Related papers: Runs of Ones in Binary Strings

We present some exact expressions for the number of paths of a given length in a perfect $m$-ary tree. We first count the paths in perfect rooted $m$-ary trees and then use the results to determine the number of paths in perfect unrooted…

A run is an inclusion maximal occurrence in a string (as a subinterval) of a repetition $v$ with a period $p$ such that $2p \le |v|$. The exponent of a run is defined as $|v|/p$ and is $\ge 2$. We show new bounds on the maximal sum of…

Let R(n,k) denote the number of permutations of {1,2,...,n} with k alternating runs. We find a grammatical description of the numbers R(n,k) and then present several convolution formulas involving the generating function for the numbers…

We determine the distributions of lengths of runs in random sequences of elements from a totally ordered set (total order) or partially ordered set (partial order). In particular, we produce novel formulae for the expected value, variance,…

We find the generating function for $C(n,k,r)$, the number of compositions of $n$ into $k$ positive parts all of whose runs (contiguous blocks of constant parts) have lengths less than $r$, using recent generalizations of the method of…

We generalize the notion of a de Bruijn sequence to a "multi de Bruijn sequence": a cyclic or linear sequence that contains every k-mer over an alphabet of size q exactly m times. For example, over the binary alphabet {0,1}, the cyclic…

Recently, Byrnes presented a formula for the maximum length of a symmetric circuit code that has a long bit run and odd spread. Here we show that the formula is also valid when the spread is even. We also establish that all maximum length…

We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions ranging from the earliest attempts…

Regularities in strings are often related to periods and covers, which have extensively been studied, and algorithms for their efficient computation have broad application. In this paper we concentrate on computing cyclic regularities of…

Enumerating the number of times one word occurs in another is a much-studied combinatorial subject. By utilizing a method that we call ``lexicographic extreme referencing'', we provide a formula for computing occurrences of one binary word…

We study the distributions of waiting times in variations of the $q$-sooner and later waiting time problem. One variation imposes length and frequency quotas on the runs of successes and failures. Another case considers binary trials for…

We briefly review superstring theories, highlighting the important concepts, developments, and open problems of the subject.

This note provides very simple, efficient algorithms for computing the number of distinct longest common subsequences of two input strings and for computing the number of LCS embeddings.

We consider the problem of binary string reconstruction from the multiset of its substring compositions, i.e., referred to as the substring composition multiset, first introduced and studied by Acharya et al. We introduce a new algorithm…

We give a complete classification of binary linear complementary dual codes of lengths up to $13$ and ternary linear complementary dual codes of lengths up to $10$.

We study the problem of cutting a length-$n$ string of positive real numbers into $k$ pieces so that every piece has sum at least $b$. The problem can also be phrased as transforming such a string into a new one by merging adjacent numbers.…

Motivated by applications in polymer-based data storage, we study the problem of reconstructing a string from part of its composition multiset. We give a full description of the structure of the strings that cannot be uniquely reconstructed…

In responding to a question on Math Stackexchange, the author formulated the problem of determining the number of strings of balls colored in most $n$ colors with a number $k$ of repeated colors. In this paper, we formulate the problem more…

We survey various classical results on invariants of polynomials, or equivalently, of binary forms, focussing on explicit calculations for invariants of polynomials of degrees 2, 3, 4.

We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…