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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.
Highlights
Combinatorial structures having a rational or an algebraic generating function play a key role in many fields: computer science, computational geometry, bioinformatics (RNA structure, pattern matching), number theory, probability theory (Markov chains, directed random walks); see e.g. [8,22,42,78]
We presented a unifying way which gives the generating functions and asymptotics of all families of lattice paths with a forbidden pattern, and we proved that the number of occurrences of a given pattern is normally distributed
It is nice that our approach gives a method to solve in an efficient way the question of the enumeration and asymptotics of words generated by a pushdown automaton
Summary
Combinatorial structures having a rational or an algebraic generating function play a key role in many fields: computer science (e.g. for the analysis of algorithms involving trees, lists, words), computational geometry (integer points in polytopes, maps, graph decomposition), bioinformatics (RNA structure, pattern matching), number theory (integer compositions, automatic sequences and modular properties, integer solutions of varieties), probability theory (Markov chains, directed random walks); see e.g. [8,22,42,78]. The words generated by these languages are in bijection with directed lattice paths, and we show how these fundamental objects can be enumerated when they have the additional constraint to avoid a given pattern. We extend the study of Banderier and Flajolet by considering lattice paths with step set S that avoid a certain pattern of length , that is, an a priori fixed path p = [a1, a2, . We use the notations W /B/M/E for generating functions enumerating paths constrained to avoid a pattern p. For the four classes of paths and for any set of steps encoded by P(u), we find the generating functions of such lattice paths that avoid a pattern p. We show the formula for the special case when p is a meander itself; in the general case, these formulas might have a different prefactor (see Theorem 3.2 below)
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