Related papers: Further results on staircase graph words
We find the two-variables generating function for the statistic which counts the number of variations in a word bounded by $1$. Thus, we refine and extend previous results concerning staircase words, which are words in which the variation…
Generalizing the notion of staircase words, introduced by Knopfmacher et.\ al, we define staircase graph words. These are functions $w$ from the vertex set $V$ of a graph into the set $\{1,2,\ldots,k\}$, such that $|w(x)-w(y)|\leq 1$, for…
We revisit staircases for words and prove several exact as well as asymptotic results for longest left-most staircase subsequences and subwords and staircase separation number, the latter being defined as the number of consecutive maximal…
A word $\sigma=\sigma_1...\sigma_n$ over the alphabet $[k]=\{1,2,...,k\}$ is said to be {\em smooth} if there are no two adjacent letters with difference greater than 1. A word $\sigma$ is said to be {\em smooth cyclic} if it is a smooth…
A staircase is the set of points in Z^2 below a given rational line in the plane that have Manhattan Distance less than 1 to the line. Staircases are closely related to Beatty and Sturmian sequences of rational numbers. Connecting the…
Generalised polynomials are maps constructed by applying the floor function, addition, and multiplication to polynomials. Despite superficial similarity, generalised polynomials exhibit many phenomena which are impossible for polynomials.…
Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n>=0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A…
A word $w_1w_2\cdots w_n$ is said to be up-down if $w_1 < w_2 >w_3 \cdots$. Carlitz and Scoville found the generating function for the number of up-down words over an alphabet of size $k$. Using properties of the Chebyshev polynomials we…
Sturmian sequences are well-known as the ones having minimal complexity over a 2-letter alphabet. They are also the balanced sequences over a 2-letter alphabet and the sequences describing discrete lines. They are famous and have been…
Covering arrays for words of length $t$ over a $d$ letter alphabet are $k \times n$ arrays with entries from the alphabet so that for each choice of $t$ columns, each of the $d^t$ $t$-letter words appears at least once among the rows of the…
For each infinite word over a given finite alphabet, we define an increasing sequence of rooted finite graphs, that can be thought as approximations of the famous Sierpinski carpet. These sequences naturally converge to an infinite rooted…
Trapezoidal words are words having at most $n+1$ distinct factors of length $n$ for every $n\ge 0$. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their…
A word is level if each letter appears in it the same number of times, plus or minus 1. We give a complete characterization of the lengths for which level ternary circular square-free words exist. Key words: combinatorics on words, circular…
Circular words are cyclically ordered finite sequences of letters. We give a computer-free proof of the following result by Currie: square-free circular words over the ternary alphabet exist for all lengths $l$ except for 5, 7, 9, 10, 14,…
Given a set of $t$ words of length $n$ over a $k$-letter alphabet, it is proved that there exists a common subsequence among two of them of length at least $\frac{n}{k}+cn^{1-1/(t-k-2)}$, for some $c>0$ depending on $k$ and $t$. This is…
Recently, a new class of words, denoted by L_n, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality given by the (n-1)-st Catalan number. Here, we consider statistics on L_n recording the number of…
The literature on word-representable graphs is quite rich, and a number of variations of the original definition have been proposed over the years. We are initiating a systematic study of such variations based on formal languages. In our…
It has become a de-facto standard to represent words as elements of a vector space (word2vec, GloVe). While this approach is convenient, it is unnatural for language: words form a graph with a latent hierarchical structure, and this…
Sturmian words form a family of one-sided infinite words over a binary alphabet that are obtained as a discretization of a line with an irrational slope starting from the origin. A finite version of this class of words called Christoffel…
We give a direct and intuitive proof (via sliding some columns up and down) of the following interesting fact: if we write out the Chebyshev polynomials in a chart and take the sums of coefficients along certain diagonals, we obtain the…