Discrete Mathematics - Vol 2 | 16. Valid Sequences Analysis by Abraham | Learn Smarter
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16. Valid Sequences Analysis

The chapter provides insights into the analysis of recurrence relations through various examples, highlighting methodologies to derive recurrence equations for different combinatorial problems. It covers a range of scenarios from bit strings to set functions, emphasizing the establishment of clear categories for effective counting and formulation. Additionally, it concludes with exercises and activities aimed at reinforcing the concepts presented.

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Sections

  • 16.1

    Valid Sequences Analysis

    This section discusses the analysis of valid sequences characterized by their strictly increasing nature and the derivation of recurrence relations for counting them.

    Learning Practice

  • 16.1.1

    Recurrence Condition

    This section introduces the concept of recurrence conditions in sequences, detailing how to determine the number of valid strictly increasing sequences.

    Learning Practice

  • 16.1.2

    Compact Recurrence Condition

    This section introduces a compact recurrence condition for strictly increasing sequences of numbers, outlining distinct categories of sequences based on constraints.

    Learning Practice

  • 16.1.3

    Categories Of Sequences

    This section discusses the recurrence conditions for strictly increasing sequences, focusing on categories based on the second last term of sequences, which maximize the exploration of valid sequence structures.

    Learning Practice

  • 16.1.4

    Degree Of Recurrence Equation

    This section explores the concept of recurrence equations, specifically focusing on deriving and compacting recurrence relations.

    Learning Practice

  • 16.1.5

    Initial Conditions

    This section introduces recurrence relations for counting valid sequences and bit strings, discussing initial conditions and various categories within those sequences.

    Learning Practice

  • 16.2

    Bit Strings With Substring '000'

    This section discusses the recurrence relations for counting bit strings of length n containing the substring '000' by considering the complementary case of strings not containing '000'.

    Learning Practice

  • 16.2.1

    Counting Bad Strings

    This section discusses how to count sequences of binary strings while considering specific patterns ('bad' strings that contain substring '000').

    Learning Practice

  • 16.2.2

    Categories Of Bad Strings

    This section discusses the categorization and recurrence relations of bad strings, particularly those that do not contain the substring '000'.

    Learning Practice

  • 16.2.3

    Recurrence Equation For Bad Strings

    This section explores the concept of recurrence relations in the context of counting bad strings that do not contain the substring '000'.

    Learning Practice

  • 16.2.4

    Initial Conditions For Bad Strings

    This section introduces the concept of bad strings and their recurrence relations, focusing on calculating the number of bit strings that contain specific substrings.

    Learning Practice

  • 16.3

    Bit Strings With Substring '01'

    This section discusses the recurrence relations for counting bit strings that contain the substring '01' based on their structure and length.

    Learning Practice

  • 16.3.1

    Recurrence Condition

    This section introduces the concept of recurrence relations in sequences, specifically focusing on valid strictly increasing sequences and the derivation of their recurrence conditions.

    Learning Practice

  • 16.3.2

    Categories Of Strings

    This section explores the categories of strictly increasing sequences of numbers represented as strings, detailing their recurrence relations.

    Learning Practice

  • 16.4

    Ternary Strings With Occurrences

    The section discusses recurrence relations governing ternary strings that must contain specific occurrences, focusing on valid sequences and their formulations.

    Learning Practice

  • 16.4.1

    Recurrence Condition

    The recurrence condition explores sequences of numbers in a strictly increasing order, establishing a relationship between valid sequences and their predecessors.

    Learning Practice

  • 16.4.2

    Categories Of Ternary Strings

    This section discusses the structure and recurrence relationships of ternary strings, specifically focusing on valid sequences comprising increasing numbers.

    Learning Practice

  • 16.5

    Onto Functions

    This section explores the concept of onto functions and provides recurrence relations for counting such functions between two sets.

    Learning Practice

  • 16.5.1

    Recurrence Condition Derivation

    This section discusses the derivation of recurrence relations for counting valid sequences in strictly increasing order and introduces concepts related to bitstrings and partitions.

    Learning Practice

  • 16.5.2

    Non-Onto Functions

    This section explores the recurrence relations for counting valid sequences and the structures of non-onto functions, emphasizing their significance in combinatorial mathematics.

    Learning Practice

  • 16.6

    Stirling Functions

    This section discusses Stirling functions, focusing on how to compute the number of valid strictly increasing sequences and presents a recurrence relation for these sequences.

    Learning Practice

  • 16.6.1

    Recurrence Condition

    The section discusses recurrence conditions for sequences, emphasizing the concept of strictly increasing sequences and deriving efficient recurrence relations.

    Learning Practice

  • 16.6.2

    Categories Of Partitions

    This section explores the concept of strictly increasing sequences within partitions and the recurrence relations that define their structure.

    Learning Practice

  • 16.7

    Combinatorial Proof Of Identity

    This section discusses a combinatorial proof of identity involving sequences and recurrence relations.

    Learning Practice

  • 16.7.1

    Lhs Expression Explanation

    This section elaborates on a recurrence relation for counting valid strictly increasing sequences ending with a specific term.

    Learning Practice

  • 16.7.2

    Rhs Expression Explanation

    This section presents a recurrence relation for valid strictly increasing sequences and the derivation of a more compact recurrence condition.

    Learning Practice

References

ch38.pdf

Class Notes

Memorization

What we have learnt

  • Recurrence relations can ef...
  • Understanding the categorie...
  • Establishing initial condit...

Final Test

Revision Tests