Combinatorial Models and Algebraic Continued Fractions Philippe Flajolet INRIA-Rocquencourt, France Orthogonal Polynomials, Special Functions and Applications Leuven, Begium, July 2009 Monday, July 20, 2009 1
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 OPS models via CFs and Orthogonal Polynomials Monday, July 20, 2009 2
LATTICE PATHS are comprised of steps Ascents ( a : ր ); Descents ( b : ց ); 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. Monday, July 20, 2009 3
FORMALITIES (i) Quasi-inverses . Let A be a ring ; by telescoping : (1 − f )(1 + f + · · · + f n ) = 1 − f n +1 . 1 1 − f = 1 + f + f 2 + f 3 + · · · . If f n → 0 , then � Distributivity : ( x + y ) n = (all words of length n ). w | w | = n Corollary A. If a = ր , b = ց , c = → , then [e.g., n ≡ 3] n blocks � �� � ( c + ab ) n = →→→ + րց →→ + → րց → + րցրցրց + · · · . Corollary B. Combining with the sum of a geometric progression any # blocks 1 � �� � � 1 − c − ab = ( → րց →→ րց ) . Monday, July 20, 2009 4
FORMALITIES (ii) 1 � Recall: 1 − c − ab = → րց →→ րց . 1 1 − d b. Then: Substitute further ab �→ a Corollary C. With a = ր , b = ց , c = → , d = ⇒ , in C [ [ a , b , c , d ] ] : 1 � = (all diagrams of height ≤ 1) 1 1 − c − a 1 − d b → ր ⇒⇒⇒ ց →→ ր ⇒ ց . � = And get Corollaries D, E, F, G, H, I, J, K, L, M,. . . ☺ Monday, July 20, 2009 5
FORMALITIES GIVE A THEOREM... ☛ Theorem [The main continued fraction theorem] 1 � = (all lattice paths) a 0 b 1 1 − c 0 − a 1 b 2 1 − c 1 − 1 − c 2 − ... a c b � � all lattice paths 1 � =P h /Q h = with height ≤ h a 0 b 1 1 − c 0 − a 1 b 2 1 − c 1 − ... 1 − c 2 − 1 − c h Monday, July 20, 2009 6
Equivalently, with weighting rules , a j �→ α j z , b j �→ β j z , c j �→ γ j z : 1 z | π | weight( π ) � ≡ α 0 β 1 · z 2 π : excursion 1 − γ 0 z − α 1 β 2 · z 2 1 − γ 1 z − 1 − γ 2 z − ... generating function universal J –fraction ≡ of [weighted] excursions Excursions : J –fraction P h —, bounded height : (convergent) Q h Paths ending at k : Q k J − P k 1 —, traversing k –strip : Q k Monday, July 20, 2009 7
• J. T ouchard [1952]: chord diagrams (i) • I.J. Good [1958]: discrete birth-death processes • A. Lenard [1961]: statistical physics • G. Szekeres [1968]: Rogers-Ramanujan identities • D. Jackson [1978]: Ising model • R. Read [1979]: chord diagrams (ii) • P . Flajolet [1978-80]: “File histories”. Discr. Math. Monday, July 20, 2009 8
What next? 1. Simple applications 2. Convergents and orthogonal polynomials 3. Arches 4. Snakes 5. Addition formulae 6. Elliptic matters Monday, July 20, 2009 9
1. Simple applications (ballots, coins) Monday, July 20, 2009 10
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: 1 z | β | = � � C n z 2 n . C := = z 2 β ballot sequence 1 − n 1 − z 2 ... We get Catalan numbers [Euler-Segner 1750; Catalan 1850] √ 1 − 4 z 2 � 2 n � 1 ⇒ C = 1 − 1 C = = = ⇒ C n = 1 − z 2 C 2 z 2 . n + 1 n The probability is C n 1 � = n + 1. � 2 n n Monday, July 20, 2009 11
COIN FOUNTAINS [Odlyzko-Wilf, 1988] C ( q ) = 1 + q + q 2 + 2 q 3 + 3 q 4 + 5 q 5 + 9 q 6 + 15 q 7 + 26 q 8 + · · · Monday, July 20, 2009 12
Do: a j �→ 1 , b j �→ q j . Then: q n 2 + n � ( − 1) n 1 (1 − q )(1 − q 2 ) · · · (1 − q n ) C ( q ) = = q n 2 . q � ( − 1) n 1 − q 2 (1 − q )(1 − q 2 ) · · · (1 − q n ) 1 − 1 − q 3 · · · Number of coin fountains: C n ∼ 0 . 31 · 1 . 73566 n . Ramanujan’s fraction: √ √ 1 5 − 1 + 5 √ = e 2 π / 5 √ � 2 e − 2 π 5 5 3 / 4 1 + 5 / 2 − 1 5 1 + √ ( 1 / 2+1 / 2 5 ) √ e − 4 π 5 1 + √ 1 + e − 6 π 5 · · · = 0 . 999999 2087 · · · . Monday, July 20, 2009 13
2. Convergent polynomials Revisiting the ballot problem Orthogonality Monday, July 20, 2009 14
“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 : 1 C [ h ] ( z ) = z 1 − h stages. ... 1 − 1 − z 1 1 1 − z 1 − 2 z F h +1 ( z ) 1 , 1 − z , 1 − 2 z , 1 − 3 z + z 2 , · · · , F h +2 ( z ) , where F h +2 = F h +1 − zF h are Fibonacci polynomials. “Constant-coe ffi cient” recurrence; Lagrange inversion. Roots are 1 / (4 cos 2 θ ), θ = k π h ; partial fractions. + Chebyshev Monday, July 20, 2009 15
Lagrange [1775] & Lord Kelvin & De Bruijn, Knuth, Rice [1973] � k π � � 2 n � · · · 4 n cos 2 n C [ h ] � � = = . n · · · h n − kh k k partial fraction / Lagrange inversion Related to Kolmogorov–Smirnov tests in statistics: Compare X 1 , . . . , X n and Y 1 , . . . , Y n ? “Sort and vote!” P´ olya [1927]: totally elementary proof of elliptic-theta transformation : � π ∞ ∞ e − ν 2 t 2 = e − π 2 ν 2 / t 2 . � � t 2 ν = −∞ ν = −∞ = Do multisection of (1 + z ) 2 n , with h = t √ n, in two ways! + asymptotics! Monday, July 20, 2009 16
Orthogonal polynomials Linear fractional transformations [homographies] get composed like 2 × 2 matrices: � a � ax + b b �→ c d . cx + d Convergent polynomials P h ( z ) Q h ( z ) satisfy a three-term recurrence 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 coe ffi cients in the expansion of the continued fraction. Monday, July 20, 2009 17
In all generality: • Orthogonal polynomials must appear in counting of paths of bounded height and in “equivalent” structures. Monday, July 20, 2009 18
3.Arches and such Colouring rules Hermite polynomials Monday, July 20, 2009 19
In how many ways can one join 2 n points on the line in pairs? ր ր ր ր ց ց ց ց ր ր ց ց 1 2 2 1 2 1 A descent from altitude j has j possibilities: d j �→ jz , a j �→ z . 1 (1 · 3 · · · (2 n − 1)) z 2 n = � 1 · z 2 . n ≥ 1 1 − 2 · z 2 1 − 1 − 3 · z 2 ... Gauss !! Monday, July 20, 2009 20
Lagarias-Odlyzko-Zagier [1985]: Which capacity do we need to arrange pairwise connections between 2 n points, with high probability? • The answer lies in the zeroes of Hermite polynomials . � ∞ f ( x ) · g ( x ) e − x 2 / 2 dx . � f , g � = −∞ 1 1 · z 2 1 − height 2 · z 2 Proof. For width h : h levels. 1 − ... 1 − h · z 2 Louchard & Janson: a Gaussian process = deterministic parabola + Brownian noise. + Airy connection Monday, July 20, 2009 21
Chord systems Join 2 n points on circle by chords. How many crossings? [ n ] = 1 − q n 1 z | σ | q # xings ( σ ) = � Sweeping: T ( z , q ) ≡ [1] · z 2 , 1 − q . 1 − σ [2] · z 2 1 − 1 − [3] · z 2 ... Theorem [Touchard]. Number of crossings has generating function 1 − √ 1 − 4 z k +1 q ( 2 )( − z k ) C 2 k +1 ; � 1 � � T ( z (1 − q ) , q ) = C := . 2 z k ≥ 0 Corollary [F-Noy 2000]: # crossings is asympt. Gaussian. Cf [Ismail, Stanton, Viennot 1987] for nice combinatorics ( q -Hermite). Monday, July 20, 2009 22
4. Snakes and curves Arnold’s snakes Stieltjes’ fraction Postnikov’s Morse links Monday, July 20, 2009 23
Arnold [ 1992 ] : How many topological types of “ smooth ” functions? D. Andr é [ 1881 ] : alternating perms = the coe ffi cients of tan ( z ) and sec ( z ) . Monday, July 20, 2009 24
ր 4 ց 3 ր 3 ց 2 ր 2 ց 1 � ∞ z (tan) 2 n +1 z 2 n +1 = � e t tan( zt ) dt � � � � = 1 · 2 z 2 . 0 1 − 1 − 2 · 3 z 3 1, 2, 16, 272, ... · · · Related to a bijection of Françon and Viennot = A continued fraction of Stieltjes Ann. Fac. Sci. T oulouse, 1894 Monday, July 20, 2009 25
A continued fraction of Postnikov (2000) = Morse links (systems of closed Morse curves) Theorem [F.2008]. The Morse–Postnikov numbers satisfy � 4 � 2 n +1 L n = 1 L n ∼ � � 2(2 n − 1)! L n , where . π E.g.: L 4 = 0 . 99949. . � L 4 Monday, July 20, 2009 26
5. Addition formulae &c. Stieltjes-Rogers Addition formulae, paths, and OPs are all belong to a single family of identities Applications to “processes”... Monday, July 20, 2009 27
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