Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects

2018 ◽  
pp. 195-206 ◽  
Author(s):  
Andrei Asinowski ◽  
Axel Bacher ◽  
Cyril Banderier ◽  
Bernhard Gittenberger
Keyword(s):  
Lattice Paths ◽  
Analytic Combinatorics ◽  
Forbidden Patterns

Analytic Combinatorics of Lattice Paths with Forbidden Patterns, the Vectorial Kernel Method, and Generating Functions for Pushdown Automata

Algorithmica ◽  
2019 ◽  
Vol 82 (3) ◽  
pp. 386-428 ◽  
Author(s):  
Andrei Asinowski ◽  
Axel Bacher ◽  
Cyril Banderier ◽  
Bernhard Gittenberger
Keyword(s):  
Kernel Method ◽  
Lattice Paths ◽  
Integer Sequences ◽  
Unified Framework ◽  
Analytic Combinatorics ◽  
On Line ◽  
Forbidden Patterns ◽  
Motzkin Paths ◽  
Self Avoiding Walks ◽  
Pushdown Automata

Abstract In this article we develop a vectorial kernel method—a powerful method which solves in a unified framework all the problems related to the enumeration of words generated by a pushdown automaton. We apply it for the enumeration of lattice paths that avoid a fixed word (a pattern), or for counting the occurrences of a given pattern. We unify results from numerous articles concerning patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. This refines the study by Banderier and Flajolet from 2002 on enumeration and asymptotics of lattice paths: we extend here their results to pattern-avoiding walks/bridges/meanders/excursions. We show that the autocorrelation polynomial of this forbidden pattern, as introduced by Guibas and Odlyzko in 1981 in the context of rational languages, still plays a crucial role for our algebraic languages. En passant, our results give the enumeration of some classes of self-avoiding walks, and prove several conjectures from the On-Line Encyclopedia of Integer Sequences. Finally, we also give the trivariate generating function (length, final altitude, number of occurrences of the pattern p), and we prove that the number of occurrences is normally distributed and linear with respect to the length of the walk: this is what Flajolet and Sedgewick call an instance of Borges’s theorem.

scholarly journals Basic analytic combinatorics of directed lattice paths

2002 ◽  
Vol 281 (1-2) ◽  
pp. 37-80 ◽  
Author(s):  
Cyril Banderier ◽  
Philippe Flajolet
Keyword(s):  
Lattice Paths ◽  
Analytic Combinatorics

scholarly journals Analytic Combinatorics of Lattice Paths: Enumeration and Asymptotics for the Area

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Cyril Banderier ◽  
Bernhard Gittenberger
Keyword(s):  
Symmetric Functions ◽  
Kernel Method ◽  
Brownian Bridge ◽  
Lattice Paths ◽  
Reflected Brownian Motion ◽  
Analytic Combinatorics ◽  
Average Area ◽  
First Case ◽  
Finite Set ◽  
International Audience

International audience This paper tackles the enumeration and asymptotics of the area below directed lattice paths (walks on $\mathbb{N}$ with a finite set of jumps). It is a nice surprise (obtained via the "kernel method'') that the generating functions of the moments of the area are algebraic functions, expressible as symmetric functions in terms of the roots of the kernel. For a large class of walks, we give full asymptotics for the average area of excursions ("discrete'' reflected Brownian bridge) and meanders ("discrete'' reflected Brownian motion). We show that drift is not playing any role in the first case. We also generalise previous works related to the number of points below a path and to the area between a path and a line of rational slope.

scholarly journals Pattern avoidance in forests of binary shrubs

2016 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
David Bevan ◽  
Derek Levin ◽  
Peter Nugent ◽  
Jay Pantone ◽  
Lara Pudwell ◽  
...  
Keyword(s):  
Growth Rate ◽  
Generating Function ◽  
Minimal Polynomial ◽  
Partially Ordered Sets ◽  
Pattern Avoidance ◽  
Lattice Paths ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Analytic Combinatorics ◽  
Partially Ordered

We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerations by exhibiting bijections with certain lattice paths bounded above by the line $y=\ell x$, for some $\ell\in\mathbb{Q}^+$, one of these being the celebrated Duchon's club paths with $\ell=2/3$. In the remaining case, we use the machinery of analytic combinatorics to determine the minimal polynomial of its generating function, and deduce its growth rate.

scholarly journals Combinatorics on lattice paths in strips

2021 ◽  
Vol 94 ◽  
pp. 103310
Author(s):  
Nancy S.S. Gu ◽  
Helmut Prodinger
Keyword(s):  
Lattice Paths

Constraints for generating graphs with imposed and forbidden patterns: an application to molecular graphs

Constraints ◽  
2019 ◽  
Vol 25 (1-2) ◽  
pp. 1-22
Author(s):  
Mohamed Amine Omrani ◽  
Wady Naanaa
Keyword(s):  
Molecular Graphs ◽  
Forbidden Patterns

Enumeration of Lattice paths with infinite types of steps and the Chung-Feller property

2021 ◽  
Vol 344 (8) ◽  
pp. 112452
Author(s):  
Lin Yang ◽  
Sheng-Liang Yang
Keyword(s):  
Lattice Paths ◽  
Feller Property

scholarly journals A Quantitative Study of Pure Parallel Processes

10.37236/3977 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
O. Bodini ◽  
A. Genitrini ◽  
F. Peschanski
Keyword(s):  
Quantitative Study ◽  
Random Sampling ◽  
Linear Time ◽  
Time Algorithm ◽  
Point Of View ◽  
Total Size ◽  
Combinatorial Explosion ◽  
Analytic Combinatorics ◽  
Parallel Processes ◽  
Uniform Random Sampling

In this paper, we study the interleaving – or pure merge – operator that most often characterizes parallelism in concurrency theory. This operator is a principal cause of the so-called combinatorial explosion that makes the analysis of process behaviours e.g. by model-checking, very hard – at least from the point of view of computational complexity. The originality of our approach is to study this combinatorial explosion phenomenon on average, relying on advanced analytic combinatorics techniques. We study various measures that contribute to a better understanding of the process behaviours represented as plane rooted trees: the number of runs (corresponding to the width of the trees), the expected total size of the trees as well as their overall shape. Two practical outcomes of our quantitative study are also presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random sampling of concurrent runs. These provide interesting responses to the combinatorial explosion problem.

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