Advanced Counting Techniques

1
Chapter 7

• Advanced Counting Techniques

2
Definitions

• A recurrence relation for the sequence an is an
  equation that expresses an in terms of one of
  more of the previous terms of the sequence,
  namely starting at some initial integer n0 all
  the remaining terms for n ? n0 can be expressed
  using the values of one or more previous terms in
  the sequence.
• A sequence is called a solution of a recurrence
  relation if its terms satisfy the recurrence
  relation.

3
Definitions

• Initial Conditions for a recurrence relation A
  set of values for the first n0 values of the
  sequence is called a set of initial values.
• Given the values of the first n0 values of the
  sequence all the remaining terms in the sequence
  for n ? n0 can be found by using the recurrence
  relationship.

4
Applications

• Compounding Interest
• Fibonacci Numbers
• Counting bit strings of length n with certain
  properties

5
Homework

• Sec 7.1
• pg. 457 1, 3, 5, 7, 9, 13

6
Sec 7.2

• Solving Linear Recurrence Relations

7
Definitions

• A linear homogeneous recurrence relation of
  degree 2 with constant coefficients is a
  recurrence relation of the form an c1an-1
  c2an-2where c1 and c2 are constants
• The characteristic equation of this recurrence
  relation is r2 c1r c2 0
• The solutions of this equation are called
  characteristic roots.

8
Theorem I

• Let c1 and c2 be real numbers. Suppose that r2
  c1r c2 0 has two distinct roots r1 and r2.
  Then the sequence an is a solution of the
  recurrence relation an c1??n-1 c2 ?n-2 if
  and only if an ?1 r1n ?2 r2n for n ? 0,
  where ?1 and ?2 are constants.

9
Theorem II

• Let c1 and c2 be real numbers, c2 ?0. Suppose
  that r2 c1r c2 0 has only one root r0.
  Then the sequence an is a solution of the
  recurrence relation an c1??n-1 c2 ?n-2 if
  and only if an ?1 r0n ?2 nr0n for n ? 0,
  where ?1 and ?2 are constants.

10
Sec 7.5

• The Principle of Inclusion/Exclusion

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Principle of Inclusion-Exclusion

• For any two sets A and B, A?B A B
  A?B
• A1?A2?A3 A1 A2 A3- A1?A2 A1?A3
  - A2-?A3 A1?A2 ?A3
• The Generalized Principle of Inclusion-Exclusion
  Let A1, A2,, An be any collection of n sets.
  Then, A1?A2? ?An A1 A2 An-
  A1?A2 A1?A3 - - An-1?An A1?A2 ?A3
  A1?A2 ?A4 An-2?An-1 ?An -
  A1?A2?A3?A4 - A1?A2?A3?A5 - -
  An-3?An-2?An-1?An . . . (-1)n1 A1?A2
  ?A3 ? ?An

12
Homework

• Sec 7.2(pg 471) 1, 3
• Sec 7.5(pg. 504) 1, 3, 5, 7, 9, 15, 23
• End Chapter 7