Related papers: Multiple twins in permutations

We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…

An $r$-matrix is a matrix with symbols in $\{0,1,\ldots,r-1\}$. A matrix is simple if it has no repeated columns. Let ${\cal F}$ be a finite set of $r$-matrices. Let $\hbox{forb}(m,r,{\cal F})$ denote the maximum number of columns possible…

A permutation $\sigma\in\mathfrak{S}_n$ is simsun if for all $k$, the subword of $\sigma$ restricted to $\{1,...,k\}$ does not have three consecutive decreasing elements. The permutation $\sigma$ is double simsun if both $\sigma$ and…

Consider n unit intervals, say [1,2], [3,4], ..., [2n-1,2n]. Identify their endpoints in pairs at random, with all (2n-1)!! = (2n-1) (2n-3) ... 3 1 pairings being equally likely. The result is a collection of cycles of various lengths, and…

The Mallows measure on the symmetric group $S_n$ is the probability measure such that each permutation has probability proportional to $q$ raised to the power of the number of inversions, where $q$ is a positive parameter and the number of…

A rank-$r$ integer matrix $A$ is $\Delta$-modular if the determinant of each $r \times r$ submatrix has absolute value at most $\Delta$. The class of $1$-modular, or unimodular, matrices is of fundamental significance in both integer…

The length is(w) of the longest increasing subsequence of a permutation w in the symmetric group S_n has been the object of much investigation. We develop comparable results for the length as(w) of the longest alternating subsequence of w,…

We determine all permutation trinomials of type $x^{2p^s+r}+x^{p^s+r}+\lambda x^r$ over the finite field $\mathbb{F}_{p^t}$ when $(2p^s+r)^4<p^t$. This partially extends a previous result by Bhattacharya and Sarkar in the case $p=2$, $r=1$.

A permutation of n letters is k-prolific if each (n-k)-subset of the letters in its one-line notation forms a unique pattern. We present a complete characterization of k-prolific permutations for each k, proving that k-prolific permutations…

Let $[x]$ be the greatest integer not exceeding $x$. In the paper we introduce the sequence $\{U_n\}$ given by $U_0=1$ and $U_n=-2\sum_{k=1}^{[n/2]}\binom n{2k}U_{n-2k}\quad(n\ge 1)$, and establish many recursive formulas and congruences…

A permutation graph is a graph that can be derived from a permutation, where the vertices correspond to letters of the permutation, and the edges represent inversions. We provide a construction to show that there are infinitely many…

The Tur\'{a}n number $T(n,\alpha+1,r)$ is the minimum number of edges in an $n$-vertex $r$-graph whose independence number does not exceed $\alpha$. For each $r\geq 2$, there exists $t_*(r)$ such that $T(n,\alpha+1,r) = t_*(r) \: n^r \:…

For a set $\Omega$ an unordered relation on $\Omega$ is a family R of subsets of $\Omega.$ If R is such a relation we let G(R) be the group of all permutations on $\Omega$ that preserves R, that is g belongs to G(R) if and only if x in R…

A permutation is defined to be cycle-up-down if it is a product of cycles that, when written starting with their smallest element, have an up-down pattern. We prove bijectively and analytically that these permutations are enumerated by the…

Let $\alpha(n)$ denote the number of perfect square permutations in the symmetric group $S_n$. The conjecture $\alpha(2n+1) = (2n+1) \alpha(2n)$, provided by Stanley[4], was proved by Blum[1] using a generating function. This paper presents…

The randomization of a complete first order theory T is the complete continuous theory T^R with two sorts, a sort for random elements of models of T, and a sort for events in an underlying probability space. We give necessary and sufficient…

A {\it superpattern} is a string of characters of length $n$ that contains as a subsequence, and in a sense that depends on the context, all the smaller strings of length $k$ in a certain class. We prove structural and probabilistic results…

We define a suffixient set for a text $T [1..n]$ to be a set $S$ of positions between 1 and $n$ such that, for any edge descending from a node $u$ to a node $v$ in the suffix tree of $T$, there is an element $s \in S$ such that $u$'s path…

For positive integers $n,r,k$ with $n\ge r$ and $k\ge2$, a set $\{(x_1,y_1),(x_2,y_2),\dots,(x_r,y_r)\}$ is called a $k$-signed $r$-set on $[n]$ if $x_1,\dots,x_r$ are distinct elements of $[n]$ and $y_1\dots,y_r\in[k]$. We say a…

Let $L(n,k,r,p)$ denote the minimum number of $k$-subsets of an $n$-set such that all the $\binom{n}{p}$ $p$-subsets are intersected by one of them in at least $r$ elements. The case $p=r$ corresponds to the covering numbers, while the case…