Related papers: Zeros of the M\"{o}bius function of permutations

We consider Riemann's Xi function $\xi(s)$ which is evaluated at $s = \frac{1}{2} + \sigma + i \omega$, given by $\xi(\frac{1}{2} + \sigma + i \omega)= E_{p\omega}(\omega)$, where $\sigma, \omega$ are real and compute its inverse Fourier…

A ballot permutation is a permutation $\pi$ such that in any prefix of $\pi$ the descent number is not more than the ascent number. By using a reversal concatenation map, we give a formula for the joint distribution (pk, des) of the peak…

Permutation $\sigma$ appears in permutation $\pi$ if there exists a subsequence of $\pi$ that is order-isomorphic to $\sigma$. The natural question is to check if $\sigma$ appears in $\pi$, and if so count the number of occurrences. We know…

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…

We say that a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n$ has a peak at index $i$ if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $\mathcal{P}(\pi)$ denote the set of indices where $\pi$ has a peak. Given a set $S$ of positive…

Let $n$ be a fixed integer with $n\geq 2$. For $i,j\in\mathbb{Z}_n$, define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of $\mathbb{Z}_n$ are written in a cycle. So $||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}$. For…

We show that if $y=(y_n)_{n\ge 1}$ is a bounded sequence with zero average along every infinite arithmetic progression then for every $N\ge 2$ there exist (unilateral or bilateral) subshifts $\Sigma$ over $N$ symbols, with entropy…

We derive a generating function for the number of integer compositions of $n$ into $k$ parts (i.e., $k$-compositions of $n$) with a given number of inversions, and obtain similar results for $k$-compositions of $n$ with a given number of…

The Mallows measure is a probability measure on $S_n$ where the probability of a permutation $\pi$ is proportional to $q^{l(\pi)}$ with $q > 0$ being a parameter and $l(\pi)$ the number of inversions in $\pi$. We show the convergence of the…

Pin permutations play an important role in the structural study of permutation classes, most notably in relation to simple permutations and well-quasi-ordering, and in enumerative consequences arising from these. In this paper, we continue…

We show that $$ \sum_{n\neq m}\frac{\mu(n)\mu(m)}{nm}E_{X}\left(\{nx\}\{mx\}\right)=-\frac{9}{2\pi^{2}}+O\left(\frac{1}{X}\right), $$ where $x$ is uniformly distributed in $[0,X]$ with $X\in \mathbb{N}$, $E_{X}(.)$ denotes the expected…

Let Sym_n denote the symmetric group of all permutations pi = a_1...a_n of {1,...,n}. An index i is a peak of pi if a_{i-1} < a_i > a_{i+1} and we let P(pi) be the set of peaks of pi. Given any set S of positive integers we define P(S;n) to…

Classical two-sample permutation tests for equality of distributions have exact size in finite samples, but they fail to control size for testing equality of parameters that summarize each distribution. This paper proposes permutation tests…

Let $\pi$ be a permutation of the set $[n]=\{1,2,\dots, n\}$. Two disjoint order-isomorphic subsequences of $\pi$ are called twins. How long twins are contained in every permutation? The well known Erd\H{o}s-Szekeres theorem implies that…

Given observations from a stationary time series, permutation tests allow one to construct exactly level $\alpha$ tests under the null hypothesis of an i.i.d. (or, more generally, exchangeable) distribution. On the other hand, when the null…

We investigate dynamical properties of the set of permutations of $\mathbb{Z}^d$ with restricted movement, i.e., permutations $\pi $ of $\mathbb{Z}^d$ such that $\pi (\mathbf{n})-\mathbf{n}$ lies, for every $\mathbf{n}\in \mathbb{Z}^d$, in…

A permutation $\tau$ in the symmetric group $S_j$ is minimally overlapping if any two consecutive occurrences of $\tau$ in a permutation $\sigma$ can share at most one element. B\'ona \cite{B} showed that the proportion of minimal…

We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation sigma = sigma_1sigma_2...sigma_n defined as the set of indices…

We are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens' distribution of a given parameter $\theta >0$, and its modification where entries equal to $1$ in the matrices…

We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.