Related papers: On column-convex and convex Carlitz polyominoes

Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$,…

We prove an asymptotic formula for the number of integral points of bounded log anticanonical height on a singular quartic del Pezzo surface over arbitrary number fields, with respect to the largest admissible boundary divisor. The…

Column-convex polyominoes were introduced in 1950's by Temperley, a mathematical physicist working on "lattice gases". By now, column-convex polyominoes are a popular and well-understood model. There exist several generalizations of…

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank $3$): (a) The number of extreme points in an…

A quadrangle in the Euclidean plane is called $n$-self-affine if it has a dissection into $n$ affine images of itself. All convex quadrangles are known to be $n$-self-affine for every $n \ge 5$. The only $2$-self-affine convex quadrangles…

Lin and Chang gave a generating function of convex polyominoes with an $m+1$ by $n+1$ minimal bounding rectangle. Gessel showed that their result implies that the number of such polyominoes is $$ \frac{m+n+mn}{m+n}{2m+2n\choose…

We consider a system of $R$ cubic forms in $n$ variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided $n\geq 25R$, we prove an asymptotic formula for the number of integer points in…

We consider the problem of enumerating integer tetrahedra of fixed perimeter (sum of side-lengths) and/or diameter (maximum side-length), up to congruence. As we will see, this problem is considerably more difficult than the corresponding…

In this paper, we study the extremal behaviour of deep holes in polyominoes. We determine the maximum number, $h_n$ of deep holes that an $n$-omino can enclose, ensuring that the boundary of each hole is disjoint from the boundaries of any…

A formula for the number of monic irreducible self-reciprocal polynomials, of a given degree over a finite field, was given by Carlitz in 1967. In 2011 Ahmadi showed that Carlitz's formula extends, essentially without change, to a count of…

Let $H_n$ be the minimal number such that any $n$-dimensional convex body can be covered by $H_n$ translates of interior of that body. Similarly $H_n^s$ is the corresponding quantity for symmetric bodies. It is possible to define $H_n$ and…

We prove an asymptotic for the number of additive triples of bijections $\{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$, that is, the number of pairs of bijections $\pi_1,\pi_2\colon \{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$ such that the pointwise…

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in $\mathbb{R}^n$ for all $n\ge2$. Our result generalizes and improves the inequalities previously obtained in M. Carozza, F. Giannetti,…

We study the asymptotic behavior of the $N$-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at $\exp(\xi/N)$ for a real number $\xi$. We show that if $\xi$ is sufficiently large, the colored Jones…

This note is motivated by an old result of Kronecker on monic polynomials with integer coefficients having all their roots in the unit disc. We call such polynomials Kronecker polynomials for short. Let $k(n)$ denote the number of Kronecker…

We consider cubic polynomials f(z)=z^3+az+b defined over the function field C(L), with a marked point of period N and multiplier L. In the case N=1, there are infinitely many such objects, and in the case N>2, only finitely many. The case…

In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of…

For all $n,\phi\in \mathbb{N}$ with $\phi\geqslant n+1$, the smallest possible isoperimetric quotient of an $n$-dimensional convex polytope that has $\phi$ facets is shown to be bounded from above and from below by positive universal…

An integer-valued matrix $\mathbf{A}$ is $\Delta$-modular if each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix has determinant at most $\Delta$ in absolute value. The column number problem is to determine the maximum…

We classify all convex polyomino whose coordinate rings are Gorenstein. We also compute the Castelnuovo-Mumford regularity of the coordinate ring of any stack polyomino in terms of the smallest interval which contains its vertices. We give…