Related papers: $k$-Foldability of Words
The combinatorics of RNA plays a central role in biology. Mathematical biologists have several commonly-used models for RNA: words in a fixed alphabet (representing the primary sequence of nucleotides) and plane trees (representing the…
An and/or tree is usually a binary plane tree, with internal nodes labelled by logical connectives, and with leaves labelled by literals chosen in a fixed set of k variables and their negations. In the present paper, we introduce the first…
A word $w=w_1\cdots w_n$ over the set of positive integers is a Motzkin word whenever $w_1=\texttt{1}$, $1\leq w_k\leq w_{k-1}+1$, and $w_{k-1}\neq w_{k}$ for $k=2, \dots, n$. It can be associated to a $n$-column Motzkin polyomino whose…
Motivated by a formula of A. Postnikov relating binary trees, we define the hook length polynomials for m-ary trees and plane forests, and show that these polynomials have a simple binomial expression. An integer value of this expression is…
In 1989 Erd\H{o}s and Sz\'ekely showed that there is a bijection between (i) the set of rooted trees with $n+1$ vertices whose leaves are bijectively labeled with the elements of $[\ell]=\{1,2,\dots,\ell\}$ for some $\ell \leq n$, and (ii)…
This article is motivated by the following satisfiability question: pick uniformly at random an and/or Boolean expression of length n, built on a set of k_n Boolean variables. What is the probability that this expression is satisfiable?…
A $k$-plane tree is a plane tree whose vertices are assigned labels between $1$ and $k$ in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are…
The Ward numbers $W(n,k)$ combinatorially enumerate set partitions with block sizes $\geq 2$ and phylogenetic trees (total partition trees). We prove that $W(n,k)$ also counts \emph{increasing Schr\"oder trees} by verifying they satisfy…
A word $w$ is called rich if it contains $| w|+1$ palindromic factors, including the empty word. We say that a rich word $w$ can be extended in at least two ways if there are two distinct letters $x,y$ such that $wx,wy$ are rich. Let $R$…
A graph $G = (V,E)$ is word-representable if there is a word $w$ over the alphabet $V$ such that $x$ and $y$ alternate in $w$ if and only if the edge $(x, y)$ is in $G$. It is known [6] that all $3$-colourable graphs are word-representable,…
Given a set $F$ of words, one associates to each word $w$ in $F$ an undirected graph, called its extension graph, and which describes the possible extensions of $w$ on the left and on the right. We investigate the family of sets of words…
A word $u$ is a scattered factor of $w$ if $u$ can be obtained from $w$ by deleting some of its letters. That is, there exist the (potentially empty) words $u_1,u_2,..., u_n$, and $v_0,v_1,..,v_n$ such that $u = u_1u_2...u_n$ and $w =…
Given a finite word $w$ over a finite alphabet $V$, consider the graph with vertex set $V$ and with an edge between two elements of $V$ if and only if the two elements alternate in the word $w$. Such a graph is said to be word-representable…
Any finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is reached, the word $w$ is called rich. The number of rich words of length $n$ over an alphabet of cardinality $q$ is denoted…
A simple graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ iff $xy\in E$. Word-representable graphs generalize several important classes of graphs. A graph…
We resolve two conjectures of Black-Drellich-Tymoczko about the numbers of valid plane trees for given primary sequences.
The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labelled by using the numbers {1,2,...,n} in such a way that the absolute differences induced on the edges are pairwise…
A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled pattern matching. In this paper we present an…
A double occurrence word $w$ over a finite alphabet $\Sigma$ is a word in which each alphabet letter appears exactly twice. Such words arise naturally in the study of topology, graph theory, and combinatorics. Recently, double occurrence…
The Fibonacci word $W$ on an infinite alphabet was introduced in [Zhang et al., Electronic J. Combinatorics 2017 24(2), 2-52] as a fixed point of the morphism $2i\rightarrow (2i)(2i+1)$, $(2i+1) \rightarrow (2i+2)$, $i\geq 0$. Here, for any…