Related papers: On the Enumerative Geometry of Pascal's Hexagram

Let $P$ be a collection of $n$ points moving along pseudo-algebraic trajectories in the plane. One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a subcubic…

The trinomial transform of a sequence is a generalization of the well-known binomial transform, replacing binomial coefficients with trinomial coefficients. We examine Pascal-like triangles under trinomial transform, focusing on the ternary…

We examine the poset $P$ of 132-avoiding $n$-permutations ordered by descents. We show that this poset is the "coarsening" of the well-studied poset $Q$ of noncrossing partitions . In other words, if $x<y$ in $Q$, then $f(y)<f(x)$ in $P$,…

The topology of the intersection of three quadrics in Euclidean 6-space is studied using Kollar results. This needs an existence of a line without real points in the complex projectivisation of quadrics. We establish the existence of such a…

We give a non-iterative solution to a particular case of the four-point three-views pose problem when three camera centers are collinear. Using the well-known Cayley representation of orthogonal matrices, we derive from the epipolar…

Let ${\mathbf P}^2$ denote the projective plane over a finite field ${\mathbb F}_q$. A pair of nonsingular conics $({\mathcal A}, {\mathcal B})$ in the plane is said to satisfy the Poncelet triangle condition if, considered as conics in…

We introduce and study the 1-planar packing problem: Given $k$ graphs with $n$ vertices $G_1, \dots, G_k$, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each $G_i$…

A linear $r$-uniform hypergraph is called acycilc if it can be constructed starting from one single edge then at each step adding a new edge that intersect the union of the vertices of the previous edges in at most one vertex. Recently,…

It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in $\mathbb{P}^4 (\mathbb{F}_{q^2})$ has at most $d(q^5+q^2) + q^3 + 1$ points in common with a threefold of degree $d$ defined over $\mathbb{F}_{q^2}$. He…

We deal with unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths $a,b+m,c,a+m,b,c+m$, where an equilateral triangle of side length $m$ has been removed from the center. We give closed formulas for the…

We characterize all algebraic numbers $\alpha$ of degree $d\in\{4,5,6,7\}$ for which there exist four distinct algebraic conjugates $\alpha_1$, $\alpha_2$, $\alpha_3$, $\alpha_4$ of $\alpha$ satisfying the relation…

In this paper, we propose to enumerate all different configurations belonging to a specific class of fractals: A binary initial tile is selected and a finite recursive tiling process is engaged to produce auto-similar binary patterns. For…

We describe the geometry of an arrangement of 24-cells inscribed in the 600-cell. In $\S$7 we apply our results to the even unimodular lattice $E_8$ and show how the 600-cell transforms $E_8$/2$E_8$, an 8-space over the field $\bf{F}$$_2$,…

We give a formula for the number of interior intersection points made by the diagonals of a regular $n$-gon. The answer is a polynomial on each residue class modulo 2520. We also compute the number of regions formed by the diagonals, by…

This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence $(S(n))_{n\ge 0}$…

Given a $2^N$-dimensional Cayley-Dickson algebra, where $3 \leq N \leq 6$, we first observe that the multiplication table of its imaginary units $e_a$, $1 \leq a \leq 2^N -1$, is encoded in the properties of the projective space PG$(N-1,2)$…

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some…

Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal…

A $(3, 6)$-fullerene is a cubic planar graph whose faces all have 3 or 6 sides. We give an exact enumeration of $(3, 6)$-fullerenes with $V$ vertices. We also enumerate $(3, 6)$-fullerenes with mirror symmetry, with 3-fold rotational…

We give explicit parametric equations for all irreducible plane projective sextic curves which have at most double points and whose total Milnor number is maximal (is equal to 19). In each case we find a parametrization over a number field…