Combinatorial Models and Algebraic Continued Fractions

Transcription

1 Combinatorial Models and Algebraic Continued Fractions Philippe Flajolet INRIA-Rocquencourt, France Orthogonal Polynomials, Special Functions and Applications Leuven, Begium, July 2009

2 ANALYTIC COMBINATORICS, by P. Flajolet & R. Sedgewick Cambridge 2009, 824p free download: algo.inria.fr/flajolet/ = exactly solvable models + asymptotics CF Continued fractions associated with power series are tightly linked to lattice paths Read off CF identities from combinatorics Solve combinatorial & discrete probabilistic models via CFs and Orthogonal Polynomials OPS 2

3 LATTICE PATHS are comprised of steps Ascents (a : ); Descents (b : ); Levels (c : ); never go below horizontal axis. Excursions start and end at 0 altitude. Main theorem: Equivalence between: Sum of all excursions encoded with altitudes; Universal continued fraction of Jacobi type. 3

4 FORMALITIES (i) Quasi-inverses. Let A be a ring; by telescoping: ( f )( + f + + f n ) = f n+. If f n 0, then f = + f + f 2 + f 3 +. Distributivity: (x + y) n = w (all words of length n). w =n Corollary A. If a =, b =, c =, then [e.g., n 3] (c+ab) n = n blocks {}}{ Corollary B. Combining with the sum of a geometric progression c ab = any # blocks {}}{ ( ). 4

5 FORMALITIES (ii) Recall: c ab =. Substitute further ab a d b. Then: Corollary C. With a =, b =, c =, d =, in C[[a, b, c, d]]: c a d b = (all diagrams of height ) =. And get Corollaries D, E, F, G, H, I, J, K, L, M,... 5

6 FORMALITIES GIVE A THEOREM... Theorem [The main continued fraction theorem] c 0 a 0 b a b 2 c c 2... = (all lattice paths) a c b c 0 c a 0 b a b 2 c 2... c h = ( all lattice paths with height h ) =Ph/Qh 6

7 Equivalently, with weighting rules, a j α j z, b j β j z, c j γ j z: γ 0 z α 0 β z 2 γ z α β 2 z 2 γ 2 z... π: excursion z π weight(π) universal J fraction generating function of [weighted] excursions Excursions :, bounded height : Paths ending at k :, traversing k strip : J fraction P h (convergent) Q h Q k J P k Q k 7

8 J. Touchard [952]: chord diagrams (i) I.J. Good [958]: discrete birth-death processes A. Lenard [96]: statistical physics G. Szekeres [968]: Rogers-Ramanujan identities D. Jackson [978]: Ising model R. Read [979]: chord diagrams (ii) P. Flajolet [978-80]: File histories. Discr. Math. 8

9 What next?. Simple applications 2. Convergents and orthogonal polynomials 3. Arches 4. Snakes 5. Addition formulae 6. Elliptic matters 9

10 . Simple applications (ballots, coins) 0

11 A SOLUTION (?!) TO THE BALLOT PROBLEM Two candidates, Alice and Bob, with (eventually) each n votes. What is the probability that Alice is always ahead or tied? Do a z; b z; c 0. By main theorem: C := z2 z2... = β ballot sequence z β = n C n z 2n. We get Catalan numbers [Euler-Segner 750; Catalan 850] C = z 2 C = C = The probability is C n ( 2n ) = n +. n 4z 2 2z 2 = C n = n + ( 2n n ).

12 COIN FOUNTAINS [Odlyzko-Wilf, 988] C(q) = + q + q 2 + 2q 3 + 3q 4 + 5q 5 + 9q 6 + 5q q 8 + 2

13 Do: a j, b j q j. Then: C(q) = q q2 q3 = ( ) n q n2 +n ( ) n Number of coin fountains: C n n. Ramanujan s fraction: + e 2π 5 + e 4π 5 + e 6π 5 = e 2π/ = ( q)( q 2 ) ( q n ) q n2 ( q)( q 2 ) ( q n ) 5 5 3/4 (/2+/2 5) 5/

14 2. Convergent polynomials Revisiting the ballot problem Orthogonality 4

15 Two candidates, Alice and Bob, with (eventually) each n votes. If Alice is always ahead (or tied), what is the probability that she never leads by more than h? The number of favorable cases has generating function (GF), with z 2 z: C [h] (z) = z h stages.... z, z, z 2z, 2z 3z + z 2,, F h+ (z) F h+2 (z), where F h+2 = F h+ zf h are Fibonacci polynomials. Constant-coefficient recurrence; Lagrange inversion. Roots are /(4 cos 2 θ), θ = kπ h ; partial fractions. + Chebyshev 5

16 Lagrange [775] & Lord Kelvin & De Bruijn, Knuth, Rice [973] C [h] n = k 4 n cos 2n ( kπ h ) = k ( ) 2n. n kh partial fraction / Lagrange inversion Related to Kolmogorov Smirnov tests in statistics: Compare X,..., X n and Y,..., Y n? Sort and vote! Pólya [927]: totally elementary proof of elliptic-theta transformation: ν= e ν2 t 2 = π t 2 ν= e π2 ν 2 /t 2. = Do multisection of ( + z) 2n, with h = t n, in two ways! + asymptotics! 6

17 Orthogonal polynomials Linear fractional transformations [homographies] get composed like 2 2 matrices: ( ) ax + b a b. cx + d c d Convergent polynomials P h(z) satisfy a three-term recurrence Q h (z) with numers/denoms of the continued fraction. Reciprocals of convergent polynomials are orthogonal with respect to f, g = f g, where moments z n are coefficients in the expansion of the continued fraction. 7

18 In all generality: Orthogonal polynomials must appear in counting of paths of bounded height and in equivalent structures. 8

19 3.Arches and such Colouring rules Hermite polynomials 9

20 In how many ways can one join 2n points on the line in pairs? A descent from altitude j has j possibilities: d j jz, a j z. ( 3 (2n ))z 2n = n Gauss!! z 2 2 z2 3 z

21 Lagarias-Odlyzko-Zagier [985]: Which capacity do we need to arrange pairwise connections between 2n points, with high probability? The answer lies in the zeroes of Hermite polynomials. f, g = f (x) g(x) e x2 /2 dx. height Proof. For width h: z 2 2 z2... h z 2 h levels. Louchard & Janson: a Gaussian process = deterministic parabola + Brownian noise. + Airy connection 2

22 Chord systems Join 2n points on circle by chords. How many crossings? Sweeping: T (z, q) σ z σ q #xings(σ) = [] z 2 [2] z2 [3] z2..., [n] = qn q. Theorem [Touchard]. Number of crossings has generating function T (z( q), q) = q (k+ 2 ) ( z k )C 2k+ ( ; C := ) 2z 4z. k 0 Corollary [F-Noy 2000]: # crossings is asympt. Gaussian. Cf [Ismail, Stanton, Viennot 987] for nice combinatorics (q-hermite). 22

23 4. Snakes and curves Arnold s snakes Stieltjes fraction Postnikov s Morse links 23

24 Arnold [992]: How many topological types of smooth functions? D. André [88]: alternating perms = the coefficients of tan(z) and sec(z). 24

25 (tan)2n+ z 2n+ =, 2, 6, 272,... 0 e t tan(zt) dt = z 2 z2 2 3 z3. Related to a bijection of Françon and Viennot = A continued fraction of Stieltjes Ann. Fac. Sci. Toulouse,

26 A continued fraction of Postnikov (2000) = Morse links (systems of closed Morse curves) Theorem [F.2008]. The Morse Postnikov numbers satisfy L n L n, where Ln = 2 (2n )! ( 4 π ) 2n+. E.g.: L 4 L4. =

27 5. Addition formulae &c. Stieltjes-Rogers Addition formulae, paths, and OPs are all belong to a single family of identities Applications to processes... 27

28 The Stieltjes Rogers Theorem Definition. φ(z) = n=0 φ z n n satisfies an addition formula if n! φ(x + y) = ω k φ k (x)φ k (y), where φ k (x) = x k k! + O(x k+ ). k Theorem. An addition formula gives automatically a continued fraction for f (z) = φ n z n = n=0 0 e t φ(zt) dt. x y = k n!z n = z (k!) 2 x k /k! y k /k! ( x) k+ ( y) k+ 2 z 2 3z 22 z 2... [Biane, Françon-Viennot] 28

29 Systems of paths and birth death processes: Number & probability of weighted paths from a to b; Discrete time processes: I.J. Good [950 s]; Continuous time processes: Karlin McGregor; F Guillemin [AAP 2000]; Combinatorial processes = file histories, [F Françon Vuillemin Puech, 980+] Paths from 0 to k have exp. gen. function ϕ k of addition formula. 29

30 Meixner s class of special OP s Classical orthogonal polynomials appear to share many properties. Theorem [Meixner 934]: If the exponential generating function satisfies a strong decomposability property, h Q h (z) tn n! = A(t)ezB(t), then there are only five possibilities. Laguerre Hermite Poisson-Charlier Meixner I Meixner II Perms Arcs Set partitions Snakes Pref. arrang. e z2 /2 e ez sec(z) z 2 e z. Computations for linear possibilities are automatic: unified theory of libraries, basic queueing systems 30

31 The Ehrenfest urn model Particles switching chambers addition formula (cosh(z)) N Stieltjes; Kac 947; Edelman-Kostlan 994 & N z 2. 2 (N ) z2 The Mabinogion urn model Spread of influence in populations: A = (B A), B = (A B). Theorem [F Huillet 2008]. Fair urn: absorption time is 2 N log N, with limit distribution of density e t e e 2t. 3

32 6. Some Elliptic matters Jacobian functions Dixonian functions Bacher s numbers 32

33 Pollaczek fractions have coefficients that are polynomials in the level = a mysterious class! Includes some Hurwitz zeta; cf Stieltjes-Apéry An interesting sporadic subclass appears to be related to elliptic functions [Pollaczek, Mem. Sc. Math., 956] 33

34 Algebraic curves of genus are doughnuts. The integrals have two periods. The inverse functions are elliptic functions; i.e., doubly periodic meromorphic. Weierstraß arises from y 2 = P 3 (z); Jacobian sn, cn arise from y 2 = ( z 2 )( k 2 z 2 ); Dixonian sm, cm arise from y 3 + z 3 =. They satisfy addition formulae! ( Stieltjes-Rogers) 34

35 Theorem [F; Dumont 980]. Jacobian elliptic functions count alternating perms w/parity of peaks. Theorem [Conrad+F, 2006]. Dixonian functions have continued fractions 0 sm(u)e u/x du = + b 0 x 3 x x 6 + b x x 6 ; levels in trees and an urn model ( Yule process), &c Theorem [Bacher+F, 2006]. Pseudofactorials a n+ = ( ) n+ ( n k) ak a n k have a CF an z n = + z z z 2 z + + 3z

36 Pseudo-factorials: a n+ = ( ) n+ ( ) n a k a n k. k Theorem [Bacher+F, 2008]. The exponential generating function of the orthogonal polynomials attached to (a n ) is η(t) cosh(zj(t)) + χ(t) sin(zj(t)), where J(t) := functions.. t 0 du 3u 2 and η, χ are algebraic + 3u4 Carlitz 960+; Ismail & Masson 999; Lomont & Brillhart 200; cf. Gilewicz et al 2006 for sm. Cf also: Flajolet--Bacher (an octic fraction, unpub.); Rivoal (deg=2(!)) relative to Γ(/3) 3 36

37 Probabilistic Processes Special Functions Urn models, branching pr., Brownian motion,... Continued Fractions Ortho Polys Meixner class, q- Hermite,... Combinatorics Permutations, chords,set partitions,... 37