Plan for second half of the course Lectures from Analytic - - PowerPoint PPT Presentation

Plan for second half of the course

1

More creative exercises

• May require code.
• (Slightly) more emphasis on Q&As.

Lectures from Analytic Combinatorics

• One or two lectures, posted weekly.
• Emphasis on lectures, with reference to the book.

AC basics Q&A 1

2

• Q. Match each description on the left to 0 or 1 of the symbolic specifications on the right

(ODD denotes the set of odd numbers).

SEQODD(SEQ(Z )) SEQ(SEQODD(Z )) SEQ(Z ) SEQM (Z ) MSET(Z ) MSETODD(Z ) PSET(Z ) MSET(SEQODD(Z )) MSETODD(SEQ(Z ))

compositions compositions into M parts compositions into odd parts compositions into an odd number of parts compositions into distinct parts partitions partitions into M parts partitions into odd parts partitions into an odd number of parts partitions into distinct parts

AC Basics Q&A 2: cyclic bitstrings

3

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

C1 = 2 C2 = 3 C3 = 4 C4 = 6

• Def. A cyclic bitstring is a cyclic sequence of bits

1 1

C5 = 8

• Q. How many N-bit cyclic bitstrings ?

Q&A example: cyclic bitstrings

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• Q. How many N-bit cyclic bitstrings ?

One possibility

• Solution is “easy”.
• Create an exam question with appropriate hints.

Another possibility

• Solution is “difficult” or “complicated”.
• Figure out a way to simplify.
• Or, think about a different problem.

1 1 1 1 1 1 1 1 1 1 1 1

C4 = 6

Third possibility

• Problem you thought of is a “classic”.
• Use OEIS.

1 1 1 1 1

C3 = 4

1

page 18

page 64

k 1 2 3 4 5 6 7 8 9

𝜒(k) 1 1 2 2 4 2 6 4 6

ln 1 1 − 2z = 2z 1 + (2z)2 2 + (2z)3 3 + (2z)4 4 + (2z)5 5 + . . . ln 1 1 − 2z2 = 2z2 1 + (2z2)2 2 + . . . ln 1 1 − 2z3 = 2z3 1 + (2z3)2 2 + . . . ln 1 1 − 2z4 = 2z4 1 + . . . ln 1 1 − 2z5 = 2z5 1 + . . .

AC Basics Q&A 2

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• Q. How many cyclic bitstrings of length 10 ?

construction OGF

N(z) =

[z10]N(z) =

result

AC Basics Q&A 3

10

• Q. How many unimodal permutations of length n ?

constructions EGFs exact result

I = increasing perms D = decreasing perms U = unimodal perms

definitions

I = SET(Z)

D =

I(z) = ez U(z) =

D(z) =

U =

Un =

AC Basics Q&A 4

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• Q. Mappings with every character appearing 2 or 0 times.

constructions EGF equations EGFs ~-approximation

Y = CY C(Z ? C)

M = SET(Y )

C(z) = z + 1 2zC(z)2

C = Z + Z ? SET2(C)

Y (z) = ln 1 1 − zC(z) M(z) = 1 1 − zC(z)

C(z) = 1 − √ 1 − 2z2 z

M(z) = 1 √ 1 − 2z2

[zn] f(z) (1 − z/ρ)α ∼ f(ρ) Γ(α)ρ−nnα−1

Γ 1 2

• = √π

n! ∼ √ 2πn ⇣n e ⌘n (2n)![z2n]M(z) ∼ (2n)! 2n √πn

∼ 2 ⇣2n e ⌘2n