Questions about combinatorics on languages of words, that is how many sequences of symbols with certain properties there are.

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3
votes
1answer
14 views

What is the closest result relating perfect shuffle closure of primitive words?

A perfect shuffle of two words $u=a_{1}a_{2} \cdots a_{n}$ and $v=b_{1}b_{2} \cdots b_{n}$ where $a_{i}$'s and $b_{j}$'s are letters from the alphabet $\Sigma$ is defined as $u \diamond v=a_{1}b_{1} ...
2
votes
2answers
54 views

How should i guide a program to perform correct things? [closed]

I want to make a small model of A.I. which can learn itself. I am inspired by 1000+ monkey theorem which states that if 1000+ monkey bangs a keyboard for enough long, then they will eventually produce ...
6
votes
0answers
66 views

What is longest string with no equidistant pairs of equal characters for a given alphabet?

Given a finite alphabet $A$ with $|A|=n$, what is the largest $l$ so that a string $s=a_1...a_l$ of length $l$ exists with $a_i\in A$ so that there is no $i,j,d$ with $i\neq j, d > 0$ so that $a_i ...
4
votes
1answer
59 views

Number of words within Hamming distance $\delta$

This is a problem I'm having reading Arora & Barak's book, page 378-379. They define: For two words $x, y \in \{0, 1\}^m$, the fractional Hamming distance of $x$ and $y$ is equal to the ...
4
votes
1answer
99 views

How many words are in this sets?

I have problems to determine the size of the following sets in dependancy of the parameters $m, n>0$: $$M_{m,n}=\{a^iwa^{m-i}\mid 0\le i \le m,\;w\in\{a,b\}^n\}$$ It is easy to see that ...
-3
votes
1answer
42 views

Is $Σ^∗$ finite? [closed]

Suppose $Σ=\{0,1\}$; then $Σ^*$ is all combinations of $Σ$. So my question: is $Σ^*$ finite?
1
vote
0answers
26 views

How many restricted length strings are there without significant repetitions

Let us fix an alphabet $\Sigma$ of size $c$, then we have the finite language $\Sigma^n$ which is the set of all $n$ length words. For each $N,M$ how many words are there in $\Sigma^n$ such that no ...
-1
votes
0answers
137 views

Prove $y'w'v'u'x' = xuvwy$ [duplicate]

This question repeats one that was closed. Let $x, u, v, w, y, x', u', v', w', y'$ be words satisfying $y'x' = xy$. $y'u'x' = xuy$. $y'v'x' = xvy$. $y'w'x' = xwy$. $y'v'u'x' = xuvy$. $y'w'v'x' = ...
3
votes
1answer
31 views

Lexicographical position of a string in its type class

I have the following problem: Given a string $x\in\{1,...,M\}^+$ of length $n$. Let $S$ be the set of all words with exactly the same numbers of occurences of smybols as in $x$. In fact, $S$ consists ...
2
votes
2answers
519 views

How Dynamic programming can be used for Coin Change problem?

As far as I can unserstand Dynamic programming stands simply for memoization (which is a fancy name for lazy evaluation or plain "caching"). Now, I read that there is we can reduce complexity of ...
1
vote
1answer
92 views

The language of all borderless words

Could anyone help me please to find who was the first person who has proved that the language of all borderless words is not regular and when was that? Could you mention the reference, please? A word ...
6
votes
4answers
375 views

What is the number of expressions containing n pairs of matching brackets with nesting limit?

I know the answer without nesting limit is the Catalan number. My question is, specifically, is there a recurrence relation that gives the number of expression containing $n$ pairs of matching ...
3
votes
2answers
6k views

Finding the number of distinct permutations of length N with n different symbols

I have one puzzle whose answer I have boiled down to finding the total number and which type of permutation they are. For example if the string is of length ten as $w = aabbbaabba$, the total number ...
7
votes
2answers
371 views

How many strings are close to a given set of strings?

This question has been prompted by Efficient data structures for building a fast spell checker. Given two strings $u,v$, we say they are $k$-close if their Damerau–Levenshtein distance¹ is small, ...
6
votes
1answer
192 views

What is a formula for the number of strings with no repeats?

I want to count the number of strings $s$ over a finite alphabet $A$, that contain no repeats, and by that I mean for any substring $t$ of $s$, $1< |t| < |s|$, there is no disjoint copy of $t$ ...
12
votes
2answers
698 views

Number of words of a given length in a regular language

Is there an algebraic characterization of the number of words of a given length in a regular language? Wikipedia states a result somewhat imprecisely: For any regular language $L$ there exist ...
11
votes
3answers
561 views

Number of words in the regular language $(00)^*$

According to Wikipedia, for any regular language $L$ there exist constants $\lambda_1,\ldots,\lambda_k$ and polynomials $p_1(x),\ldots,p_k(x)$ such that for every $n$ the number $s_L(n)$ of words of ...
15
votes
1answer
386 views

Does every large enough string have repeats?

Let $\Sigma$ be some finite set of characters of fixed size. Let $\alpha$ be some string over $\Sigma$. We say that a nonempty substring $\beta$ of $\alpha$ is a repeat if $\beta = \gamma \gamma$ for ...