Related papers: Polyominoes with maximally many holes

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…

We show that there are fewer than (e^2+3) 2^(k choose 2) n^k/4 non-degenerate positive solutions to a fewnomial system consisting of n polynomials in n variables having a total of n+k+1 distinct monomials. This is significantly smaller than…

Let k and n be positive integers. We mainly show that $$(ln+1) | k\binom{kn+ln}{kn},$$ $$2\binom{kn}n | \binom {2n}{n}C_{2n}^{(k-1)}$$, $$\binom{kn}n | (2k-1)C_n\binom{2kn}{2n},$$ $$\binom{2n}n | (k+1)C_n^{(k-1)}\binom{2kn}{kn},$$…

We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has 2^{Omega(n)}…

A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and…

Let $m,k$ be fixed positive integers. Determining the generating function for the number of tilings of an $m\times n$ rectangle by $k\times 1$ rectangles is a long-standing open problem to which the answer is only known in certain special…

Given a right-angled triangle of squares in a grid whose horizontal and vertical sides are $n$ squares long, let N(n) denote the maximum number of dots that can be placed into the cells of the triangle such that each row, each column, and…

Motivated by coding applications,two enumeration problems are considered: the number of distinct divisors of a degree-m polynomial over F = GF(q), and the number of ways a polynomial can be written as a product of two polynomials of degree…

We study the problems of covering or partitioning a polygon $P$ (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to…

Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of…

We consider a problem concerning tilings of rectangular regions by a finite library of polyominoes. We specifically look at rectangular regions of dimension $n\times m$ and ask whether or not a tiling of this region can be rearranged so…

For $n\in\nats$ and $3\leq k\leq n$ we compute the exact value of $E_k(n)$, the maximum number of edges of a simple planar graph on $n$ vertices where each vertex bounds an $\ell$-gon where $\ell\geq k$. The lower bound of $E_k(n)$ is…

Put n nonoverlapping squares inside the unit square. Let f(n) and g(n) denote the maximum values of the sum of the edge lengths of the n small squares, where in the case of f(n) the maximum is taken over all arbitrary packings of the unit…

Let a tribonacci sequence be a sequence of integers satisfying $a_k=a_{k-1}+a_{k-2}+a_{k-3}$ for all $k\ge 4$. For any positive integers $k$ and $n$, denote by $f_k(n)$ the number of tribonacci sequences with $a_1, a_2, a_3>0$ and with…

In this paper, we show that for given integers $h$ and $d$ with $h \geq 1$ and $d \geq 3$, there exists a non-normal very ample integral convex polytope of dimension $d$ which has exactly $h$ holes.

For n=1,2,3,... let N_n(q) denote the number of monic irreducible polynomials over the finite field F_q. We mainly show that the sequence N_n(q)^{1/n} (n>e^{3+7/(q-1)^2}) is strictly increasing and the sequence…

A new formula for the partition function $p(n)$ is developed. We show that the number of partitions of $n$ can be expressed as the sum of a simple function of the two largest parts of all partitions. Specifically, if $a_1 + >... + a_k = n$…

We derive tight expressions for the maximum number of $k$-faces, $0\le k\le d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$, as a function of the number of vertices of the…

Fine and Gill (1973) introduced the geometric representation for those comparative probability orders on n atoms that have an underlying probability measure. In this representation every such comparative probability order is represented by…

A matching of graph $G$ is maximal if it cannot be expanded by adding any edge to create a larger matching. In this paper, for a hexagonal ring $H$ with $n$ hexagons, we show that the number of maximal matchings of $H$ equals to the trace…