Related papers: Permutation Complexity of the Thue-Morse Word
We study the palindromic complexity of infinite words $u_\beta$, the fixed points of the substitution over a binary alphabet, $\phi(0)=0^a1$, $\phi(1)=0^b1$, with $a-1\geq b\geq 1$, which are canonically associated with quadratic non-simple…
We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds…
We provide an exact estimate on the maximal subword complexity for quasiperiodic infinite words. To this end we give a representation of the set of finite and of infinite words having a certain quasiperiod q via a finite language derived…
These lecture notes provide an introduction to combinatorics on words and its interactions with dynamics, algebra, and arithmetic. The central theme is the notion of low factor complexity for infinite words. We investigate the following…
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.
In this article we undertake a study of extension complexity from the perspective of formal languages. We define a natural way to associate a family of polytopes with binary languages. This allows us to define the notion of extension…
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…
This is my dissertation. Its research object is a symmetric group of permutations acting on a finite set. The density of permutations with a given cycle structure pattern is explored when the group order tends to infinity. New and sharper…
A reduced word of a permutation $w$ is a minimal length expression of $w$ as a product of simple transpositions. We examine the computational complexity, formulas and (randomized) algorithms for their enumeration. In particular, we prove…
Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines,…
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 consider two orthogonal points of view on finite permutations, seen as pairs of linear orders (corresponding to the usual one line representation of permutations as words) or seen as bijections (corresponding to the algebraic point of…
We investigate the asymptotic properties of permutations drawn from the Luce model, a natural probabilistic framework in which permutations are generated sequentially by sampling without replacement, with selection probabilities…
I am going to compare well-known properties of infinite words with those of infinite permutations, a new object studied since middle 2000s. Basically, it was Sergey Avgustinovich who invented this notion, although in an early study by Davis…
We study the complexity of multiplication of two elements in a finite field extension given by their coordinates in a normal basis. We show how to control this complexity using the arithmetic and geometry of algebraic curves.
We introduce a notion of pattern occurrence that generalizes both classical permutation patterns as well as poset containment. Many questions about pattern statistics and avoidance generalize naturally to this setting, and we focus on…
The Thue-Morse sequence is an aperiodically ordered infinite binary sequence. It is used as a one-dimensional way to model the structure of a quasicrystal. For example, taking autocorrelations of these sequences (roughly, measuring how…
We consider rotations on the torus $\mathbb{T}^2$, and we classify them with respect to the complexity functions. In dimension one, a minimal rotation can be coded by a sturmian word. A sturmian word has complexity $n+1$ by the…
In this paper, we study the additive complexity $\rho^{+}_{\mathbf{t}}(n)$ of a Thue-Morse like sequence $\mathbf{t}=\sigma^{\infty}(0)$ with the morphism $\sigma: 0\to 01, 1\to 12, 2\to 20$. We show that…
Two permutations of the natural numbers diverge if the absolute value of the difference of their elements in the same position goes to infinity. We show that there exists an infinite number of pairwise divergent permutations of the…