summations (portions from concrete mathematics by graham, … · 2005. 10. 31. · concrete...

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 )}


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 secondkind

S(n,k) =


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

12345


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 )?


Summing a Sequence• Specific sum

1+2+3+ · · ·+n−1+n

• General summation of a sequence ofterms:

a1+a2+ · · ·+an


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


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)2Summations (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 =


Zero Terms are OKWhich is better?

n−1

∑i=2

k(k−1)(n− k)

or

n

∑i=0

k(k−1)(n− k)


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]


Products• Pi notation is similar to Sigma notation

n

∏k=1

k = 1×2×3×·· ·× (n−1)×n


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


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

∑0≤k≤n

k =n(n+1)

2


Example

∑0≤k≤n

(a+bk) =


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.


Perturbation Method Example

Sn = ∑0≤k≤n

xk


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


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


Standard Closed FormsHarmonic series

Hn =n

∑k=0

1k≈ lnn

n 0 1 2 3 4 5 6Hn 0 1 32

116

2512

13760

4920


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?


FunctionsNumber of PartitionsCalculating $S(n, k$)Number of FunctionsSumming a SequenceSigma NotationGeneralized Sigma NotationChanging the Index VariableZero Terms are OKUsing Iverson NotationProductsWorking with SummationsClosed Forms for SummationsExamplePerturbation MethodPerturbation Method ExampleStandard Closed FormsStandard Closed FormsStandard Closed FormsApplication of Harmonic Series

summations (portions from concrete mathematics by graham, … · 2005. 10. 31. · concrete...

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