C. Banderier and P. Flajolet. Basic analytic combinatorics of directed lattice paths. Theoret. Comput. Sci., 281(1-2):37--80, 2002.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a family of pattern classes. It can also be used in the context of generating functions for generating trees, thereby generalizing a number of the theorems in [5] about the existence of algebraic generating functions. It must be noted though that the results of that paper and similar results in [6] provide much more explicit detail concerning the generating functions that they produce. 13 One of the striking features of the equations for the various irreducible and indecomposable subsets of A(321) is their simplicity. In some sense then the enumerative coincidences that we observed are ....

Cyril Banderier and Philippe Flajolet. Basic analytic combinatorics of directed lattice paths. Theoret. Comput. Sci., 281(1-2):37--80, 2002. Selected papers in honour of Maurice Nivat.

.... combinatorics shows that the algebraic structure of the generating function depends entirely on the roots (converging in z = 0) of 1 zP (u) 0, where P (u) the characteristic polynomial associated to the walk: P (u) a X k= b p k u k : X i2P u i : 1) This approach was used in [1, 2, 4]) and allows to make explicit the generating function and the asymptotics of its coecients in 7 steps: 1. nd a recurrence 1 for fn (u) 2. translate this recurrence into a functional equation (via generating functions) 3. solve this functional equation (via the kernel method) 4. express the ....

....As the right hand side of the previous equation can be considered (rewritten) as a polynomial in u of degree 2, for which one knows the two roots (u 1 and u 2 ) one can factorize it. This leads to u c (1 zP (u) F (z; u) u u 1 ) u u 2 ) 4. The simple factor is A = u u 1 (z) and we refer to [4] for the steps 5, 6, 7. Figure 2 gives a summary of the most notable results for these one dimensional walks. To answer to a question of Bousquet M elou, it is noteworthy that these results have some non trivial applications on two dimensional walks: using her results from [11] and then using ....

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Cyril Banderier and Philippe Flajolet. Basic analytic combinatorics of directed lattice paths. Theoretical Computer Science, June 2001. Submitted, 37 pages.

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Cyril Banderier and Philippe Flajolet. Basic analytic combinatorics of directed lattice paths. To appear in Theoretical Computer Science, June 2001.

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Cyril Banderier and Philippe Flajolet. Basic analytic combinatorics of directed lattice paths. Theoretical Computer Science, 281:37-80, 2002.

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C. Banderier and P. Flajolet. Basic analytic combinatorics of directed lattice paths. Theoretical Computer Science, 281(1-2):37--80, 2002.

....uniform distribution on all paths of length n. When the sets P k s are equal to a xed set P (the simplest interesting case being P = f1; 1g) the corresponding walks have been deeply studied both in combinatorics (Dyck paths, and in probability theory (coin ipping, We refer to [4] for enumerative and analytical studies of such walks on N with a nite set of jumps . When the sets P k s are unbounded, both enumeration and asymptotics become cumbersome: contrary to the previous case, the walks are not space homogeneous (the set of available jumps depends on the position) ....

.... root of the associated generating tree) In what follows, we often (but not always) consider the case for which e(k; i) e k i (where (e k ) k2Z is a xed sequence) If the sequence (e(k; i) k (for a xed i) is ultimately 0, then the situation covers the case of walks with a nite set of jumps [4]. If the sequence is ultimately 1, then this covers the case of factorial rules which are of great interests for 9 the generation of combinatorial objects [8] and for which it was proved in [3] that the associated generating functions are algebraic. We still note f n;k the number of walks on N ....

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Cyril Banderier and Philippe Flajolet. Basic analytic combinatorics of directed lattice paths. Theoretical Computer Science, 281(1-2):37-80, 2002. Selected papers in honour of Maurice Nivat.

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C. Banderier and P. Flajolet. Basic analytic combinatorics of directed lattice paths. Theoretical Computer Science, 281(1-2):37 80, 2002.

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) Cyril Banderier and Philippe Flajolet. Basic analytic combinatorics of directed lattice paths. Theoretical Computer Science, June 2002.

....model under consideration here is the uniform distribution on all paths of length n. When the sets P k s are equal to a xed set P (the simplest interesting case being P = f1; 1g) the corresponding walks have been deeply studied both in combinatorics and in probability theory. We refer to [3] for asymptotic properties of such walks on N with a nite set of jumps . When the sets P k s are unbounded, both enumeration and asymptotics become cumbersome: contrary to the previous case, the walks are not space homogeneous (the set of available jumps depends on the position) and it is not ....

.... u fn (1) g stands for the sum of the monomials in u with a negative degree. Multiplying by z (1 zP (u) F (z; u) 1 z 4 1 r k (u)F k (z) z u F (z; 1) 3) and t i (u) are (Laurent) polynomials which can be made explicit. One can use the kernel method (we refer to [3, 5] for recent applications of this method) to solve this equation. We call 1 zP (u) the kernel of the equation. Solving 1 zP (u) 0 with respect to u gives 4 roots u 1 (z) u 2 (z) u 3 (z) and u 4 (z) which are Puiseux series in z 1=4 and which tend to zero in 0. There are also 23 others roots ....

Cyril Banderier and Philippe Flajolet. Basic analytic combinatorics of directed lattice paths. To appear in Theoretical Computer Science, June 2001.

....were the rst general way to solve Ballot like problems. I give also a tentative short biography. 1. Classical lattice paths A good reference for classical results on lattice paths is [35] more recent works show that lattice paths are still a subject of an intensive activity in combinatorics [24, 25, 27, 34, 42, 3, 38, 22, 26, 1] and in probability theory [2, 23] Most of the classical number sequences or lattice paths have a name which is related to some famous mathematicians such as the Italian Leonardo Fibonacci ( 1170 1250) the French Blaise Pascal (1623 1662) the Swiss Jacob Bernoulli (1654 1705) the Scottish ....

Cyril Banderier and Philippe Flajolet. Basic analytic combinatorics of directed lattice paths. Theoretical Computer Science, June 2001. To appear.

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C. Banderier and P. Flajolet. Basic analytic combinatorics of directed lattice paths. Theoret. Comput. Sci., 281(1-2):37--80, 2002.

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C. Banderier and P. Flajolet. Basic analytic combinatorics of directed lattice paths. Theoret. Comput. Sci., 281(1-2):37-80, 2002.

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C. Banderier and P. Flajolet, 2001, Basic Analytic Combinatorics of Directed Lattice Paths. To appear in Theo. Comp. Sci.

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C. Banderier and P. Flajolet, 2001, Basic Analytic Combinatorics of Directed Lattice Paths. To appear in Theo. Comp. Sci.

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Cyril Banderier and Philippe Flajolet. Basic analytic combinatorics of directed lattice paths. Theoretical Computer Science, 281(1-2):3780, 2002.

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Banderier (Cyril) and Flajolet (Philippe). { Basic analytic combinatorics of directed lattice paths. { Available from http://algo.inria.fr/flajolet/Publications/BaFl01.ps.gz, August 2001. 39 pages. Accepted for publication in Theoretical Computer Science.

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C. Banderier and P. Flajolet, Basic analytic combinatorics of directed lattice paths, to appear in Theoret. Comput. Sci.

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