Related papers: Recent results on permutations without short cycle…

We study the automorphism group of an infinite minimal shift $(X,\sigma)$ such that the complexity difference function, $p(n+1)-p(n)$, is bounded. We give some new bounds on $\mbox{Aut}(X,\sigma)/\langle \sigma \rangle$ and also study the…

Let $\mathbb{S}_n$ denote the symmetric group on $[n]=\{1,\ldots,n\}$ with the uniform probability measure. For a permutation $\pi \in \mathbb{S}_n$ let $X_{\pi}$ denote the simplicial complex on the vertex set $[n]$ whose simplices are all…

We show that in the point process limit of the bulk eigenvalues of $\beta$-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size $\lambda$ is given by \[\bigl(\…

To eliminate gravitational non-Gaussianity, we introduce the $\mathcal{Z}$-$\kappa$ transform, a simple local nonlinear transform of the matter density field that emulates the inverse of nonlinear gravitational evolution. Using $N$-body…

We establish the following quantitative form of the Green--Tao theorem: if a set $\mathcal{A}$ of relative density $\delta$ within the primes up to $N$ contains no nontrivial arithmetic progressions of length $k\geq 4$, then $\delta\ll…

The idea of left(right) palindromic permutations(LPPs,RPPs) and left(right) generalized Smarandache palindromic permutations(LGSPPs,RGSPPs) are introduced in symmetric groups S_n of degree n. It is shown that in S_n, there exist a LPP and a…

In this note we study certain sufficient conditions for a set of minimal klt pairs $(X,\Delta)$ with $\kappa(X,\Delta)=\dim(X)-1$ to be bounded.

A permutation p is realized by the shift on N symbols if there is an infinite word on an N-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as p. The set of realized permutations…

Recurrence quantification analysis is a method for measuring the complexity of dynamical systems. Recurrence determinism is a fundamental characteristic of it, closely related to correlation sum. In this paper, we study asymptotic behavior…

We consider the testing and estimation of change-points, locations where the distribution abruptly changes, in a sequence of multivariate or non-Euclidean observations. We study a nonparametric framework that utilizes similarity information…

Given a positive integer $N$ and real number $\alpha\in [0, 1]$, let $m(\alpha,N)$ denote the minimum, over all sets $A\subset \mathbb{Z}/N\mathbb{Z}$ of size at least $\alpha N$, of the normalized count of 3-term arithmetic progressions…

A record in a permutation is a maximum or a minimum, from the left or from the right. The entries of a permutation can be partitioned into two types: the ones that are records are called external points, the others are called internal…

Let $S_n$ denote the symmetric group of order $n$. Say that two subsets $x, y\subseteq S_n$ are \emph{equivalent} if there exist permutations $g_1, g_2\in S_n$ such that $g_1xg_2=y$, where multiplication is understood elementwise. Recently,…

We derive the density evolution equations for non-binary low-density parity-check (LDPC) ensembles when transmission takes place over the binary erasure channel. We introduce ensembles defined with respect to the general linear group over…

Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n? The distribution of this k!-dimensional vector of pattern…

For $k\geqslant 1$, denote by $p_k(n)$ the number of partitions of an integer $n$ into $k$-th powers. In this note, we apply the saddle-point method to provide a new proof for the well-known asymptotic expansion of $p_k(n)$. This approach…

We use moment method to understand the cycle structure of the composition of independent invariant permutations. We prove that under a good control on fixed points and cycles of length 2, the limiting joint distribution of the number of…

We propose a low-complexity method to find quasi-cyclic low-density parity-check block codes with girth 10 or 12 and shorter length than those designed through classical approaches. The method is extended to time-invariant spatially coupled…

We show that almost all permutations have some power that is a cycle of prime length. The proof includes a theorem giving a strong upper bound on the proportion of elements of the symmetric group having no cycles with length in a given set.

Given a finite group with a generating subset there is a well-established notion of length for a group element given in terms of its minimal length expression as a product of elements from the generating set. Recently, certain quantities…