435 research outputs found
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
Staircase polygons: moments of diagonal lengths and column heights
We consider staircase polygons, counted by perimeter and sums of k-th powers
of their diagonal lengths, k being a positive integer. We derive limit
distributions for these parameters in the limit of large perimeter and compare
the results to Monte-Carlo simulations of self-avoiding polygons. We also
analyse staircase polygons, counted by width and sums of powers of their column
heights, and we apply our methods to related models of directed walks.Comment: 24 pages, 7 figures; to appear in proceedings of Counting Complexity:
An International Workshop On Statistical Mechanics And Combinatorics, 10-15
July 2005, Queensland, Australi
Area limit laws for symmetry classes of staircase polygons
We derive area limit laws for the various symmetry classes of staircase
polygons on the square lattice, in a uniform ensemble where, for fixed
perimeter, each polygon occurs with the same probability. This complements a
previous study by Leroux and Rassart, where explicit expressions for the area
and perimeter generating functions of these classes have been derived.Comment: 18 pages, 3 figure
A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape
Tableau sequences of bounded height have been central to the analysis of
k-noncrossing set partitions and matchings. We show here that familes of
sequences that end with a row shape are particularly compelling and lead to
some interesting connections. First, we prove that hesitating tableaux of
height at most two ending with a row shape are counted by Baxter numbers. This
permits us to define three new Baxter classes which, remarkably, do not
obviously possess the antipodal symmetry of other known Baxter classes. We then
conjecture that oscillating tableau of height bounded by k ending in a row are
in bijection with Young tableaux of bounded height 2k. We prove this conjecture
for k at most eight by a generating function analysis. Many of our proofs are
analytic in nature, so there are intriguing combinatorial bijections to be
found.Comment: 10 pages, extended abstrac
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