P. Flajolet. Combinatorial aspects of continued fractions. Discrete Mathematics, 32:125--161, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be labeled from 1 to y i and the East steps going from (i; y i ) to (i 1; y i ) can be labeled from 1 to b y i . Moreover, given two sequences f n g n1 and fb n g n0 , let M n be the number of labeled Motzkin walks and M(z) n0 M n z the associated generating function. 14 Proposition 4 [9, 10, 26] The generating function M(z) is a continued fraction. Its expression M(z) 1 . Labeled Motzkin walks are in relation with several well studied combinatorial objects [9, 26, 27] and in particular with permutations. The walks we will deal with are labeled as follows: ffl each ....

Ph. Flajolet. Combinatorial aspects of continued fractions. Ann. Discrete Math., 9:217222, 1980.

....be labeled from 1 to y i and the East steps going from (i; y i ) to (i 1; y i ) can be labeled from 1 to b y i . Moreover, given two sequences f n g n1 and fb n g n0 , let M n be the number of labeled Motzkin walks and M(z) n0 M n z the associated generating function. 14 Proposition 4 [9, 10, 26] The generating function M(z) is a continued fraction. Its expression M(z) 1 . Labeled Motzkin walks are in relation with several well studied combinatorial objects [9, 26, 27] and in particular with permutations. The walks we will deal with are labeled as follows: ffl each ....

....labeled Motzkin walks and M(z) n0 M n z the associated generating function. 14 Proposition 4 [9, 10, 26] The generating function M(z) is a continued fraction. Its expression M(z) 1 . Labeled Motzkin walks are in relation with several well studied combinatorial objects [9, 26, 27] and in particular with permutations. The walks we will deal with are labeled as follows: ffl each South East step (i; y i ) i 1; y i Gamma 1) is labeled by an integer between 1 and y i (or, equivalently, by a pair of integers, each one between 1 and y i ) ffl each East step (i; y i ) ....

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Ph. Flajolet. Combinatorial aspects of continued fractions. Discrete Math., 32:125-161, 1980.

....continued fractions to the statistics e k , whereas others relate these statistics to various other combinatorial structures. 3.1. A continued fraction of Ramanujan. The continued fraction R(q; t) 1 qt 5 7 . was studied by Ramanujan (see [10, p. 126] It was shown in [2] that the coe cient to t in the expansion of R(q; t) is the number of Dyck paths of length 2n and area k. Using the converse part of Theorem 2, we would like to nd the linear combinations of the statistics e k s that have as bivariate generating function the continued fraction R(q; t) ....

P. Flajolet. Combinatorial aspects of continued fractions. Discrete Math., 32(2):125161, 1980.

....as the number of Dyck paths from (0; 0) to (2n 2; 0) with height k 1. Proof. The expression (4.3) occurs in [3, Proposition 3.B] There, replace n by n 1, n j by i h j 1 i h j , a j b j 1 by x, j = 0; 1; h, and extract the coecient of x . If this is combined with Corollary 2 in [3], then it follows that the expression (4.3) is equal to the number of Dyck paths from (0; 0) to (2n 2; 0) with height at most h 1. Clearly, since by Corollary 4.3 we know that it also equals the number of ideals in I with class of nilpotence at most h, this implies the result. An ....

P. Flajolet, Combinatorial aspects of continued fractions, Discrete Math. 32 (1980), 125-161.

....as s: hl(1) l(n) n)i. This implies that the word m can be factorized in y(n 1) s) and y(n) s) Proof. The proof can be found in [6] Some others results about the combinatorial properties of mechanical words can be found in [8] and concerning the continued fractions in [3]. 3.2 Output process with a Sturmian input When the input in the system is Sturmian, the output process of each queue can be described more precisely. Indeed the structure of the output process induced by a mechanical input in the first queue is kept after passing through several queues. From ....

P. Flajolet. Combinatorial aspects of continued fractions. Discrete Mathematics, 32:125--161, 1980.

....the South East steps of height i can be labeled from 1 to i 1 and the East steps of height i can be labeled from 1 to b i . Moreover, given two sequences f n g n1 and fb n g n0 , let M n be the number of labeled Motzkin walks and n0 M n z the associated generating function. Proposition 4 [9, 25] The generating function M(z) is a continued fraction. Its expression is 1 . Labeled Motzkin walks are in relation with several well studied combinatorial objects [9, 25] and in particular with permutations. The walks we will deal with are labeled as follows: ffl each South East ....

....M n be the number of labeled Motzkin walks and n0 M n z the associated generating function. Proposition 4 [9, 25] The generating function M(z) is a continued fraction. Its expression is 1 . Labeled Motzkin walks are in relation with several well studied combinatorial objects [9, 25] and in particular with permutations. The walks we will deal with are labeled as follows: ffl each South East step of height i is labeled by an integer between 1 and (i 1) or, equivalently, by a pair of integers, each one between 1 and i 1) ffl each East step of height i is labeled by ....

[Article contains additional citation context not shown here]

Ph. Flajolet. Combinatorial aspects of continued fractions. Discrete Math., 32:125-161, 1980.

....This probability is the same for all vertices in the case of uniform random walks [15] our results are based on a general formal expression that describes the movement of the walk before visiting all vertices. The technique of formal rational fraction have been initially introduced by Flajolet [12], Arqu es and Francon [3] for other purposes. Although the exhibited applications are often carried out on simple graphs, but the method seems sufficiently powerful to take into account larger classes of graphs. The authors hope that more investigation will be done in this direction. Further ....

....borrowing only vertices of the interval [l; k] We will use the formal notation K n0 K over the powerset of fa i ; a i ; 0 i n Gamma 1g. Since the a i s and a i s are intended to be replaced by their probabilities, we shortly write K 1 GammaK , where 1 is the empty word, see [3, 12, 11]. Proposition 3.2 For j belonging to the interval [i; k] F i;j [l; k] may be written under the fractional form: F i;j [l;k] 1 Gamma(a i . i a 0 i Gamma1 1 Gamma2 . a l a 1 Gamma2 a i Gamma1 ) R i;j [l;k] 1) where R i;j [l;k] a i . a ....

P. Flajolet. Combinatorial aspects of continued fractions. Discrete Mathematics, 32:125--161, 1980. 14

....in the introduction. Notably, no (classical) analytical methods are being used here. Such methods might be used, though, at the point where our methods stops: many of the obtained stream expressions are suited for further analytical treatment. The use of continued fractions has been inspired by [Fla80]; see also [GJ83, Chapter 5.2] The formal treatment of such continued fractions seems somewhat easier in the present setting of coinductive stream calculus. Also, some of the combinatorial interpretations of the various fractions discussed here, seem to be 27 automaton represents Fibonacci ....

P. Flajolet. Combinatorial aspects of continued fractions. Discrete Mathematics, 32:125--161, 1980.

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P. Flajolet. Combinatorial aspects of continued fractions. Discrete Mathematics, 32:125--161, 1980.

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Philippe Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics 32 (1980), 125-161.

....two kinds of system at once. Among the systems with bounded jumps, those for which e i (k) k belongs to f1; 0; 1g for all i k have a nice property: the generating function for the corresponding excursions (walks starting and ending at level 0) can be written as the following continued fraction [15]: 1 1 b 0 z 1 b 1 z 1 b 2 z where the coecients a k ; b k and c k are the multiplicities appearing in the rules, which read (k) k 1) a k (k) b k c k Example 16. Arrangements The system (k) k) k 1) with axiom (s 0 ) 2) generates a sequence that starts with 1; 2; ....

....a walk model with axiom (0) and rules (k 1) k 1: The corresponding bivariate generating function F (z; u) satis es the functional di erential equation 1 z(u u = 1 z(1 u )F (z; 0) zu F u (z; u) which is certainly not obvious to solve. However, as observed in [15], it is easy to obtain a continued fraction expansion of the excursion generating function: F (z; 0) 1 z 2z = 1 1 z 1 z 1 3z . 1 B(z) 22 Axiom System Name Id. Generating Function Rational OGF OGF ( k mod 2) 1) Ex. 3: Fibonacci M0692 (k 1) Ex. ....

P. Flajolet. Combinatorial aspects of continued fractions. Discrete Mathematics, 32:125{ 161, 1980.

.... A decomposition by slices taken at an angle of (on the example, this gives 1,2,2,2,1,2,3,1,1,2,3,3,4) is then expressed by an in nite speci cation (not a priori covered by the standard paradigm) S(ZS(Z S( The OGF is consequently given by the continued fraction (see also [23]) O(z) 1 At top level, the singular Boltzmann sampler of Theorem 7 applies (write O = S(Q) and O(z) 1 Q(z) this even though O is not nitely speci able. The root of Q(z) 1 is easily found to 50D, 0:5761487691427566022978685737199387823547246631189; see [53] ....

Flajolet, P. Combinatorial aspects of continued fractions. Discrete Mathematics 32 (1980), 125-161.

....see Kac s account [31] for a vivid discussion. The urn models balls randomly switching between two urns. There are some interesting combinatorial aspects related to continued fractions, and in particular to zu cosh u du = 1 1 1 m z 1 1 . which is due to Stieltjes; see [13, 19] and especially the paper of Goulden and Jackson [25] The balance is s = 0, while a = 1 and b = 1. We x here an integer parameter m. Start with a 0 = t 0 = m, for simplicity. One has (u) 1 u . The basic integral I(u) is thus simply a hyperbolic integral (of genus 0 but with the ....

.... to (w) sinh w: Then, the BGF solution to the model is H(z; u) 1 u m=2 sinh (z atanh u) which simpli es to H(z; u) sinh z u cosh z) This last formula is in agreement with what is classically known and is susceptible to a direct combinatorial interpretation [13, 19, 25]. The paper by Edelman and Kostlan [16] makes for pleasant collateral reading. The functions obtained are here elementary since only integration on an algebraic curve of genus 0 (a conic) is needed. 5.3. Urns with subtraction: the elliptic cases. In this subsection, we list all the cases of ....

Flajolet, P. Combinatorial aspects of continued fractions. Discrete Mathematics 32 (1980), 125-161.

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P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics 32 (1980), 125-161.

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P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics 32 (1980), 125--161.

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P. Flajolet, Combinatorial aspects of continued fractions, Discrete Math. 32 (1980), 125-161. 12

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P. Flajolet. Combinatorial aspects of continued fractions, Discr. Math., 32(1980), 125-161.

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P. Flajolet, "Combinatorial aspects of continued fractions", Discrete Math. 32, 1980, 125 -- 161.

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P. Flajolet. Combinatorial aspects of continued fractions. Discrete Math., 32:125--161, 1980.

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P. Flajolet. Combinatorial aspects of continued fractions. Discrete Math., 32:125-161, 1980.

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P. Flajolet. Combinatorial aspects of continued fractions. Annals of Discrete Mathematics, 8:217--222, 1980.

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P. Flajolet, Combinatorial aspects of continued fractions, Disc. Math. 32 (1980), 125-161.

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P. Flajolet, "Combinatorial aspects of continued fractions", Discrete Math. 32, 1980, 125 -- 161.

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Philippe Flajolet. Combinatorial aspects of continued fractions. Discrete Mathematics, 32(2):125-161, 1980.

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P. Flajolet. Combinatorial aspects of continued fractions. Discrete Math., 32(2):125-161, 1980.

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