271 research outputs found
Hypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter Plane
International audienceWe study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or . We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane , counted by their length and by the position of their ending point. Bousquet-Mélou and Mishna [Contemp. Math., pp. 1--39, Amer. Math. Soc., 2010] identified 19 models of walks that possess a D-finite generating function; linear differential equations have then been guessed in these cases by Bostan and Kauers [FPSAC 2009, Discrete Math. Theor. Comput. Sci. Proc., pp. 201--215, 2009]. We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their combinatorially meaningful specializations only four are algebraic functions
Asymptotics of lattice walks via analytic combinatorics in several variables
We consider the enumeration of walks on the two dimensional non-negative
integer lattice with short steps. Up to isomorphism there are 79 unique two
dimensional models to consider, and previous work in this area has used the
kernel method, along with a rigorous computer algebra approach, to show that 23
of the 79 models admit D-finite generating functions. In 2009, Bostan and
Kauers used Pad\'e-Hermite approximants to guess differential equations which
these 23 generating functions satisfy, in the process guessing asymptotics of
their coefficient sequences. In this article we provide, for the first time, a
complete rigorous verification of these guesses. Our technique is to use the
kernel method to express 19 of the 23 generating functions as diagonals of
tri-variate rational functions and apply the methods of analytic combinatorics
in several variables (the remaining 4 models have algebraic generating
functions and can thus be handled by univariate techniques). This approach also
shows the link between combinatorial properties of the models and features of
its asymptotics such as asymptotic and polynomial growth factors. In addition,
we give expressions for the number of walks returning to the x-axis, the
y-axis, and the origin, proving recently conjectured asymptotics of Bostan,
Chyzak, van Hoeij, Kauers, and Pech.Comment: 10 pages, 3 tables, as accepted to proceedings of FPSAC 2016 (without
conference formatting
Automatic Classification of Restricted Lattice Walks
We propose an experimental mathematics approach leading to the
computer-driven discovery of various structural properties of general counting
functions coming from enumeration of walks
Lattice paths of slope 2/5
We analyze some enumerative and asymptotic properties of Dyck paths under a
line of slope 2/5.This answers to Knuth's problem \\#4 from his "Flajolet
lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in
June 2014.Our approach relies on the work of Banderier and Flajolet for
asymptotics and enumeration of directed lattice paths. A key ingredient in the
proof is the generalization of an old trick of Knuth himself (for enumerating
permutations sortable by a stack),promoted by Flajolet and others as the
"kernel method". All the corresponding generating functions are algebraic,and
they offer some new combinatorial identities, which can be also tackled in the
A=B spirit of Wilf--Zeilberger--Petkov{\v s}ek.We show how to obtain similar
results for other slopes than 2/5, an interesting case being e.g. Dyck paths
below the slope 2/3, which corresponds to the so called Duchon's club model.Comment: Robert Sedgewick and Mark Daniel Ward. Analytic Algorithmics and
Combinatorics (ANALCO)2015, Jan 2015, San Diego, United States. SIAM, 2015
Proceedings of the Twelfth Workshop on Analytic Algorithmics and
Combinatorics (ANALCO), eISBN 978-1-61197-376-1, pp.105-113, 2015, 2015
Proceedings of the Twelfth Workshop on Analytic Algorithmics and
Combinatorics (ANALCO
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 : 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: there
are 11074225 non-trivial and non-equivalent step sets in
(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.Comment: Final version, to appear in Annals of Combinatorics. 36 page
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