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8
votes
1answer
286 views

Number of trivializations of a trivial word in the free group

Let $M$ be the free monoid on $2n$ generators $x_1,X_1,...,x_n,X_n$ and consider the set $T$ of all those elements of $M$ which map to 1 of the free group on $x_1,...,x_n$ under the homomorphism $\pi$ ...
17
votes
0answers
407 views

partition of infinite word onto permitted words

Consider words over binary alphabet $\{0,1\}$. Let $M$ be a set of finite words such that $M$ contains at least $c\cdot 2^n$ words of length $n$ for all large enough $n$ (for a constant $c$, ...
16
votes
0answers
448 views

Avoidable words

Let $u(x_1,...,x_n)$ be a word, $k\in \mathbb{N}$. We say that $u$ is $k$-avoidable if there exists an infinite word in $k$ letters $\{a_1,...,a_k\}$ which does not contain values of $u$ (i.e. words ...
11
votes
0answers
303 views

Word complexity of primes mod 4

For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...
10
votes
0answers
156 views

Computing exact or asymptotics for number of strings over an alphabet of size $n$ that have no non-trivial substrings that appear more than once

I ran across a seemingly relatively simple combinatorics problem that appears open. For an alphabet of size $n$, let $A(n)$ be the number of strings over the alphabet that have no substring of length ...
7
votes
0answers
128 views

Combinatorics of palindromic decompositions

This is sort of a companion to my question Number of trivializations of a trivial word in the free group (which in turn is motivated by my earlier question here). It turns out that that question may ...
3
votes
0answers
51 views

Covariance matrix for number of powers in a word

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 $w$. ...
3
votes
0answers
156 views

Generalised de Bruijn Graph

I have encountered sets of the following type, consisting of words over a finite aphabet $A$. If $S$ is such a set, then $S$ is finite, No word in $S$ is part of another element of $S$, and every ...
3
votes
0answers
343 views

Software for Combinatorial Algebra sought

I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task: I have two graded (generally ...
2
votes
0answers
98 views

Counting strings with alternating letters with generating functions

It is a classical problem that of finding the generating function (GF) of the number of strings with length $n$ having $m$ different letters (basically, the problem reduces to that of writing the ...