SLIDE 1
ANALYTIC COMBINATORICS
Philippe FLAJOLET
Bologna, June 2010
1 Wednesday, June 2, 2010
Counting...
2 Wednesday, June 2, 2010
ANALYTIC COMBINATORICS Philippe FLAJOLET Bologna, June 2010 - - PowerPoint PPT Presentation
ANALYTIC COMBINATORICS Philippe FLAJOLET Bologna, June 2010 - - PowerPoint PPT Presentation
ANALYTIC COMBINATORICS Philippe FLAJOLET Bologna, June 2010 Wednesday, June 2, 2010 1 Counting... Wednesday, June 2, 2010 2 Counting (and asymptotics) Binary trees => Catalan numbers Formula is Growth rate is ( asymptotics ) Wednesday,
SLIDE 2
SLIDE 3
Counting (and asymptotics)
Binary trees => Catalan numbers Formula is Growth rate is (asymptotics)
3 Wednesday, June 2, 2010
SLIDE 4
increasing tree plane tree
Counting (and probabilities)
4 Wednesday, June 2, 2010
SLIDE 5
E.g. binary trees: 1,1,2,5,14,42,... Bijective combinatorics = first principles Generating function methods ... Algebraic methods (e.g., symmetric fns, operator)
Counting (methods)
5 Wednesday, June 2, 2010
SLIDE 6
Generating Functions (GFs)
Combinatorial class C; counting sequence (Cn): C = ⇒ C(z) =
- Cnzn
(OGF) ˆ C(z) =
- Cn
zn n! (EGF) Get GFs combinatorics algebra of special fns Look at GFs as mappings of complex plane, z ∈ C algebra of special fns complex analysis For parameters, add extra variables complex analysis perturbation theory
1 / 1
6 Wednesday, June 2, 2010
SLIDE 7
Discrete Continuous
(a digital tree aka trie of size 500) (a generating function in the complex plane)
A Calculus of Discrete Structures
7 Wednesday, June 2, 2010
SLIDE 8
FREE algo.inria.fr/flajolet
8 Wednesday, June 2, 2010
SLIDE 9
Analytic Combinatorics
- A. Combinatorial structures
- B. Analytic structures
- C. Randomness properties
for objects given by constructions
9 Wednesday, June 2, 2010
SLIDE 10
Quotations (1)
Laplace discovered the remarkable correspondence between set theoretic operations and operations on formal power series and put it to great use to solve a variety of combinatorial problems.— G.–C. ROTA La méthode des fonctions génératrices, qui a exercé ses ravages pendant un siècle, est tombée en désuétude... — Claude BERGE
10 Wednesday, June 2, 2010
SLIDE 11
Quotations (2)
Despite all appearances they [generating functions] belong to algebra and not to analysis. Combinatorialists use recurrence, generating functions, and such transformations as the Vandermonde convolution;
- thers to my horror, use contour integrals,
differential equations, and other resources of mathematical analysis. — John RIORDAN
11 Wednesday, June 2, 2010
SLIDE 12
PART I
Symbolic Methods
*1. Unlabelled structures & OGFs * 2. Labelled structures and EGFs * 3. Parameters and multivariate GFs
Embed a fragment of set theory into a language of constructions; map to algebra(s) of special functions.
12 Wednesday, June 2, 2010
SLIDE 13
Chapter I
Unlabelled structures and OGFs
13 Wednesday, June 2, 2010
SLIDE 14
Embed a fragment of set theory into a language of constructions; map combinatorics to algebra(s) of special functions.
Symbolic Methods
14 Wednesday, June 2, 2010
SLIDE 15
15 Wednesday, June 2, 2010
SLIDE 16
16 Wednesday, June 2, 2010
SLIDE 17
17 Wednesday, June 2, 2010
SLIDE 18
18 Wednesday, June 2, 2010
SLIDE 19
Outline
19 Wednesday, June 2, 2010
SLIDE 20
20 Wednesday, June 2, 2010
SLIDE 21
21 Wednesday, June 2, 2010
SLIDE 22
22 Wednesday, June 2, 2010
SLIDE 23
Summary:
23 Wednesday, June 2, 2010
SLIDE 24
24 Wednesday, June 2, 2010
SLIDE 25
Roots...
A modicum of Pólya theory (1937) Schützenberger: languages and GFs (~1960) Rota-Stanley = MIT School (1970s) Goulden-Jackson = constructions (~1980) Joyal’s theory of species +BLL (1980s)
25 Wednesday, June 2, 2010
SLIDE 26
26 Wednesday, June 2, 2010
SLIDE 27
Catalan numbers again!
27 Wednesday, June 2, 2010
SLIDE 28
28 Wednesday, June 2, 2010
SLIDE 29
29 Wednesday, June 2, 2010
SLIDE 30
30 Wednesday, June 2, 2010
SLIDE 31
31 Wednesday, June 2, 2010
SLIDE 32
A variety of classes => a variety of “special functions”
32 Wednesday, June 2, 2010
SLIDE 33
Algebraic functions (1)
Arise from specifications (CF grammars), with +, x, Seq Elimination: system -> single equation P(x,y)=0 Coefficients are “combinatorial sums”
[e.g., Sokal, SLC 2009]
33 Wednesday, June 2, 2010
SLIDE 34
MAPS: Tutte’s quadratic method;
cf Cori, Bousquet-Mélou et al., Bordeaux School...
EXCURSIONS: the kernel method;
cf Lalley 1993, Banderier-F 2001, MBM
Algebraic functions (2)
34 Wednesday, June 2, 2010
SLIDE 35
Chapter 2
Labelled structures and EGFs
35 Wednesday, June 2, 2010
SLIDE 36
36 Wednesday, June 2, 2010
SLIDE 37
37 Wednesday, June 2, 2010
SLIDE 38
38 Wednesday, June 2, 2010
SLIDE 39
39 Wednesday, June 2, 2010
SLIDE 40
40 Wednesday, June 2, 2010
SLIDE 41
41 Wednesday, June 2, 2010
SLIDE 42
42 Wednesday, June 2, 2010
SLIDE 43
(End of proof of Theorem)
43 Wednesday, June 2, 2010
SLIDE 44
44 Wednesday, June 2, 2010
SLIDE 45
45 Wednesday, June 2, 2010
SLIDE 46
46 Wednesday, June 2, 2010
SLIDE 47
47 Wednesday, June 2, 2010
SLIDE 48
48 Wednesday, June 2, 2010
SLIDE 49
49 Wednesday, June 2, 2010
SLIDE 50
50 Wednesday, June 2, 2010
SLIDE 51
labelled
51 Wednesday, June 2, 2010
SLIDE 52
runs in perms
Chapter 3. Parameters and Multivariate GFs
52 Wednesday, June 2, 2010
SLIDE 53
53 Wednesday, June 2, 2010
SLIDE 54
54 Wednesday, June 2, 2010
SLIDE 55
55 Wednesday, June 2, 2010
SLIDE 56
PRINCIPLE: Add variables marking parameters at appropriate places and recycle:
56 Wednesday, June 2, 2010
SLIDE 57
Conclusions (Part I)
[Chapter 3]: Multivariate GFs give access to parameters; those that can be
- btained by “marking” in combinatorial
constructions. [Chapters 1-2-3]: Exploit all this asymptotically? counting; mean, variance, distribution?
57 Wednesday, June 2, 2010