Related papers: On the M\"obius Function of Permutations With One …

The M\"obius polynomial is an invariant of ranked posets, closely related to the M\"obius function. In this paper, we study the M\"obius polynomial of face posets of convex polytopes. We present formulas for computing the M\"obius…

In this paper we derive some identities and inequalities on the M\"obius mu function. Our main tool is phi functions for intervals of positive integers and their unions.

A barcode is a finite multiset of intervals on the real line. Jaramillo-Rodriguez (2023) previously defined a map from the space of barcodes with a fixed number of bars to a set of multipermutations, which presented new combinatorial…

Let $P^-(n)$ denote the smallest prime factor of a natural integer $n>1$. Furthermore let $\mu$ and $\omega$ denote respectively the M\"obius function and the number of distinct prime factors function. We show that, given any set ${{\scr…

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 establish an interesting upper bound for the moments of truncated Dirichlet convolution of M\"obius function, a function noted $M(n,z)$. Our result implies that $M(n,j)$ is usually quite small for $j \in \{1,\dots,n\}$. Also, we…

We investigate Sarnak's conjecture on the M\"obius function in the special case when the test function is the indicator of the set of integers for which a real additive function assumes a given value.

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.…

An infinite permutation is a linear ordering of the set of non-negative integers. Generally, the properties of infinite permutations analogous to those of infinite words show some resemblances and some differences between permutations and…

We give multiple proofs of two formulas concerning the enumeration of permutations avoiding a monotone consecutive pattern with a certain value for the inverse peak number or inverse left peak number statistic. The enumeration in both cases…

Using the stratifications of Deligne-Mumford moduli spaces $\overline{\mathcal M}_{g,n}$ indexed by stable graphs, we introduce a partially ordered set of stable graphs by defining a partial ordering on the set of connected stable graphs of…

We give a graded version of the M\"obius inversion formula in the framework of trace monoids. The formula is based on a graded version of the M\"obius transform, related to the notion of height deriving from the Cartier-Foata normal form of…

We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the M\"obius function we show that there are cancellations…

A permutation $\sigma$ describing the relative orders of the first $n$ iterates of a point $x$ under a self-map $f$ of the interval $I=[0,1]$ is called an \emph{order pattern}. For fixed $f$ and $n$, measuring the points $x\in I$ (according…

The Mertens function is defined as $M(x) = \sum_{n \leq x} \mu(n)$, where $\mu(n)$ is the M\"obius function. The Mertens conjecture states $|M(x)/\sqrt{x}| < 1$ for $x > 1$, which was proven false in 1985 by showing $\liminf M(x)/\sqrt{x} <…

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…

In $\mathcal L$, the semilattice of faces of an $n$-cube, we count the number of automorphisms of $\mathcal L$ that fix a given subalgebra -- either pointwise or as a subalgebra. By using M\"obius inversion we get a formula for the number…

Let $G$ be the simple group ${\rm PSU}(2,2^{2^n})$, $n>0$. For any subgroup $H$ of $G$, we compute the M\"obius function $\mu_L(H,G)$ of $H$ in the subgroup lattice $L$ of $G$, and the M\"obius function $\mu_{\bar L}([H],[G])$ of $[H]$ in…

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P$ that avoid the pattern $\pi$ is denoted $Av_P(\pi)$. We…

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…