Related papers: Higher dimensional floorplans and Baxter d-permuta…
Let $\Delta =\{ \delta_1,\delta_2,...,\delta_m \} $ be a finite set of 2-connected patterns, i.e. graphs up to vertex relabelling. We study the generating function $D_{\Delta }(z,u_1,u_2,...,u_m),$ which counts polygon dissections and marks…
Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to $d$-angulations (plane graphs with…
An (n,d)-permutation code is a subset C of Sym(n) such that the Hamming distance d_H between any two distinct elements of C is at least equal to d. In this paper, we use the characterisation of the isometry group of the metric space…
We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is…
We introduce a symmetry class for higher dimensional partitions - fully complementary higher dimensional partitions (FCPs) - and prove a formula for their generating function. By studying symmetry classes of FCPs in dimension 2, we define…
We study the existence of plane substructures in drawings of the $d$-dimensional hypercube graph $Q_d$. We construct drawings of $Q_d$ which contain no plane subgraph with more than $2d-2$ edges, no plane path with more than $2d-3$ edges,…
We investigate geometric and topological properties of $d$-majorization -- a generalization of classical majorization to positive weight vectors $d \in \mathbb{R}^n$. In particular, we derive a new, simplified characterization of…
A sweep of a point configuration is any ordered partition induced by a linear functional. Posets of sweeps of planar point configurations were formalized and abstracted by Goodman and Pollack under the theory of allowable sequences of…
New reparametrisation invariant field equations are constructed which describe $d$-brane models in a space of $d+1$ dimensions. These equations, like the recently discovered scalar field equations in $d+1$ dimensions, are universal, in the…
We consider the problem of enumerating d-irreducible maps, i.e. planar maps whose all cycles have length at least d, and such that any cycle of length d is the boundary of a face of degree d. We develop two approaches in parallel: the…
A positive integer $d$ is a floor quotient of $n$ if there is a positive integer $k$ such that $d = \lfloor n/k \rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies the internal…
A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The first author recently showed that the…
Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori, Jacquard and Schaeffer…
In this paper we analyze the geometric structure and properties of a certain class of subsets of $\Bbb R^d$, known in the literature as 1-multicones, and here simply called multicones, which are quite natural generalizations of the…
Generalised Sierpinski carpets are planar sets that generalise the well-known Sierpinski carpet and are defined by means of sequences of patterns. We study the structure of the sets at the kth iteration in the construction of the…
There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have…
In this paper we introduce the notion of a $d$-dimensional cycle which is a homological generalization of the idea of a graph cycle to higher dimensions. We examine both the combinatorial and homological properties of this structure and use…
A family of $k$ point sets in $d$ dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of…
Fixing $n$ general points $p_i$ in the plane, what is the dimension of the space of plane curves of degree $d$ having multiplicity $m_i$ at $p_i$ for each $i$? In this article we propose an approach to attack this problem, and demonstrate…
The dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if $P$ has dimension $d$, then to know whether $x \leq y$ in $P$ it is enough to check whether $x\leq y$ in each of the…