Singularity Analysis Via the Iterated Kernel Method
Stephen Melczer,Marni Mishna +1 more
65
TL;DR: It is proved that the univariate generating functions marking the number of walks of a given length are not D-finite, and exact and asymptotic enumerative formulas for the number the walks, and an efficient algorithm for exact enumeration are described.
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Abstract: In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach featured in recent works by Fayolle, Kurkova and Raschel. Here we consider these five models, called the singular models, and prove that the univariate generating functions marking the number of walks of a given length are not D-finite. Furthermore, we provide exact and asymptotic enumerative formulas for the number of such walks, and describe an efficient algorithm for exact enumeration.
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Citations
On the nature of the generating series of walks in the quarter plane
TL;DR: In this article, the Galois theory of difference equations is used to study the nature of the generating series of walks in the quarter plane and to recover many of the recent results about these series.
79
On the functions counting walks with small steps in the quarter plane
Irina Kurkova,Kilian Raschel +1 more
TL;DR: For all non-singular models of walks, the functions $x \mapsto Q(x,y,z)$ and $y\mapstof Q(0,y;z) are shown to be holonomic for any $z$ from one of them and non-holonomic from the other as mentioned in this paper.
•Posted Content
Non-D-finite excursions in the quarter plane
TL;DR: In this paper, it was shown that the trivariate generating function of the number of walks with given length and prescribed ending point is not D-finite, and the asymptotics of the generator are shown to be O(n)finite.
56
On the nature of the generating series of walks in the quarter plane
TL;DR: In this article, the Galois theory of difference equations is used to study the nature of the generating series of walks in the quarter plane and to recover many of the recent results about these series.
46
A human proof of Gessel's lattice path conjecture
TL;DR: Bostan and Koutschan as mentioned in this paper gave a computer-aided proof of the Gessel walks' algebraic properties. But they used Weierstrass zeta functions.
40
References
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The On-Line Encyclopedia of Integer Sequences.
TL;DR: The On-Line Encyclopedia of Integer Sequences (OEIS) as mentioned in this paper is a database of 13,000 number sequences and is freely available on the Web (http://www.att.com/~njas/sequences/) and is widely used.
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The On-Line Encyclopedia of Integer Sequences
Neil J. A. Sloane
- 27 Jun 2007
TL;DR: The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.
•Book
Analytic Combinatorics
Philippe Flajolet,Robert Sedgewick +1 more
- 01 Jan 2009
TL;DR: This text can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study, and is certain to become the definitive reference on the topic.
3.9K
Analytic Combinatorics: RANDOM STRUCTURES
Philippe Flajolet,Robert Sedgewick +1 more
- 01 Jan 2009
2.1K
A survey of max-type recursive distributional equations
David Aldous,Antar Bandyopadhyay +1 more
TL;DR: In this paper, the authors consider the problem of endogeny in the recursive tree process X_i and draw attention to the theoretical question of whether the X-i measurable functions of the innovations process (X_i) are also measurable in the tree process.