Jamie Simpson

How Many Squares Can a String Contain

All our words (strings) are over afixedalphabet. A square is a subword of the formuu=u2, whereuis a nonempty word. Two squares aredistinctif they are of different shape, not just translates of each other. A worduisprimitiveifucannot be written in the formu=vjfor somej?2. A squareu2withuprimitive isprimitive rooted. LetM(n) denote the maximum number of distinct squares,P(n) the maximum number of distinct primitive rooted squares in a word of length n. We prove: no position in any word can be the beginning of the rightmost occurrence of more than two squares, from which we deduceM(n) 0, andP(n)=n?o(n) for infinitely manyn.

How many runs can a string contain

Given a string x=x[1..n], a repetition of period p in x is a substring u^r=x[i+1..i+rp], p=|u|, r>=2, where neither u=x[i+1..i+p] nor x[i+1..i+(r+1)p+1] is a repetition. The maximum number of repetitions in any string x is well known to be @Q(nlogn). A run or maximal periodicity of period p in x is a substring u^rt=x[i+1..i+rp+|t|] of x, where u^r is a repetition, t a proper prefix of u, and no repetition of period p begins at position i of x or ends at position i+rp+|t|+1. In 2000 Kolpakov and Kucherov showed that the maximum number @r(n) of runs in any string x[1..n] is O(n), but their proof was nonconstructive and provided no specific constant of proportionality. At the same time, they presented experimental data to prompt the conjecture: @r(n)

The Exact Number of Squares in Fibonacci Words

All our words (sequences) are binary. A square is a subword of the form uu (concatenation). Two squares are distinct if they are of different shape, not just translates of each other. Otherwise they are repeated. Fibonacci words are defined by f>sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub sub /sub< j)] < 0.7962).

Intersecting periodic words

Let a=a[1..2v] be a word which is a square and for which a[1..2u] is also a square and a[k+1..min(k+2w,2v)] has period w where u

More results on overlapping squares

Three recent papers (Fan et al., 2006; Simpson, 2007; Kopylova and Smyth, 2012) [5,11,8] have considered in complementary ways the combinatorial consequences of assuming that three squares overlap in a string. In this paper we provide a unifying framework for these results: we show that in 12 of 14 subcases that arise the postulated occurrence of three neighboring squares forces a breakdown into highly periodic behavior, thus essentially trivial and easily recognizable. In particular, we provide a proof of Subcase 4 for the first time, and we simplify and refine the previously established results for Subcases 11-14.

Euclidean strings

A string p = p0p1...pn-1 of non-negative integers is a Euclidean string if the string (p0 + 1)p1...(pn-1 - 1) is rotationally equivalent (i.e., conjugate) to p. We show that Euclidean strings exist if and only if n and p0 + p1 + ... + pn-1 are relatively prime and that, if they exist, they are unique. We show how to construct them using an algorithm with the same structure as the Euclidean algorithm, hence the name. We show that Euclidean strings are Lyndon words and we describe relationships between Euclidean strings and the Stem-Brocot tree, Fibonacci strings, Beatty sequences, and Sturmian sequences. We also describe an application to a graph embedding problem.

On Weak Circular Squares in Binary Words

A weak square in a binary word is a pair of adjacent nonempty blocks of the same length, having the same number of 1s. A weak circular square is a weak square which is possibly wrapped around the word: the tail protruding from the right end of the word reappears at the left end. Two weak circular squares are equivalent if they have the same length and contain the same number of ones. We prove that the longest word with only k inequivalent weak circular squares contains 4k+2 bits and has the form (01)2k+1 or its complement. Possible connections to tandem repeats in the human genome are pointed out.

Palindromes in circular words

There is a very short and beautiful proof that the number of distinct non-empty palindromes in a word of length n is at most n. In this paper we show, with a very complicated proof, that the number of distinct non-empty palindromes with length at most n in a circular word of length n is less than 5n/3. For n divisible by 3 we present circular words of length n containing 5n/3-2 distinct palindromes, so the bound is almost sharp. The paper finishes with some open problems.

The total run length of a word

A run in a word is a periodic factor whose length is at least twice its period and which cannot be extended to the left or right (by a letter) to a factor with greater period. In recent years a great deal of work has been done on estimating the maximum number of runs that can occur in a word of length n. A number of associated problems have also been investigated. In this paper we consider a new variation on the theme. We say that the total run length (TRL) of a word is the sum of the lengths of the runs in the word and that @t(n) is the maximum TRL over all words of length n. We show that n^2/8<@t(n)<47n^2/72+2n for all n. We also give a formula for the average total run length of words of length n over an alphabet of size @a, and some other results.

An Extension of the Periodicity Lemma to Longer Periods (Invited Lecture)

The well-known periodicity lemma of Fine and Wilf states that if the word x of length n has periods p, q satisfying p + q - d ? n, then x has also period d, where d = gcd(p, q). Here we study the case of long periods, namely p+ q - d > n, for which we construct recursively a sequence of integers p = p1 > p2 > ... > pj-1 > 2, such that x1, up to a certain prefix of x1, has these numbers as periods. We further compute the maximum alphabet size |A| = p+ q - n of A over which a word with long periods can exist, and compute the subword complexity of x over A.