Abstract
The main objective of this paper is to analyze symmetric and asymmetric peaks in Dyck paths with air pockets (DAPs). These paths are formed by combining each maximal run of down-steps in ordinary Dyck paths into a larger, single down-step. To achieve this, we present a trivariate generating function that counts the number of DAPs based on their length and the number of symmetric and asymmetric peaks they contain. We determine the total numbers of symmetric and asymmetric peaks across all DAPs, providing an asymptotic for the ratio of these two quantities. Recursive relations and closed formulas are provided for the number of DAPs of length n, as well as for the total number of symmetric peaks, weight of symmetric peaks, and height of symmetric peaks. Furthermore, a recursive relation is established for the overall number of DAPs, similar to that for classic Dyck paths. A DAP is said to be non-decreasing if the sequence of ordinates of all local minima forms a non-decreasing sequence. In the last section, we focus on the sets of non-decreasing DAPs and examine their symmetric and asymmetric peaks.
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References
Asakly, W.: Enumerating symmetric and non-symmetric peaks in words. Online J. Anal. Comb. 13 (2018). https://api.semanticscholar.org/CorpusID:53611358
Barcucci, E., Del Lungo, A., Fezzi, A., Pinzani, R.: Nondecreasing Dyck paths and \(q\)-Fibonacci numbers. Discrete Math. 170, 211–217 (1997). https://doi.org/10.1016/S0012-365X(97)82778-1
Baril, J.-L., Kirgizov, S., Maréchal, R., Vajnovszki, V.: Enumeration of Dyck paths with air pockets. J. Integer Seq. 26 (2023), Article 23.3.2. https://cs.uwaterloo.ca/journals/JIS/VOL26/Kirgizov/kirg5.pdf
Baril, J.-L., Kirgizov, S., Maréchal, R., Vajnovszki, V.: Grand Dyck paths with air pockets. Art Discrete Appl. Math. 7 (2024) #P1.07. https://doi.org/10.26493/2590-9770.1587.b2a
Baril, J.-L., Barry, P.: Two kinds of partial Motzkin paths with air pockets. Ars Math. Contemp. (2023). https://doi.org/10.26493/1855-3974.3035.6ac
Baril, J.-L., Ramírez, J.L.: Knight’s paths towards Catalan numbers. Discrete Math. 346, 113372 (2023). https://doi.org/10.1016/j.disc.2023.113372
Cameron, N., Sullivan, E.: Peakless Motzkin paths with marked level steps at fixed height. Discrete Math. 344, 112154 (2021). https://doi.org/10.1016/j.disc.2020.112154
Czabarka, E., Flórez, R., Junes, L.: Some enumerations on non-decreasing Dyck paths. Electron. J. Combin. 22 (2015), # P1.3, 1–22. https://doi.org/10.37236/3941
Czabarka, E., Flórez, R., Junes, L., Ramírez, J.L.: Enumerations of peaks and valleys on non-decreasing Dyck paths. Discrete Math. 341, 2789–2807 (2018). https://doi.org/10.1016/j.disc.2018.06.032
Deutsch, E., Prodinger, H.: A bijection between directed column-convex polyominoes and ordered trees of height at most three. Theoret. Comput. Sci. 307, 319–325 (2003). https://doi.org/10.1016/S0304-3975(03)00222-6
Deutsch, E.: Dyck path enumeration. Discrete Math. 204, 167–202 (1999). https://doi.org/10.1016/S0012-365X(98)00371-9
Donaghey, R.: Automorphisms on Catalan trees and bracketings. J. Combin. Theory Ser. B 29, 75–90 (1980). https://doi.org/10.1016/0095-8956(80)90045-3
Elizalde, S.: Symmetric peaks and symmetric valleys in Dyck paths. Discrete Math. 34, 112364 (2021). https://doi.org/10.1016/j.disc.2021.112364
Elizalde, S., Flórez, R., Ramírez, J.L.: Enumerating symmetric peaks in nondecreasing Dyck paths. Ars Math. Contemp. 21, 1–23 (2021)
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Flórez, R., Ramírez, J.L.: Enumerating symmetric pyramids in Motzkin paths. Ars Math. Contemp. 23 (2023), # 4.06. https://doi.org/10.26493/1855-3974.3061.5bf
: Flórez, R., Junes, L., Ramírez, J. L.: Counting asymmetric weighted pyramids in non-decreasing Dyck paths. Australas. J. Combin. 79, 123–140 (2021). https://ajc.maths.uq.edu.au/pdf/79/ajc_v79_p123.pdf
Flórez, R., Ramírez, J.L.: Enumerating symmetric and asymmetric peaks in Dyck paths. Discrete Math. 343, 112118 (2020). https://doi.org/10.1016/j.disc.2020.112118
Flórez, R., Ramírez, J.L., Velandia, F.A., Villamizar, D.: A refinement of Dyck paths: a combinatorial approach. Discrete Math. Algorithms Appl. 14, 2250026 (2022). https://doi.org/10.1142/S1793830922500264
Krinik, A., Rubino, G., Marcus, D., Swift, R.J., Kasfy, H., Lam, H.: Dual processes to solve single server systems. J. Statist. Plann. Inference 135, 121–147 (2005). https://doi.org/10.1016/j.jspi.2005.02.010
Merlini, D., Sprugnoli, R., Verri, M.C.: Lagrange inversion: when and how. Acta Appl. Math. 94, 233–249 (2006). https://doi.org/10.1007/s10440-006-9077-7
Orlov, A.G.: On asymptotic behavior of the Taylor coefficients of algebraic functions. Sib. Math. J. 25(5), 1002–1013 (1994). https://doi.org/10.1007/BF02104578
Sloane, N.J.A.: The On-line Encyclopedia of Integer Sequences. Available electronically at http://oeis.org
Prodinger, H.: Words, dyck paths, trees, and bijections. In: Words, Semigroups, and Transductions, pp. 369–379. World Scientific (2001). https://doi.org/10.1142/9789812810908_0028
Stein, P.R., Waterman, M.S.: On some new sequences generalizing the Catalan and Motzkin numbers. Discrete Math. 26, 261–272 (1979). https://doi.org/10.1016/0012-365X(79)90033-5
Sun, Y., Shi, W., Zhao, D.: Symmetric and asymmetric peaks or valleys in (partial) Dyck paths. Enumer. Combin. Appl. 2 (2022). article #S2R24. https://doi.org/10.54550/ECA2022V2S3R24
Acknowledgements
The authors are grateful to the anonymous referees for helpful comments. The first author was partially supported by University of Burgundy, France. The second author was partially supported by the Citadel Foundation, Charleston SC. The third author was partially supported by Universidad Nacional de Colombia.
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Baril, JL., Flórez, R. & Ramírez, J.L. Symmetries in Dyck paths with air pockets. Aequat. Math. (2024). https://doi.org/10.1007/s00010-024-01043-7
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DOI: https://doi.org/10.1007/s00010-024-01043-7