MCS 256 Exam 4
Friday, May 9
Note the new date.
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Exam 4 will be a closed-book examination, but you may use one new page
of handwritten notes (in addition to your previous exam note pages).
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Exam 4 will consist of problems and questions concerning
generating functions and recurrence relations and other applications
such as enumerative
combinatorics corresponding to the material we have covered in class
and in the problem sets.
Topics: what you should know and be able to do
- Generating functions
There is a one-to-one correspondence between sequences
and generating functions.
"THE MOST POWERFUL WAY to deal with sequences of numbers,
as far as anybody knows,
is to manipulate infinite series that 'generate' those sequences."
[GKP 320]
- Ordinary generating functions
- A dictionary of ordinary generating functions
See Problem 1, Set 12, and GKP 335.
- Basic maneuvers
See class notes/GKP 334.
- Exponential generating functions
A dictionary of exponential generating functions
- ex = ...
- etx = ...
- (ex+e-x)/2 = ...
- (ex-e-x)/2 = ...
- Generating functions and enumerative combinatorics
By their nature, generating functions give an entire sequence
of counting values, one answer for each r, say.
- Use ordinary generating functions to enumerate
"selections" of r items.
- Use exponential generating functions to enumerate distributions
or ordered arrangements of r items.
- Generating functions and recurrence relations
- Given a recurrence relation for a sequence and appropriate
initial conditions, determine an algebraic equation for the
generating function of the sequence.
- In nice cases, solve for the generating function.
- Use your knowledge of generating functions
and, perhaps, partial fractions to extract
the coefficient of the nth power of the variable.
- Partial fractions: See GKP 340-341.
- Linear recurrence relations with constant coefficients
In principle, their solution can be reduced to solving
polynomial equations and systems of linear equations.
- Homogeneous recurrences
- Use the characteristic polynomial/characteristic equation
to find the general solution.
- Use the initial conditions to get specific values for the
arbitrary constants that appear in the general solution.
- Nonhomogeneous recurrences
- Find the general solution of the corresponding homogeneous
recurrence.
- Find a particular solution of the nonhomogeneous equation
by the method of undetermined coefficients.
See the document
"Finding particular solutions of nonhomogeneous recurrences."
- Find the solution of the complete recurrence equation
as the sum of the above two solutions, using
the initial conditions to get specific values for the
arbitrary constants that appear in the general solution.
- "Divide and conquer" recurrences
These commonly arise in computer science.
Changing the sequence index to a geometric sequence can convert
such a recurrence to a linear recurrence.