Related papers: A Note on 3-free Permutations

Let $\theta(n)$ be the number of permutations of $\{1,\dots,n\}$ with no $3$-term arithmetic progressions. We prove that $\lim_{n\to\infty}\theta(n)^{1/n}$ does not exist.

We improve the lower bound on the number of permutations of {1,2,...,n} in which no 3-term arithmetic progression occurs as a subsequence, and derive lower bounds on the upper and lower densities of subsets of the positive integers that can…

We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic…

There has been much work on the following question: given n how large can a subset of {1,...,n} be that has no arithmetic progressions of length 3. We call such sets 3-free. Most of the work has been asymptotic. In this paper we sketch…

A poset is {\it $(\3+\1)$-free} if it contains no induced subposet isomorphic to the disjoint union of a 3-element chain and a 1-element chain. These posets are of interest because of their connection with interval orders and their…

We consider the triangle-free process: given an integer n, start by taking a uniformly random ordering of the edges of the complete n-vertex graph K_n. Then, traverse the ordered edges and add each traversed edge to an (initially empty)…

A set is said to be \emph{3-free} if no three elements form an arithmetic progression. Given a 3-free set $A$ of integers $0=a_0<a_1<\cdots<a_t$, the \emph{Stanley sequence} $S(A)=\{a_n\}$ is defined using the greedy algorithm: For each…

The topic of the article is the parametric study of the complexity of algorithms on arrays of pairwise distinct integers. We introduce a model that takes into account the non-uniformness of data, which we call the Ewens-like distribution of…

A graceful n-permutation is a graceful labeling of an n-vertex path P_n. In this paper we improve the asymptotic lower bound on the number of such permutations from (5/3)^n to 2.37^n. This is a computer-assisted proof based on an effective…

The problem of looking for subsets of the natural numbers which contain no 3-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960…

A permutation P on {1,..,N} is a_fast_forward_permutation_ if for each m the computational complexity of evaluating P^m(x)$ is small independently of m and x. Naor and Reingold constructed fast forward pseudorandom cycluses and involutions.…

Define $S_n(R;T)$ to be the number of permutations on $n$ letters which avoid all patterns in the set $R$ and contain each pattern in the multiset $T$ exactly once. In this paper we enumerate $S_n(\{\alpha\};\{\beta\})$ and…

We prove new lower bounds on the maximum size of sets $A\subseteq \mathbb{F}_p^n$ or $A\subseteq \mathbb{Z}_m^n$ not containing three-term arithmetic progressions (consisting of three distinct points). More specifically, we prove that for…

We consider the number of linear extensions of an N-free order P. We give upper and lower bounds on this number in terms of parameters of the corresponding arc diagram. We propose a dynamic programming algorithm to calculate the number. The…

For a permutation $\pi: [K]\rightarrow [K]$, a sequence $f: \{1,2,\cdots, n\}\rightarrow \mathbb R$ contains a $\pi$-pattern of size $K$, if there is a sequence of indices $(i_1, i_2, \cdots, i_K)$ ($i_1<i_2<\cdots<i_K$), satisfying that…

In this paper, we characterize and enumerate pattern-avoiding permutations composed of only 3-cycles. In particular, we answer the question for the six patterns of length 3. We find that the number of permutations composed of $n$ 3-cycles…

Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…

A superpermutation is a sequence that contains every permutation of $n$ distinct symbols as a contiguous substring. For instance, a valid example for three symbols is a sequence that contains all six permutations. This paper introduces a…

For every fixed integer $t\geq 3$, we construct an $n$-vertex $K_{2,t+1}$-free graph containing $\Omega_t(n^2)$ copies of $K_{t,t}$. Combined with a simple counting argument, this shows that \[…

Let $f_{s,k}(n)$ be the maximum possible number of $s$-term arithmetic progressions in a sequence $a_1<a_2<\ldots<a_n$ of $n$ integers which contains no $k$-term arithmetic progression. For all integers $k > s \geq 3$, we prove that…