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ANALYTIC COMBINATORICS Philippe FLAJOLET Bologna, June 2010 - PowerPoint PPT Presentation

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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,

  1. ANALYTIC COMBINATORICS Philippe FLAJOLET Bologna, June 2010 Wednesday, June 2, 2010 1

  2. Counting... Wednesday, June 2, 2010 2

  3. Counting (and asymptotics) Binary trees => Catalan numbers Formula is Growth rate is ( asymptotics ) Wednesday, June 2, 2010 3

  4. Counting (and probabilities) plane tree increasing tree Wednesday, June 2, 2010 4

  5. Counting (methods) E.g. binary trees: 1,1,2,5,14,42,... Bijective combinatorics = first principles Generating function methods ... Algebraic methods (e.g., symmetric fns, operator) Wednesday, June 2, 2010 5

  6. Generating Functions (GFs) Combinatorial class C ; counting sequence ( C n ):  � C ( z ) = C n z n (OGF)  = C ⇒ z n ˆ � C ( z ) = (EGF) C n  n ! 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 Wednesday, June 2, 2010 6

  7. A Calculus of Discrete Structures Discrete Continuous (a digital tree aka trie of size 500) (a generating function in the complex plane) Wednesday, June 2, 2010 7

  8. FREE algo.inria.fr/flajolet Wednesday, June 2, 2010 8

  9. Analytic Combinatorics A . Combinatorial structures B . Analytic structures C . Randomness properties for objects given by constructions Wednesday, June 2, 2010 9

  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 Wednesday, June 2, 2010 10

  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; others to my horror, use contour integrals, differential equations, and other resources of mathematical analysis. — John RIORDAN Wednesday, June 2, 2010 11

  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. Wednesday, June 2, 2010 12

  13. Chapter I Unlabelled structures and OGFs Wednesday, June 2, 2010 13

  14. Symbolic Methods Embed a fragment of set theory into a language of constructions; map combinatorics to algebra(s) of special functions. Wednesday, June 2, 2010 14

  15. Wednesday, June 2, 2010 15

  16. Wednesday, June 2, 2010 16

  17. Wednesday, June 2, 2010 17

  18. Wednesday, June 2, 2010 18

  19. Outline Wednesday, June 2, 2010 19

  20. Wednesday, June 2, 2010 20

  21. Wednesday, June 2, 2010 21

  22. Wednesday, June 2, 2010 22

  23. Summary: Wednesday, June 2, 2010 23

  24. Wednesday, June 2, 2010 24

  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) Wednesday, June 2, 2010 25

  26. Wednesday, June 2, 2010 26

  27. Catalan numbers again! Wednesday, June 2, 2010 27

  28. Wednesday, June 2, 2010 28

  29. Wednesday, June 2, 2010 29

  30. Wednesday, June 2, 2010 30

  31. Wednesday, June 2, 2010 31

  32. A variety of classes => a variety of “special functions” Wednesday, June 2, 2010 32

  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] Wednesday, June 2, 2010 33

  34. Algebraic functions (2) MAPS : Tutte’s quadratic method; cf Cori, Bousquet-Mélou et al., Bordeaux School... EXCURSIONS: the kernel method; cf Lalley 1993, Banderier-F 2001, MBM Wednesday, June 2, 2010 34

  35. Chapter 2 Labelled structures and EGFs Wednesday, June 2, 2010 35

  36. Wednesday, June 2, 2010 36

  37. Wednesday, June 2, 2010 37

  38. Wednesday, June 2, 2010 38

  39. Wednesday, June 2, 2010 39

  40. Wednesday, June 2, 2010 40

  41. Wednesday, June 2, 2010 41

  42. Wednesday, June 2, 2010 42

  43. (End of proof of Theorem) Wednesday, June 2, 2010 43

  44. Wednesday, June 2, 2010 44

  45. Wednesday, June 2, 2010 45

  46. Wednesday, June 2, 2010 46

  47. Wednesday, June 2, 2010 47

  48. Wednesday, June 2, 2010 48

  49. Wednesday, June 2, 2010 49

  50. Wednesday, June 2, 2010 50

  51. labelled Wednesday, June 2, 2010 51

  52. Chapter 3. Parameters and Multivariate GFs runs in perms Wednesday, June 2, 2010 52

  53. Wednesday, June 2, 2010 53

  54. Wednesday, June 2, 2010 54

  55. Wednesday, June 2, 2010 55

  56. PRINCIPLE: Add variables marking parameters at appropriate places and recycle: Wednesday, June 2, 2010 56

  57. Conclusions (Part I) [Chapter 3]: Multivariate GFs give access to parameters; those that can be obtained by “marking” in combinatorial constructions. [Chapters 1-2-3]: Exploit all this asymptotically? counting; mean, variance, distribution ? Wednesday, June 2, 2010 57

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