Related papers: A Note on Ternary Sequences of Strings of 0 and 1
A permutation is called {\it {block-wise simple}} if it contains no interval of the form $p_1\oplus p_2$ or $p_1 \ominus p_2$. We present this new set of permutations and explore some of its combinatorial properties. We present a generating…
We study pairs and triples consisting of triangular numbers such that the product of any two distinct elements decreased by 1 is a perfect square. For a positive integer $n$, we establish a necessary condition for the $n$-th triangular…
We study the relationship between the number of minus signs in a generalized sumset, $A+...+A-...-A$, and its cardinality; without loss of generality we may assume there are at least as many positive signs as negative signs. As addition is…
Given a finite permutation group $G$ with domain $\Omega$, we associate two subsets of natural numbers to $G$, namely $\mathcal{I}(G,\Omega)$ and $\mathcal{M}(G,\Omega)$, which are the sets of cardinalities of all the irredundant and…
Let $G$ be a finite abelian group. Let $g(G)$ be the smallest positive integer $t$ such that every subset of cardinality $t$ of the group $G$ contains a subset of cardinality $\mathrm{exp}(G)$ whose sum is zero. In this paper, we show that…
A recent unlabeled sampling result by Unnikrishnan, Haghighatshoar and Vetterli states that with probability one over iid Gaussian matrices $A$, any $x$ can be uniquely recovered from an unknown permutation of $y = A x$ as soon as $A$ has…
We provide a bijective proof of a formula of Auli and the author expressing the number of inversion sequences with no three consecutive equal entries in terms of the number of non-derangements, that is, permutations with fixed points.…
A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B_2[g]-set is a set A of integers such that no integer has…
Let $A$ and $B$ be sets of $k\ge5$ elements in $F=\mathbb{Z}/p\mathbb{Z}$ the field with $p>2k-2$ elements. We denote by $A\dot{+}B$ the set of different elements of $F$ that can be written in the form $a+b$, where $a\in A$, $b\in B$,…
We present a bijection between non-crossing partitions of the set $[2n+1]$ into $n+1$ blocks such that no block contains two consecutive integers, and the set of sequences $\{s_{i}\}_{1}^{n}$ such that $1 \leq s_{i} \leq i$, and if…
The conjecture of Peter Horak and Alex Rosa (generalizing that of Marco Buratti) states that a multiset L of v-1 positive integers not exceeding [v/2] is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with…
In this paper we show that the following problem is NP-complete: Given an alphabet $\Sigma$ and two strings over $\Sigma$, the question is whether there exists a permutation of $\Sigma$ which is a subsequence of both of the given strings.
A Cayley permutation is a word of positive integers such that if a letter appears in this word, then all positive integers smaller than that letter also appear. We initiate a systematic study of pattern avoidance on Cayley permutations…
A permutation group is {\it binary} if its orbits on $k$-tuples, for any integer $k\geq 2$, can be deduced from its orbits on $2$-tuples. Cherlin conjectured that a finite primitive binary permutation group $G$ must lie in one of three…
This report presents an expression for the number of a multiset's sub-multisets of a given cardinality as a function of the multiplicity of its elements. This is also the number of distinct samples of a given size that may be produced by…
Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem} asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9…
A relational structure ${\mathbb X}$ is said to be reversible iff every bijective endomorphism $f:X\rightarrow X$ is an automorphism. We define a sequence of non-zero cardinals $\langle \kappa_i :i\in I\rangle$ to be reversible iff each…
Given a bivariate system of polynomial equations with fixed support sets $A, B$ it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity $i$ for all $i$ in the…
A celebrated unresolved conjecture of Peter Frankl states that every finite union-closed collection of sets ($B$), with non-empty universe, admits an abundant element. The best result in the literature states that if $|B|=n$, then there…
An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our…