Abstract
This article presents a powerful method for the enumeration of pattern-avoiding words generated by an automaton or a context-free grammar. It relies on methods of analytic combinatorics, and on a matricial generalization of the kernel method. Due to classical bijections, this also gives the generating functions of many other structures avoiding a pattern (e.g., trees, integer compositions, some permutations, directed lattice paths, and more generally words generated by a push-down automaton). We focus on the important class of languages encoding lattice paths, sometimes called generalized Dyck paths. We extend and refine the study by Banderier and Flajolet in 2002 on lattice paths, and we unify several dozens of articles which investigated patterns like peaks, valleys, humps, etc., in Dyck and Motzkin words. Indeed, we obtain formulas for the generating functions of walks/bridges/meanders/excursions avoiding any fixed word (a pattern). We show that the autocorrelation polynomial of this forbidden pattern (as introduced by Guibas and Odlyzko in 1981, in the context of regular expressions) still plays a crucial role for our algebraic functions. We identify a subclass of patterns for which the formulas have a neat form. 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. Our approach also opens the door to establish the universal asymptotics and limit laws for the occurrence of patterns in more general algebraic languages.
Work funded by the Austrian Science Fund (FWF) project SFB F50 “Algorithmic and Enumerative Combinatorics”.
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Notes
- 1.
Some weights (or probabilities, or multiplicities) could be associated with each jump, but we omit them in this article to keep readability. All the proofs would be similar.
- 2.
A similar notion also appears in the work of Schützenberger on synchronizing words [34].
References
Asinowski, A., Bacher, A., Banderier, C., Gittenberger, B.: Analytic combinatorics of lattice paths with forbidden patterns: asymptotic aspects. In preparation (2017)
Bacher, A., Bousquet-Mélou, M.: Weakly directed self-avoiding walks. J. Comb. Theory Ser. A 118(8), 2365–2391 (2011)
Banderier, C., Drmota, M.: Formulae and asymptotics for coefficients of algebraic functions. Comb. Probab. Comput. 24(1), 1–53 (2015)
Banderier, C., Flajolet, P.: Basic analytic combinatorics of directed lattice paths. Theoret. Comput. Sci. 281(1–2), 37–80 (2002)
Banderier, C., Gittenberger, B.: Analytic combinatorics of lattice paths: enumeration and asymptotics for the area. Discrete Math. Theor. Comput. Sci. Proc. AG: 345–355 (2006)
Banderier, C., Nicodème, P.: Bounded discrete walks. Discrete Math. Theor. Comput. Sci. AM: 35–48, 2010
Banderier, C., Wallner, M.: The kernel method for lattice paths below a rational slope. In: Lattice Paths Combinatorics And Applications, Developments in Mathematics Series, pp. 1–36. Springer (2018)
Baril, J.-L.: Avoiding patterns in irreducible permutations. Discret. Math. Theor. Comput. Sci. 17(3), 13–30 (2016)
Baril, J.-L., Petrossian, A.: Equivalence classes of Motzkin paths modulo a pattern of length at most two. J. Integer Seq. 18(11), 1–17 (2015). Article no. 15.7.1
Bernini, A., Ferrari, L., Pinzani, R., West, J.: Pattern-avoiding Dyck paths. Discrete Math. Theor. Comput. Sci. Proc. 683–694 (2013)
Bóna, M., Knopfmacher, A.: On the probability that certain compositions have the same number of parts. Ann. Comb. 14(3), 291–306 (2010)
Bousquet-Mélou, M.: Rational and algebraic series in combinatorial enumeration. In: International Congress of Mathematicians, vol. III, pp. 789–826. EMS (2006)
Bousquet-Mélou, M., Jehanne, A.: Polynomial equations with one catalytic variable, algebraic series and map enumeration. J. Comb. Theory Ser. B 96(5), 623–672 (2006)
Brennan, C., Mavhungu, S.: Visits to level \(r\) by Dyck paths. Fund. Inform. 117(1–4), 127–145 (2012)
Chomsky, N., Schützenberger, M.-P.: The algebraic theory of context-free languages. In: Computer Programming and Formal Systems, pp. 118–161, North-Holland, Amsterdam (1963)
Dershowitz, N., Zaks, S.: More patterns in trees: up and down, young and old, odd and even. SIAM J. Discret. Math. 23(1), 447–465 (2008/2009)
Deutsch, E., Shapiro, L.W.: A bijection between ordered trees and 2-Motzkin paths and its many consequences. Discret. Math. 256(3), 655–670 (2002)
Ding, Y., Du, R.R.X.: Counting humps in Motzkin paths. Discret. Appl. Math. 160(1–2), 187–191 (2012)
Duchon, P.: On the enumeration and generation of generalized Dyck words. Discret. Math. 225(1–3), 121–135 (2000)
Eu, S.-P., Liu, S.-C., Yeh, Y.-N.: Dyck paths with peaks avoiding or restricted to a given set. Stud. Appl. Math. 111(4), 453–465 (2003)
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Guibas, L.J., Odlyzko, A.M.: String overlaps, pattern matching, and nontransitive games. J. Combin. Theory Ser. A 30(2), 183–208 (1981)
Hofacker, I.L., Reidys, C.M., Stadler, P.F.: Symmetric circular matchings and RNA folding. Discret. Math. 312(1), 100–112 (2012)
Jin, E.Y., Reidys, C.M.: Asymptotic enumeration of RNA structures with pseudoknots. Bull. Math. Biol. 70(4), 951–970 (2008)
Labelle, J., Yeh, Y.N.: Generalized Dyck paths. Discret. Math. 82(1), 1–6 (1990)
Manes, K., Sapounakis, A., Tasoulas, I., Tsikouras, P.: Strings of length 3 in Grand-Dyck paths and the Chung-Feller property. Electron. J. Combin. 19(2), 10 (2012). Paper 2
Manes, K., Sapounakis, A., Tasoulas, I., Tsikouras, P.: Equivalence classes of ballot paths modulo strings of length 2 and 3. Discret. Math. 339(10), 2557–2572 (2016)
Mansour, T.: Statistics on Dyck paths. J. Integer Seq. 9, 1–17 (2006). Article no. 06.1.5
Mansour, T., Shattuck, M.: Counting humps and peaks in generalized Motzkin paths. Discret. Appl. Math. 161(13–14), 2213–2216 (2013)
Merlini, D., Rogers, D.G., Sprugnoli, R., Verri, M.C.: Underdiagonal lattice paths with unrestricted steps. Discret. Appl. Math. 91(1–3), 197–213 (1999)
Park, Y., Park, S.K.: Enumeration of generalized lattice paths by string types, peaks, and ascents. Discret. Math. 339(11), 2652–2659 (2016)
Righi, C.: Number of “udu”s of a Dyck path and ad-nilpotent ideals of parabolic subalgebras of \(sl_{l+1}\)(\({\mathbb{C}}\)). Sém. Lothar. Combin. 59, 17 (2007/2010). Article no. B59c
Schützenberger, M.-P.: On context-free languages and push-down automata. Inf. Control 6, 246–264 (1963)
Schützenberger, M.-P.: On the synchronizing properties of certain prefix codes. Inf. Control 7, 23–36 (1964)
Stanley, R.P.: Enumerative Combinatorics: Volume 1. Cambridge Studies in Advanced Mathematics, vol. 49, 2nd edn. Cambridge University Press, Cambridge (2011)
Sun, Y.: The statistic “number of udu’s” in Dyck paths. Discret. Math. 287(1–3), 177–186 (2004)
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Asinowski, A., Bacher, A., Banderier, C., Gittenberger, B. (2018). Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects. In: Klein, S., Martín-Vide, C., Shapira, D. (eds) Language and Automata Theory and Applications. LATA 2018. Lecture Notes in Computer Science(), vol 10792. Springer, Cham. https://doi.org/10.1007/978-3-319-77313-1_15
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