1 / 10

4.6 Generating functions

4.6 Generating functions. 4.6.1 Generating functions Let S={n 1 •a 1 ,n 2 •a 2 ,…,n k •a k }, and n=n 1 +n 2 +…+n k =|S| ， then the number N of r-combinations of S equals (1)0 when r>n (2)1 when r=n (3) N=C(k+r-1,r) when n i  r for each i=1,2,…,n.

meaghan
Download Presentation

4.6 Generating functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 4.6 Generating functions • 4.6.1 Generating functions • Let S={n1•a1,n2•a2,…,nk•ak}, and n=n1+n2+…+nk=|S|，then the number N of r-combinations of S equals • (1)0 when r>n • (2)1 when r=n • (3) N=C(k+r-1,r) when nir for each i=1,2,…,n. • (4)If r<n, and there is, in general, no simple formula for the number of r-combinations of S. • A solution can be obtained by the inclusion-exclusion principle and technique of generating functions. • 6-combination a1a1a3a3a3a4

  2. xi1xi2…xik= xi1+i2+…+ik=xr • r-combination of S • Definition 1: The generating function for the sequence a0,a1,…,an,… of real numbers is the infinite series f(x)=a0+a1x+a2x2+…+anxn+…, and if only if ai=bi for all i=0,1, …n, …

  3. We can define generating function for finite sequences of real numbers by extending a finite sequences a0,a1,…,an into an infinite sequence by setting an+1=0, an+2=0, and so on. • The generating function f(x) of this infinite sequence {an} is a polynomial of degree n since no terms of the form ajxj, with j>n occur, that is f(x)=a0+a1x+a2x2+…+anxn.

  4. Example: (1)Determine the number of ways in which postage of r cents can be pasted on an envelope using 1 1-cent,1 2-cent, 1 4-cent, 1 8-cent and 1 16-cent stamps. • (2)Determine the number of ways in which postage of r cents can be pasted on an envelope using 2 1-cent, 3 2-cent and 2 5-cent stamps. • Assume that the order the stamps are pasted on does not matter. • Let ar be the number of ways in which postage of r cents. Then the generating function f(x) of this sequence {ar} is • (1)f(x)=(1+x)(1+x2)(1+x4)(1+x8)(1+x16) • (2)f(x)=(1+x+x2)(1+x2+(x2)2+(x2)3)(1+x5+(x5)2)） • =1+x+2x2+x3+2x4+2x5+3x6+3x7+2x8+2x9+2x10+3x11 +3x12+2x13+ 2x14+x15+2x16+x17+x18。

  5. Example: Use generating functions to determine the number of r-combinations of multiset S={·a1,·a2,…, ·ak }. • Solution: Let br be the number of r-combinations of multiset S. And let generating functions of {br} be f(y)， • (1+y+y2+…)k=? f(y)

  6. Example: Use generating functions to determine the number of r-combinations of multiset S={n1·a1,n2·a2,…,nk·ak}. • Solution: Let generating functions of {br} be f(y)， • f(y)=(1+y+y2+…+yn1)(1+y+y2+…+yn2)…(1+y+y2+…+ynk) • Example: Let S={·a1,·a2,…,·ak}. Determine the number of r-combinations of S so that each of the k types of objects occurs even times. • Solution: Let generating functions of {br} be f(y)， • f(y)=(1+y2+y4+…)k=1/(1-y2)k • =1+ky2+C(k+1,2)y4+…+C(k+n-1,n)y2n+…

  7. Example: Determine the number of 10-combinations of multiset S={3·a,4·b,5·c}. • Solution: Let generating functions of {ar} be f(y)， • f(y)=(1+y+y2+y3)(1+y+y2+y3+y4)(1+y+y2+y3+y4+y5) • =1+3y+6y2+10y3+14y4+17y5+18y6+17y7+14y8+10y9+6y10+3y11+y12

  8. Example: What is the number of integral solutions of the equation • x1+x2+x3=5 • which satisfy 0x1,0x2,1x3? • Let x3'=x3-1， • x1+x2+x3'=4, where 0x1,0x2,0x3'

  9. Exponential generating functions • Recurrence Relations P13, P100

  10. Exercise 1. Let S be the multiset {·e1,·e2,…, ·ek}. Determine the generating function for the sequence a0, a1, …,an, … where an is the number of n-combinations of S with the added restriction: • 1) Each ei occurs an odd number of times. • 2) the element e2 does not occur, and e1 occurs at most once. • 2. Determine the generating function for the number an of nonnegative integral solutions of 2e1+5e2+e3+7e4=n

More Related