Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings

Félix Balado, Guénolé C. M. Silvestre

Abstract

We give recurrences, generating functions and explicit exact expressions for the enumeration of fundamental quantities involving runs in binary strings. We first focus on enumerations concerning runs of ones, and we then analyse the same enumerations when runs of ones and runs of zeros are jointly considered. We give the connections between these two types of run enumeration, and with the problem of compositions. We also analyse the same enumerations with a Hamming weight constraint. We discuss which of the many number sequences that emerge from these problems are already known and listed in the OEIS. Additionally, we extend our main enumerative results to the probabilistic scenario in which binary strings are outcomes of independent and identically distributed Bernoulli variables.

Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings

Abstract

We give recurrences, generating functions and explicit exact expressions for the enumeration of fundamental quantities involving runs in binary strings. We first focus on enumerations concerning runs of ones, and we then analyse the same enumerations when runs of ones and runs of zeros are jointly considered. We give the connections between these two types of run enumeration, and with the problem of compositions. We also analyse the same enumerations with a Hamming weight constraint. We discuss which of the many number sequences that emerge from these problems are already known and listed in the OEIS. Additionally, we extend our main enumerative results to the probabilistic scenario in which binary strings are outcomes of independent and identically distributed Bernoulli variables.
Paper Structure (151 sections, 2 theorems, 434 equations, 1 figure)

This paper contains 151 sections, 2 theorems, 434 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the Paper
  3. Definitions
  4. Notation
  5. Number of $n$-Strings that Contain Prescribed Quantities of Runs Under Different Constraints
  6. Number of $n$-Strings that Contain Exactly $m$ ($\ushort{k}\le { \macc@depth1 \frozen@everymath{\mathgroup\macc@group} \macc@set@skewchar \macc@nested@a111{} }$)-Runs
  7. Recurrences
  8. Generating Functions
  9. Explicit Expressions
  10. Number of $n$-Strings that Contain Exactly $m$ $k$-Runs
  11. Recurrences
  12. Generating Functions
  13. Explicit Expression
  14. OEIS
  15. Number of $n$-Strings that Contain Exactly $m$ ($\ge\!k$)-Runs
  16. ...and 136 more sections

Key Result

Theorem 5.1

Proof. Substitute eq:ogf_w_n_m_klek into eq:ogfs_chave_das_nozes. $\square$

Figures (1)

  • Figure 1: Partitioning of an $n$-string into nonempty substrings versus runs.

Theorems & Definitions (51)

  • Definition 1.1: Run, Length of a Run
  • Definition 1.2: $n$-String
  • Definition 1.3: $k$-Run
  • Definition 1.4: Null Run, Nonnull Run
  • remark 1
  • Definition 1.5: ($\ushort{k}\le { \macc@depth1 \frozen@everymath{\mathgroup\macc@group} \macc@set@skewchar \macc@nested@a111{} }$)-Run, ($\ge\! k$)-Run, ($\le\!k$)-Run
  • Definition 1.6: Odd and Even Runs, $p$-Parity Runs
  • remark 2
  • remark 3
  • remark 4
  • ...and 41 more