CMSC 603 Class Notes, 3/12/99

Problem

We are given a 3 level deep, complete binary tree. We have to assign one of 3 types of processors to each node of the tree. In how many ways can this be done?

Solution

The generator for the permutation group for the tree is G = <f,L,R>,
where f = flip, L = flip left subtree and R = flip right subtree

(Assume tree nodes are labeled from A to G in inorder manner)

The group elements in cycle notation are:
e
(D)(BF)(AE)(CG)
(D)(B)(F)(AC)(EG)
(D)(B)(F)(A)(C)(EG)
(D)(BF)(AG)(CE)
(D)(BF)(AGCE)
(D)(BF)(AECG)

Let X = Set of all possible essentially different processor assignments for the nodes

|X| = 1/|G| x Sum of the sizes of all fixes of G.
= 513

Generating Functions

Given a sequence S = (c₁,c₂,c₃, ....)
p(x) = Sum from i=0 to infinity of c_i.x_i

p(x) is the generating function for the sequence S

Cycle Index

Multivariate polynomial representation of the cycle notation for permutation groups. Terms are of the form x_i^e_i , where e_i = number of cycles of size i

For example, consider the permutation group in the problem above. The cycle index for this group would be.

I = 1/|G|( x₁⁷ + 2.x₁^5.x₂ + x₁^3.x₂² + 2.x₁.x₂³ + 2.x₁.x₂.x₄)

The number of k colorings for this group can now be easily computed using the cycle index.

# of k colorings of group G = I_G(k).

Polya's Theorem

Biggs, pg 453, Theorem 20.4
Biggs, pg 457, Theorem 20.5
Biggs, pg 459, Theorem 20.6 - Polya's theorem, general case of the k-coloring problem

Solution to Assignment 6, Problem 2

(Number of solutions of an equation in 4 variables, given upper bounds on the variable)

Let S = the set of good solutions

x' = (x₁, x₂, x₃, x₄)

V(x') = x₁ + x₂ + x₃+ x₄

S* = { x': V(x') = n}

|S*| = (n-1) choose 3

S_j = {x': x_j > k}

Good solutions = Total solutions - bad solutions

S = S* - (S₁ U S₂ U S₃ U S₄)

| S₁ U S₂ U S₃ U S₄| = Sum(from i=1 to 4) (-1)ⁱ⁺¹. (4 choose i) | Intersection of S₁ through S_i|

| Intersection of S₁ through S_i| = (n-4-k_i+3) choose (n-4-k_i) = (n-4-k_i) choose 3.