Analytic Combinatorics of Lattice Paths with Forbidden Patterns, the Vectorial Kernel Method, and Generating Functions for 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. Keywords Lattice paths • Dyck paths • Motzkin paths • Łukasiewicz paths • Pattern avoidance • Autocorrelation • Finite automata • Markov chains • Pushdown automata • Generating functions • Wiener-Hopf factorization • Kernel method • Asymptotic analysis • Gaussian limit law • Borges' theorem We dedicate this article to the memory of Philippe Flajolet, our cheerful and inspiring mentor, founder of analytic combinatorics.

Computational Aspects of Sturdy and Flimsy Numbers

“…Our method starts with a pushdown automaton (PDA) recognizing the k-flimsy numbers, and by a series of steps, it is converted into an asymptotic series expansion for the number of k-flimsy numbers with N bits. Previously, the basic approach has been used for a wide variety of combinatorial enumerations; see, for example, [4,5,2,3]. We have implemented all the steps, and the flow of control is explained in the diagram below.…”

Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats

“…Study of Dyck paths avoiding runs of given lengths (as well as peaks and valleys avoiding certain heights) is conducted in [EZ] with automated computational methods and in [BDB] with formal grammar techniques. Avoiding a pattern in lattice paths was considered in [ABBG20] and later for multiple patterns in [AB20,ABR20]. These latter works make use of the so called vectorial kernel method which modifies the kernel method to allow for pattern avoidance.…”

Counting embeddings of rooted trees into families of rooted trees

“…This is a vast field where many different types of structures have been considered. We only mention subgraph avoidance (and characterizing whole graph classes like series-parallel or planar graphs in that way) or subgraph counts in random graphs (see [25,1]), pattern avoidance in permutations (see [4]) or in trees (see [9]), or pattern avoidance in lattice paths and words, where many particular patterns have been treated separately (see [8] or the introduction of [2] for a survey) and eventually put under a unifying umbrella in [2].…”