On 3-Dimensional Lattice Walks Confined to the Positive Octant

Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in { 0 , ± 1 } 2 : the generating function is D-finite if and only if a certain group associated with the step set is f...

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Vydané v:	Annals of combinatorics Ročník 20; číslo 4; s. 661 - 704
Hlavní autori:	Bostan, Alin, Bousquet-Mélou, Mireille, Kauers, Manuel, Melczer, Stephen
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Cham Springer International Publishing 01.12.2016 Springer Nature B.V Springer Verlag
Predmet:	Combinatorics Computer algebra Differential equations Enumeration Group theory Mathematics Mathematics and Statistics Quadrants D-finite series exact enumeration 05A15 lattice walks Exact enumeration Lattice walks
ISSN:	0218-0006, 0219-3094
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Abstract	Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in { 0 , ± 1 } 2 : the generating function is D-finite if and only if a certain group associated with the step set is finite. We explore in this paper the analogous problem for 3- dimensional walks confined to the positive octant. The first difficulty is their number: we have to examine no less than 11074225 step sets in { 0 , ± 1 } 3 (instead of 79 in the quadrant case). We focus on the 35548 that have at most six steps. We apply to them a combined approach, first experimental and then rigorous. On the experimental side, we try to guess differential equations. We also try to determine if the associated group is finite. The largest finite groups that we find have order 48 — the larger ones have order at least 200 and we believe them to be infinite. No differential equation has been detected in those cases. On the rigorous side, we apply three main techniques to prove D-finiteness. The algebraic kernel method, applied earlier to quadrant walks, works in many cases. Certain, more challenging, cases turn out to have a special Hadamard structure which allows us to solve them via a reduction to problems of smaller dimension. Finally, for two special cases, we had to resort to computer algebra proofs. We prove with these techniques all the guessed differential equations. This leaves us with exactly 19 very intriguing step sets for which the group is finite, but the nature of the generating function still unclear.
AbstractList	Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in $\{0, \pm 1\}^2$: the generating function is D-finite if and only if a certain group associated with the step set is finite. We explore in this paper the analogous problem for 3-dimensional walks confined to the positive octant. The first difficulty is their number: we have to examine no less than 11074225 step sets in $\{0, \pm 1\}^3$ (instead of 79 in the quadrant case). We focus on the 35548 that have at most six steps. We apply to them a combined approach, first experimental and then rigorous. On the experimental side, we try to guess differential equations. We also try to determine if the associated group is finite. The largest finite groups that we find have order 48 -- the larger ones have order at least 200 and we believe them to be infinite. No differential equation has been detected in those cases. On the rigorous side, we apply three main techniques to prove D-finiteness. The algebraic kernel method, applied earlier to quadrant walks, works in many cases. Certain, more challenging, cases turn out to have a special Hadamard structure, which allows us to solve them via a reduction to problems of smaller dimension. Finally, for two special cases, we had to resort to computer algebra proofs. We prove with these techniques all the guessed differential equations. This leaves us with exactly 19 very intriguing step sets for which the group is finite, but the nature of the generating function still unclear. Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in { 0 , ± 1 } 2 : the generating function is D-finite if and only if a certain group associated with the step set is finite. We explore in this paper the analogous problem for 3- dimensional walks confined to the positive octant. The first difficulty is their number: we have to examine no less than 11074225 step sets in { 0 , ± 1 } 3 (instead of 79 in the quadrant case). We focus on the 35548 that have at most six steps. We apply to them a combined approach, first experimental and then rigorous. On the experimental side, we try to guess differential equations. We also try to determine if the associated group is finite. The largest finite groups that we find have order 48 — the larger ones have order at least 200 and we believe them to be infinite. No differential equation has been detected in those cases. On the rigorous side, we apply three main techniques to prove D-finiteness. The algebraic kernel method, applied earlier to quadrant walks, works in many cases. Certain, more challenging, cases turn out to have a special Hadamard structure which allows us to solve them via a reduction to problems of smaller dimension. Finally, for two special cases, we had to resort to computer algebra proofs. We prove with these techniques all the guessed differential equations. This leaves us with exactly 19 very intriguing step sets for which the group is finite, but the nature of the generating function still unclear. Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in { 0 , ± 1 } 2 : the generating function is D-finite if and only if a certain group associated with the step set is finite. We explore in this paper the analogous problem for 3- dimensional walks confined to the positive octant. The first difficulty is their number: we have to examine no less than 11074225 step sets in { 0 , ± 1 } 3 (instead of 79 in the quadrant case). We focus on the 35548 that have at most six steps. We apply to them a combined approach, first experimental and then rigorous. On the experimental side, we try to guess differential equations. We also try to determine if the associated group is finite. The largest finite groups that we find have order 48 — the larger ones have order at least 200 and we believe them to be infinite. No differential equation has been detected in those cases. On the rigorous side, we apply three main techniques to prove D-finiteness. The algebraic kernel method, applied earlier to quadrant walks, works in many cases. Certain, more challenging, cases turn out to have a special Hadamard structure which allows us to solve them via a reduction to problems of smaller dimension. Finally, for two special cases, we had to resort to computer algebra proofs. We prove with these techniques all the guessed differential equations. This leaves us with exactly 19 very intriguing step sets for which the group is finite, but the nature of the generating function still unclear.
Author	Melczer, Stephen Bostan, Alin Kauers, Manuel Bousquet-Mélou, Mireille
Author_xml	– sequence: 1 givenname: Alin surname: Bostan fullname: Bostan, Alin organization: INRIA Saclay Île-de-France – sequence: 2 givenname: Mireille surname: Bousquet-Mélou fullname: Bousquet-Mélou, Mireille email: bousquet@labri.fr organization: CNRS, LaBRI, Université de Bordeaux – sequence: 3 givenname: Manuel surname: Kauers fullname: Kauers, Manuel organization: Institute for Algebra, Johannes Kepler University – sequence: 4 givenname: Stephen surname: Melczer fullname: Melczer, Stephen organization: Cheriton School of Computer Science, University of Waterloo, U. Lyon, CNRS, ENS de Lyon, Inria, UCBL, Laboratoire LIP
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Snippet	Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete...
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SubjectTerms	Combinatorics Computer algebra Differential equations Enumeration Group theory Mathematics Mathematics and Statistics Quadrants
Title	On 3-Dimensional Lattice Walks Confined to the Positive Octant
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