Summations (portions from Concrete Mathematics by Graham...

Summations (portions fromConcrete Mathematicsby Graham,

Knuth, Patashnik)CSE 20— Nov. 1, 2005, Day 12

Neil Rhodes

UC San Diego

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.1

FunctionsGiven f : A → B:

Inverse Image of f

f−1(b) = {a : a ∈ A and f (a) = b}

Coimage is the partition ofA where all elements in ablock map to the same element ofB.

Coimage(f ) = { f−1(b) : b ∈ Image(f )}

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.2

Number of PartitionsGivenS = {a1,a2, . . . ,an}:

• How many partitions ofS are there?• How many partitions of sizek are there?

• If an is in a new block:• If an is in an existing block:

• S(n,k) or {nk} is theStirling number of the second

kind

S(n,k) =

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.3

Calculating S(n,k)1 2 3 4 5

12345

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.4

Number of FunctionsGiven setsA andB, how many functions are there ofthe form f : A → B where|Image(f )| = k?

• How many partitions ofA are there of sizek?• How many images are there of sizek?• How many ways to map the blocks of

Coimage(f ) to Image(f )?

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.5

Summing a Sequence• Specific sum

1+2+3+ · · ·+n−1+n

• General summation of a sequence ofterms:

a1+a2+ · · ·+an

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.6

Sigma Notation• Three-dots notation is vague and wordy.• Sigma notation is more compact:

n

∑k=1

k

• Parts of the notation• Summand• Index variable• Lower limit• Upper limit

• Sigma notation inline:∑nk=1k

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.7

Generalized Sigma Notation• Specify a condition that the index variable must

satisfy

∑1≤k≤n

k

• Compare:

∑1≤k≤100

k odd

k2

to:

49

∑k=0

(2k +1)2

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.8

Changing the Index Variable• Can always change from one variable to another

∑1≤k≤n

ak =

• What if we want to switch fromk to k +1?Compare:

∑1≤k≤n

ak =

to:

n

∑k=1

ak =

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.9

Zero Terms are OKWhich is better?

n−1

∑i=2

k(k−1)(n− k)

or

n

∑i=0

k(k−1)(n− k)

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.10

Using Iverson Notation• Iverson notation. True-or-false statement

enclosed in brackets. Value is 0 or 1

[p odd] =

• Example

∑1≤k≤100

k odd

k2 = ∑k

k2[1≤ k ≤ 100][k odd]

• 0 in Iverson notation isvery strongly zero

∑k

1k[k > 0]

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.11

Products• Pi notation is similar to Sigma notation

n

∏k=1

k = 1×2×3×·· ·× (n−1)×n

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.12

Working with SummationsDistributive law

∑k∈K

cak = c ∑k∈K

ak

Associative law

∑k∈K

ak +bk = ∑k∈K

ak + ∑k∈K

bk

Commutative law

∑k∈K

ak = ∑p(k)∈K

ak

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.13

Closed Forms for SummationsClosed form is a formula for a summation, with thesummation removed.

∑0≤k≤n

k =n(n+1)

2

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.14

Example

∑0≤k≤n

(a+bk) =

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.15

Perturbation MethodGivenSn = ∑0≤k≤n ak, rewriteSn+1 by splitting offfirst and last term:

Sn +an+1 = a0+ ∑1≤k≤n+1

ak

= a0+ ∑1≤k+1≤n+1

ak+1

= a0+ ∑0≤k≤n

ak+1

Then, work on last sum and express in terms ofSn.

Finally, solve forSn.

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.16

Perturbation Method Example

Sn = ∑0≤k≤n

xk

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.17

Standard Closed FormsArithmetic series

n

∑k=0

k =12

n(n+1)

Sum of squares and cubes

n

∑k=0

k2 =n(n+1)(2n+1)

6n

∑k=0

k3 =n2(n+1)2

4

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.18

Standard Closed FormsGeometric series (x 6= 1)

n

∑k=0

xk =xn+1−1

x−1

Infinite Geometric series (|x| < 1)

∞

∑k=0

xk =1

1− x

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.19

Standard Closed FormsHarmonic series

Hn =n

∑k=0

1k≈ lnn

n 0 1 2 3 4 5 6Hn 0 1 3

2116

2512

13760

4920

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.20

Application of Harmonic SeriesGiven a set of 52 playing cards (of length 2 units). Can

you stack them overlapping so the top card completely

overhangs the table?

Summations (portions from Concrete Mathematics by Graham, Knuth, Patashnik) – p.21