Comprehensive Guide to Derangements and Generating Functions

2.Derangements • A derangement of {1,2,…,n} is a permutation i1i2…in of {1,2,…,n} in which no integer is in its natural position: • i11,i22,…,inn. • We denote by Dn the number of derangements of {1,2,…,n}. • Theorem 4.15：For n1,

Proof: Let S={1,2,…,n} and X be the set of all permutations of S. Then |X|=n!. • For j=1,2,…,n, let pj be the property that in a permutation, j is in its natural position. Thus the permutation i1,i2,…,in of S has property pj provided ij=j. A permutation of S is a derangement if and only if it has none of the properties p1,p2,…,pn. • Let Aj denote the set of permutations of S with property pj ( j=1,2,…,n).

Example:(1)Determine the number of permutations of {1,2,3,4,5,6,7,8,9} in which no odd integer is in its natural position and all even integers are in their natural position. • (2) Determine the number of permutations of {1,2,3,4,5,6,7,8,9} in which four integers are in their natural position.

3. Permutations with relative forbidden position • A Permutations of {1,2,…,n} with relative forbidden position is a permutation in which none of the patterns i,i+1(i=1,2,…,n) occurs. We denote by Qn the number of the permutations of {1,2,…,n} with relative forbidden position. • Theorem 4.16：For n1, • Qn=n!-C(n-1,1)(n-1)!+C(n-1,2)(n-2)!-…+(-1)n-1 C(n-1,n-1)1!

Proof: Let S={1,2,…,n} and X be the set of all permutations of S. Then |X|=n!. • j(j+1), pj(1,2,…,n-1) • Aj: pj • Qn=Dn+Dn-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

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, …

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.

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。

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)

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+…

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

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'

4.6.2 Exponential generating functions • The number of r-combinations of multiset S={·a1,·a2,…,·ak} : C(r+k-1,r), • generating function: The number of r-permutation of set S={a1,a2,…, ak} :p(n,r), generating function:

C(n,r)=p(n,r)/r! Definition 2: The exponential generating function for the sequence a0,a1,…,an,…of real numbers is the infinite series

Theorem 4.17: Let S be the multiset {n1·a1,n2·a2,…,nk·ak} where n1,n2,…,nk are non-negative integers. Let br be the number of r-permutations of S. Then the exponential generating function g(x) for the sequence b1, b2,…, bk,… is given by • g(x)=gn1(x)·g n2(x)·…·gnk(x)，where for i=1,2,…,k, • gni(x)=1+x+x2/2!+…+xni/ni! . • (1)The coefficient of xr/r! in gn1(x)·g n2(x)·…·gnk(x) is

Example: Let S={1·a1,1·a2,…,1·ak}. Determine the number r-permutations of S. • Solution: Let pr be the number r-permutations of S, and

Example: Let S={·a1,·a2,…,·ak}，Determine the number r-permutations of S. • Solution: Let pr be the number r-permutations of S, • gri(x)=(1+x+x2/2!+…+xr/r!+…)，then • g(x)=(1+x+x2/2!+…+xr/r!+…)k=(ex)k=ekx

Example：Let S={2·x1,3·x2}，Determine the number 4-permutations of S. • Solution: Let pr be the number r-permutations of S, • g(x)=(1+x+x2/2!)(1+x+x2/2!+x3/3!) • Note: pr is coefficient of xr/r!. • Example：Let S={2·x1,3·x2,4·x3}. Determine the number of 4-permutations of S so that each of the 3 types of objects occurs even times. • Solution: Let p4 be the number 4-permutations, g(x)=(1+x2/2!)(1+x2/2!)(1+x2/2!+x4/4!)

Example: Let S={·a1,·a2, ·a3}，Determine the number of r-permutations of S so that a3 occurs even times and a2 occurs at least one time. • Solution: Let pr be the number r-permutations of S so that a3 occurs even times and a2 occurs at least one time, • g(x)=(1+x+x2/2!+…+xr/r!+…)(x+x2/2!+…+xr/r! +…) (1+x2/2!+x4/4!+…)=ex(ex-1)(ex+e-x)/2 • =(e3x-e2x+ex-1)/2

Next: • Recurrence Relations P13, P100

Exercise : • 1.Determine the number of permutations of {1,2,3,4,5,6,7,8} in which no even integer is in its natural position. • 2.Determine the number of permutations of {1,2,…,n} in which exactly k integers are in their natural positions. • 3.Eight boys are seated around a carousel. In how many ways can they change seats so that each has a different boy in front of him? • 4.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. • 5.Determine the generating function for the number an of nonnegative integral solutions of 2e1+5e2+e3+7e4=n • 6.Determine the number of n digit numbers with all digits at least 4, such that 4 and 6 each occur an even number of times, and 5 and 7 each occur at least once, there being no restriction on the digits 8 and 9.

Comprehensive Guide to Derangements and Generating Functions