Abstract We provide a piecewise linear isomorphism from the normal fan of the pivot polytope of a product of simplices to the normal fan of a shuffle of associahedra.

Wiggly permutations were introduced by Bapat and Pilaud (Wigglyhedron Mathematische Zeitschrift 2025). We positively answer one of their conjectures by finding a Hamilton path in the wiggly flip graph that is isomorphic to the wigglyhedron. Our path provides a Gray code in which successive wiggly permutations are obtained by a single jump or hop, meaning that one or two consecutive symbols move past some number of smaller symbols. The Gray code has a simple greedy description that produces a recursive zig-zag pattern reminiscent of plain changes for permutations. More broadly, our results extend Algorithm J and the series of papers on zig-zag languages initiated by Hartung, Hoang, Mütze and Williams (Combinatorial Generation via Permutation Languages SODA 2020). Finally, we use wiggly changes as the basis for an 𝒪(n)-time delay generation algorithm.

Abstract We study deformations of graphical zonotopes. Deformations of the classical permutahedron (which is the graphical zonotope of the complete graph) have been intensively studied in recent years under the name of generalized permutahedra. We provide an irredundant description of the deformation cone of the graphical zonotope associated to a graph G, consisting of independent equations defining its linear span (in terms of non-cliques of G) and of the inequalities defining its facets (in terms of common neighbors of neighbors in G). In particular, we deduce that the faces of the standard simplex corresponding to induced cliques in G form a linear basis of the deformation cone, and that the deformation cone is simplicial if and only if G is triangle-free.

We extend the facial weak order from finite Coxeter groups to central hyperplane arrangements. The facial weak order extends the poset of regions of a hyperplane arrangement to all its faces. We provide four non-trivially equivalent definitions of the facial weak order of a central arrangement: (1) by exploiting the fact that the faces are intervals in the poset of regions, (2) by describing its cover relations, (3) using covectors of the corresponding oriented matroid, and (4) using certain sets of normal vectors closely related to the geometry of the corresponding zonotope. Using these equivalent descriptions, we show that when the poset of regions is a lattice, the facial weak order is a lattice. In the case of simplicial arrangements, we further show that this lattice is semidistributive and give a description of its join-irreducible elements. Finally, we determine the homotopy type of all intervals in the facial weak order.

Given k labelings of a finite d-dimensional cubical grid, define the combined distance between two labels to be the sum of the $\ell_1$-distance between the two labels in each labeling. We want to construct k labelings which maximize the minimum combined distance between any two labels. When $d=1$, this can be interpreted as placing n nonattacking rooks in a k-dimensional chessboard of size n in such a way to maximize the minimum $\ell_1$-distance between any two rooks. Rook placements are also known as Latin hypercube designs in the literature. In this paper, we revisit this problem with a more geometric approach. Instead of providing explicit but complicated formulas, we construct rook placements in a k-dimensional chessboard of size n as certain lattice-like structures for certain well-chosen values of n. Then, we extend these constructions to any values of n using geometric arguments. With this method, we present a clean and geometric description of the known optimal rook placements in the two-dimensional square grid. Furthermore, we provide asymptotically optimal constructions of k labelings of d-dimensional cubical grids which maximize the minimum combined distance. Finally, we discuss the extension of this problem to labelings of an arbitrary graph. We prove that deciding whether a graph has two labelings with combined distance at least 3 is at least as hard as graph isomorphism.