MATH 310 FALL 2003 Combinatorial Problem Solving Lecture
MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 27, Wednesday, November 5
6. 3. Partitions n Homework (MATH 310#9 W): • Read 6. 4. • Do 6. 3: all odd numberes problems • Turn in 6. 3: 2, 4, 16, 20, 22 • Volunteers: • ____________ • Problem: 16.
Partitions n n n n n A partition of a group of r identical objects divides the group into a collection of unordered subsets of various sizes. Analogously, we define a partition of the interger r to be a collection of positive integers whose sum is r. Normally we write this object as a sum ans list the integers in increasing order. 5 5 5 5 = = = = 1 1 2 1 5 +1+ +2+ +1+ +3 +4 1+1+1 1+2 2 3
The Generating Function n The generating function for partitions can be written as the infinite product n g(x) =1/[(1 – x)(1 – x 2). . . (1 – xr). . . ]
Example 1 Find the generating function for ar, the number of ways to express r as a sum of distinct integers. n Answer: n g(x) = (1+x)(1 + x 2). . . (1 + xk). . . n
Example 2 Find a generating function for ar, the number of ways that we can choose 2¢, 3¢, and 5¢ stamps adding to the net value of r cents. n Answer: 1/[(1 – x 2)(1 – x 3)(1 – x 5)] n
Example 3. n n n Show with generating functions that every positive integer can be written as a unique sum of distinct powers of 2. Answer: The generating function g*(x) = (1 + x)(1 + x 2)(1 + x 4)(1 + x 8). . . (1 – x) g*(x) = (1 – x)(1 + x 2). . . = (1 – x 2)(1 + x 4). . . = (1 – x 4)(1 + x 8). . . = (1 – x 8) (1 + x 8). . . = 1 + 0 x 2 +. . . 0 xk +. . . = 1.
Ferrers Diagram and Conjugate Partitions n Example: n 15 = 7 + 3 + 2 + 1 n n n Ferrers Diagram is shown on the left. We we transpose the diagram we obtain the conjugate partition 15 = 5 + 4 + 2 + 1 + 1+ 1
Example 4 n Show that the number of partitions of an integer r as a sum of m positive integers is equal to the number of partitions of r as a sum of integers, the largest of which is equal to m.
- Slides: 9