Talk by Daniel Zwillinger
5 April 2001
This file contains:
(1) a list of slide titles (hyperlinked), and
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Balls into cells
Example: Switching functions (I)
Example: Switching functions (II)
Example: Radar filtering (I)
Example: Radar filtering (II)
Example: Radar filtering (III)
Example: Radar filtering (IV)
Counting transition diagrams
Permutation groups: Fast review
Permutation groups: Example
Example: 4-bead necklaces
Example: Necklaces with colored beads (I)
Example: Necklaces with colored beads (II)
Example: Rotations of an equilateral triangle
Example: Rotations of an equilateral triangle with colored beads (I)
Example: Rotations of an equilateral triangle with colored beads (II)
Special case of Polya's Theorem
Example: Necklaces with colored beads (III)
Example: Necklaces with colored beads (IV)
Example: Rotations of an equilateral triangle with colored beads (III)
Permutation groups: Cycle index
Example: Necklaces with colored beads (V a)
Example: Necklaces with colored beads (V b)
Example: Necklaces with colored beads (V c)
Example: Necklaces with colored beads (V d)
Example: Necklaces with colored beads (V e)
Example: Rotations of an equilateral triangle with colored beads (IV)
(With a radar application motivation)
Daniel Zwillinger, PhD
5 April 2001
This lecture will show how Polya theory can be used in counting
objects, which is often the design basis for statistical tests.
Specifically, Polya theory determines the number of distinct
equivalence classes of objects. It can also give counts for specific
types of patterns within equivalence classes. While sounding
esoteric, this simple-to-apply technique easily counts non-isomorphic
(i.e., different) graphs, and many other combinatorial structures. An
application to radar filtering of hard-limited data will be used to
motivate the topic.
- Who am I? Why am I talking?
- Talk is about combinatorial counting method
- Problem motivation
- Switching functions
- Radar filtering
using 3 beads of 3 different colors,
how many different necklaces are there?
- Polya theory
- Permutation groups
- Burnside's Lemma
- Education: MIT, Caltech
- Joined Raytheon in 2001 (Sr Principal Engineer)
- Formerly at: RPI,
JPL, BBN, IDA, MITRE,
- Broad expertise in applied mathematics
(continuous and discrete)
- Books on:
table of integrals
- Editor-in-chief of edition of CRC's
Standard Mathematical Tables and Formulae
- Home page: http://www.mathtable.com/zwillinger
Choosing a sample of size r from m objects
- Combinatorial optimization
- How to define a problem?
- How many solutions are there?
- How to identify the solutions?
- How to find the best solution?
- Today's focus is on the number of solutions
- Standard counting problems (next 2 slides)
- Generating functions
- Standard problems (assignment problem, etc.)
- Polya theory
||The sample is called an
||Number of ways to choose the sample
||r-combination with replacement
||r-permutation with replacement
|Distinguish the cells
||Distinguish the balls
||Can cells be empty
||Number of ways to place n balls into k cells
- is a Stirling cycle number
- pk(n) is the number of partitions of n into k parts
- A switching function (Boolean function) assigns a 0 or 1 to each
bit string of length n
- There are 22n different switching functions
(For n=4 have 65,536 switching functions)
- Allowing symmetries, there are
only 3,984 different switching functions (see )!
- Allowing an inverter and symmetries, there are
only 222 different switching functions (see )!
- Polya theory easily gives the number 222.
- D be a ``detection'', N is a ``no detection''
- R is a ``report'', ``-'' is ``no report''
- Sample filter result for ``2 out of 3'' filter
- ``2 out of 3'' filter has 4 states:
- State ``W'', last two input elements are ``NN''
- State ``X'', last two input elements are ``ND''
- State ``Y'', last two input elements are ``DN''
- State ``Z'', last two input elements are ``DD''
Transition diagram is
``N'' is shown as a solid line
``D'' is shown as a dashed line
- For iid inputs (D has a probability of p) have transition matrix
- Usual eigen-analysis yields steady state distribution:
Need to know the number of directed graphs with out-degree one for different
numbers of vertices.
- Find (using Polya theory)
|number of states S
|naive counting S2S
|number of actual filters
Use Sloane  to identify sequence A054050:
10, 129, 2836, 83061, 3076386,
136647824, ...:``Number of non-isomorphic binary n-state automata''
- Consider permutations of
Operation: means: permute by then by
- can be closed to form a group G
- ``a'' is invariant if ; ``'' is number of elements invariant under
- Write as unique product of disjoint cycles;
``'' is number of cycles in
- An equivalence relation S (on G) is defined by ``''
iff there exists a such that
Then the number of equivalence classes in S is
- Burnside's lemma: equivalence classes
- The 2 equivalence classes are and
- , : Burnside
- , : Burnside
- The 6 equivalence classes are:
- Note: 4 classes of 1 element, 2 classes of 2 elements
- , ,
- Burnside lemma: equivalence class
- The single equivalence class is
- Three different colored beads
(Black, White, Gray) on an equilateral triangle. Allow rotations.
Then the number of distinct colorings in S(D,C) is
Number of equivalence classes for various k and m
Let color c have ``weight'' w(c). The pattern inventory describes
the colorings, of a certain kind, as functions of the weights.
Then the pattern inventory of colorings in S(D,C) is
Note: for w(c)=1 have
If m=2 colors (Black, White) available:
If m=2 colors (B, W) are available, then
If m=3 colors (Black, White, Shade) available:
- For k=2 have 6 distinct colorings:
- For k=3 have 18 distinct colorings:
- Many applications of Polya theory:
- bracelets (rotations of an n-gon)
For n=6 find
- photomasks for chips
- molecular structures
- non-isomorphic graphs
- seating arrangements
- Polya theory is more advanced than shown here
- Daniel Zwillinger can help in formulating and solving discrete
and continuous mathematical problems
(Sudbury 1-1-565, telephone 978-440-1660)
For pictures of colored necklaces, bracelets, unlabeled necklaces,
(be careful of terms!)
Harvard Computation Laboratory Staff,
Synthesis of electronic computing and control circuits,
Harvard University Press, Cambridge, MA, 1951.
Fred S. Roberts,
Sloane's on-line encyclopedia:
F. Harary and E. Palmer,
``Number of non-isomorphic binary n-state automata'',
Graphical Enumeration, 1973.
CRC Standard Mathematical Tables and Formulae,
Boca Raton, FL,