bijections to geometric objects like point-symmetric triangulations and new Hamiltonian traversals on type $B$ quotientopes. Overall, the work connects hyperplane geometry, lattice theory, and combinatorial generation to produce broad, algorithmically implementable Hamiltonian paths/cycles across multiple combinatorial families.

Abstract

For an arrangement $\mathcal{H}$ of hyperplanes in $\mathbb{R}^n$ through the origin, a region is a connected subset of $\mathbb{R}^n\setminus\mathcal{H}$. The graph of regions $G(\mathcal{H})$ has a vertex for every region, and an edge between any two vertices whose corresponding regions are separated by a single hyperplane from $\mathcal{H}$. We aim to compute a Hamiltonian path or cycle in the graph $G(\mathcal{H})$, i.e., a path or cycle that visits every vertex (=region) exactly once. Our first main result is that if $\mathcal{H}$ is a supersolvable arrangement, then the graph of regions $G(\mathcal{H})$ has a Hamiltonian cycle. More generally, we consider quotients of lattice congruences of the poset of regions $P(\mathcal{H},R_0)$, obtained by orienting the graph $G(\mathcal{H})$ away from a particular base region $R_0$. Our second main result is that if $\mathcal{H}$ is supersolvable and $R_0$ is a canonical base region, then for any lattice congruence $\equiv$ on $P(\mathcal{H},R_0)=:L$, the cover graph of the quotient lattice $L/\equiv$ has a Hamiltonian path. [...]

Figures (27)

• Figure 1: (a) Steinhaus-Johnson-Trotter listings of permutations for $n=1,\ldots,4$, with the largest element $n$ highlighted; (b) visualization of the listing for $n=4$ as a Hamiltonian cycle in the permutahedron.
• Figure 2: (a) The coordinate arrangement for $n=3$ with regions labeled by binary strings; (b) its stereographic projection from the south pole with the normal vectors for each hyperplane; (c) its graph of regions, realized as a polytope, namely the 3-dimensional hypercube.
• Figure 3: (a) The braid arrangement (=type $A$ Coxeter arrangement) for $n=4$ projected to three dimensions with the regions labeled by permutations; (b) its stereographic projection; (c) its graph of regions, realized as a polytope, namely the 3-dimensional permutahedron of type $A$.
• Figure 4: (a) The type $B$ Coxeter arrangement for $n=3$ with the regions labeled by signed permutations (barred entries have a negative sign); (b) its stereographic projection; (c) its graph of regions, realized as a polytope, namely the 3-dimensional $B$-permutahedron.
• Figure 5: Stereographic projections of two hyperplane arrangements $\mathcal{H}$ in $\mathbb{R}^3$ for which $G(\mathcal{H})$ does not admit a Hamiltonian path (nor cycle): (a) $\mathcal{H}$ consists of 4 hyperplanes and the two partition classes of $G(\mathcal{H})$ have sizes 6 (black vertices) and 8 (white vertices); (b) $\mathcal{H}$ consists of 7 hyperplanes and both partition classes of $G(\mathcal{H})$ have size 22.

Theorems & Definitions (47)

• Theorem 1
• Theorem 2
• Corollary 3
• Corollary 4
• Lemma 5
• Lemma 6
• Lemma 7
• Theorem 8
• Theorem 9
• Lemma 10