Related papers: On the M\"{o}bius function of permutations under t…

A permutation $\pi: [n] \rightarrow [n]$ is a Baxter permutation if and only if it does not contain either of the patterns $2-41-3$ and $3-14-2$. Baxter permutations are one of the most widely studied subclasses of general permutation due…

In this paper, we study permutations $\pi \in S_n$ with exactly $m$ transpositions. In particular, we are interested in the expected value of $\pi(1)$ when such permutations are chosen uniformly at random. When $n$ is even, this expected…

Permutation Pattern Matching (or PPM) is a decision problem whose input is a pair of permutations $\pi$ and $\tau$, represented as sequences of integers, and the task is to determine whether $\tau$ contains a subsequence order-isomorphic to…

We provide some upper bounds for the Mertens function ($M(n)$: the cumulative sum of the M$\ddot{\mathrm{o}}$bius function) by an approach of statistical mechanics, in which the M$\ddot{\mathrm{o}}$bius function is taken as a particular…

The paper studies entire functions of finite order of growth for which a representation of the form $\psi(z) = 1+ O(|z|^{-\mu}), \mu >0,$ as $z\to \infty$, is valid on a fixed ray of the complex plane. The main result is the following.…

This paper discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of neat power series for the prime counting function, $\pi(x)$, and the prime-power counting…

In this paper, we study the counting functions $\psi_\mathcal{P}(x)$, $N_\mathcal{P}(x)$ and $M_\mathcal{P}(x)$ of a generalized prime system $\mathcal{N}$. Here $M_\mathcal{P}(x)$ is the partial sum of the M\"{o}bius function over…

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…

There is a deep connection between permutations and trees. Certain sub-structures of permutations, called sub-permutations, bijectively map to sub-trees of binary increasing trees. This opens a powerful tool set to study enumerative and…

Given a set $V$, a subset $S$, and a permutation $\pi$ of $V$, we say that $\pi$ permutes $S$ if $\pi (S) \cap S = \emptyset$. Given a collection $\cS = \{V; S_1,\ldots , S_m\}$, where $S_i \subseteq V ~~(i=1,\ldots ,m)$, we say that $\cS$…

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. The authors of [2] showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ was $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$,…

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…

This is a brief survey of some open problems on permutation patterns, with an emphasis on subjects not covered in the recent book by Kitaev, \emph{Patterns in Permutations and words}. I first survey recent developments on the enumeration…

Let $m$ be a positive integer and let $\rho(m,n)$ be the proportion of permutations of the symmetric group ${\rm Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $\rho(n,m)n^{1-\frac{\phi(m)}{m}}\sim \kappa_m$ where…

We show that the sum function of the M\"{o}bius function of a Beurling number system must satisfy the asymptotic bound $M(x)=o(x)$ if it satisfies the prime number theorem and its prime distribution function arises from a monotone…

We consider partitions $p_{w}(n)$ of a positive integer $n$ arising from the generating functions \[ \sum_{n=1}^\infty p_{w}(n) z^n = \prod_{m \in \mathbb{N}} (1-z^m)^{-w(m)}, \] where the weights $w(m)$ are M\"{o}bius convolutions. We…

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…

This paper shows how to construct a discrete Morse function with a relatively small number of critical cells for the order complex of any finite poset with $\hat{0} $ and $\hat{1}$ from any lexicographic order on its maximal chains.…

In this paper, we investigate the M{\"o}bius function $\mu\_{\mathcal{S}}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of $\mathbb{Z}^m$. We introduce and develop a new approach to study…

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…