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A word over the alphabet $\{0,1\}$ of length $n$ may contain squares, cubes, and generally $k$th powers, where $2\le k\le n$. Let $O_k(w)$ denote the number of $k$th power occurrences in the word $w$. …

Also not an answer, but may be useful. A somewhat similar kind of word is mentioned at the end of section 1.5 of Salomaa: Jewels of Formal Language Theory: There is an infinite word over a 3-letter …

A word is cubefree if it cannot be written as $xyyyz$ where $y$ has positive length. Let $h$ be the morphism from $\{0,1,2,3,4\}^*$ to $\{0,1\}^*$ given for words of length 1 as follows ($a\to h(a)$) …

Let $R_{n,k,b}$ be the number of $b$-ary strings of length $n$ that contain some run of length at least $k$ from some $(b-1)$-ary subalphabet. Let $N_{n,k,b}=b^n-R_{n,k,b}$ be the size of the compleme …

The version with overlaps allowed was studied in: On words containing all short subwords Ioan Tomescu http://dl.acm.org/citation.cfm?id=276278

Yes, $C(s)$ is an example of a cylinder set. More specifically, $C(s)$ is called a basic open cylinder (since other cylinder sets are unions of such sets). See e.g. Andre Nies' monograph Computabili …

To get started you can use oeis.org to investigate this. For instance, from plugging in the numerators corresponding to some of your data it seems that $$\#P_n^{(2)}(1)=n(n-1)$$ ("the oblong numbers") …

Consider the longest runs $\ell_\sigma(x)$ of the pattern $\sigma$ for $\sigma\in \{0, 1, 01, 10, 001,\dots\}$ etc. in a binary sequence $x=x_1\dots x_n$. For example, $\ell_{001}(0001110010011001)=2 …

Suppose $x$ is a word over the alphabet $\{0,1\}$. Let $a$, $b$ be elements of the group Dih$_k$ for some $k$. Let $\varphi=\varphi_{a,b,k}$ be the map from words over $\{0,1\}$ to elements of the di …

According to the 2013 paper "Computing the Partial Word Avoidability Indices of Ternary Patterns" by Blanchet-Sadri, Lohr, and Scott, The problem of deciding whether a given pattern is avoidable h …