Related papers: Locally Convex Words and Permutations
An operator convex function on (0,\infty) which satisfies the symmetry condition k(1/x) = x k(x) can be used to define a type of non-commutative multiplication by a positive definite matrix (or its inverse) using the primitive concepts of…
This paper addresses the uniform random generation of words from a context-free language (over an alphabet of size $k$), while constraining every letter to a targeted frequency of occurrence. Our approach consists in a multidimensional…
An involution in a Coxeter group has an associated set of involution words, a variation on reduced words. These words are saturated chains in a partial order first considered by Richardson and Springer in their study of symmetric varieties.…
We investigate a multivariate growth series $\Gamma_L({\bf z}), {\bf z} \in \mathbb{C}^d$ associated with a regular language $L$ over an alphabet of cardinality $d.$ Our focus is on languages coming from subgroups of the free group and from…
Let $f$ be a smooth real function with strictly monotone first $k$ derivatives. We show that for a finite set $A$, with $|A+A|\leq K|A|$, $|2^kf(A)-(2^k-1)f(A)|\gg_k |A|^{k+1-o(1)}/K^{O_k(1)}$. We deduce several new sum-product type…
Two words are $k$-binomially equivalent if each subword of length at most $k$ occurs the same number of times in both words. The $k$-binomial complexity of an infinite word is a counting function that maps $n$ to the number of $k$-binomial…
A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a $\textit{zigzag word}$. A…
In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions with deterministic terms are…
In this text, we consider random permutations which can be written as free words in several independent random permutations: firstly, we fix a non trivial word $w$ in letters $g_1,g_1^{-1},..., g_k,g_k^{-1}$, secondly, for all $n$, we…
We investigate the problem of percolation of words in a random environment. To each vertex, we independently assign a letter $0$ or $1$ according to Bernoulli r.v.'s with parameter $p$. The environment is the resulting graph obtained from…
In this paper we determine a closed formula for the number of convex permutominoes of size n. We reach this goal by providing a recursive generation of all convex permutominoes of size n+1 from the objects of size n, according to the ECO…
We study generating functions for the number of involutions in $S_n$ avoiding (or containing once) 132, and avoiding (or containing once) an arbitrary permutation $\tau$ on $k$ letters. In several interesting cases the generating function…
In this paper we examine decision problems associated with various classes of convex languages, studied by Ang and Brzozowski (under the name "continuous languages"). We show that we can decide whether a given language L is prefix-,…
We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are…
We introduce notions of a constraint metric approximation and of a constraint stability of a metric approximation. This is done in the language of group equations with coefficients. We give an example of a group which is not constraintly…
We focus on infinite words with languages closed under reversal. If frequencies of all factors are well defined, we show that the number of different frequencies of factors of length n+1 does not exceed 2C(n+1)-2C(n)+1.
We define two new statistics on words: the k-connector and the gk-connector. For a word $\pi = \pi_1\pi_2\cdots\pi_n$ of length $n$ over the alphabet $[k]$, a k-connector is defined as an ordered pair $(\pi_j, \pi_{j+1})$ where $1 \leq j…
The recently confirmed Dejean's conjecture about the threshold between avoidable and unavoidable powers of words gave rise to interesting and challenging problems on the structure and growth of threshold words. Over any finite alphabet with…
We prove a lower bound of exp(-C (log(2/alpha))^7)N^{k-1} to the number of solutions of an invariant equation in k variables, contained in a set of density alpha. Moreover, we give a Behrend-type construction for the same problem with the…
Call a permutation $k$-inflatable if the sequence of its tensor products with uniform random permutations of increasing lengths has uniform $k$-point pattern densities. Previous work has shown that nontrivial $k$-inflatable permutations do…