Enumerating symmetric and asymmetric peaks in Dyck paths. (English) Zbl 1448.05106
Summary: A Dyck path is a lattice path in the first quadrant of the \(x y\)-plane that starts at the origin and ends on the \(x\)-axis and has even length. This is composed of the same number of North-East \((X)\) and South-East \((Y)\) steps. A peak and a valley of a Dyck path are the subpaths \(XY\) and \(YX\), respectively. A peak is symmetric if the valleys determining the maximal pyramid containing the peak are at the same level. In this paper we give recursive relations, generating functions, as well as closed formulas to count the total number of symmetric peaks and asymmetric peaks. We also give an asymptotic expansion for the number of symmetric peaks.
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OEISReferences:
| [1] | Asakly, W., Enumerating symmetric and non-symmetric peaks in words, Online J. Anal. Comb., 13 (2018) · Zbl 1394.05003 |
| [2] | Bacher, A.; Bernini, A.; Ferrari, L.; Gunby, B.; Pinzani, R.; West, J., The Dyck pattern poset, Discrete Math., 321, 12-23 (2014) · Zbl 1281.05009 |
| [3] | Barcucci, E.; Del Lungo, A.; Fezzi, A.; Pinzani, R., Nondecreasing Dyck paths and \(q\)-Fibonacci numbers, Discrete Math., 170, 211-217 (1997) · Zbl 0886.05012 |
| [4] | Baril, J.-L.; Genestier, R.; Kirgizov, S., Pattern distributions in Dyck paths with a first return decomposition constrained by height, Discrete Math., 343, Article 111995 pp. (2020) · Zbl 1443.05009 |
| [5] | Blecher, A.; Brennan, C.; Knopfmacher, A., Water capacity of Dyck paths, Adv. Appl. Math., 112, Article 101945 pp. (2020) · Zbl 1427.05005 |
| [6] | Czabarka, E.; Flórez, R.; Junes, L., Some enumerations on non-decreasing dyck paths, Electron. J. Combin., 22, 1-22 (2015), # P1.3 · Zbl 1305.05009 |
| [7] | Czabarka, E.; Flórez, R.; Junes, L.; Ramírez, J., Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math., 341, 2789-2807 (2018) · Zbl 1393.05154 |
| [8] | Deutsch, E., Dyck path enumeration, Discrete Math., 204, 167-202 (1999) · Zbl 0932.05006 |
| [9] | Elezović, N., Asymptotic expansions of central binomial coefficients and Catalan numbers, J. Integer Seq., 17, Article 14.2.1 pp. (2014) · Zbl 1317.11023 |
| [10] | Luke, Y. L., The Special Functions and their Approximations, Vol. I (1969), Academic Press: Academic Press New York · Zbl 0193.01701 |
| [11] | Olver, F., Asymptotics and Special Functions (1997), A.K. Peters · Zbl 0982.41018 |
| [12] | Sapounakis, A.; Tasoulas, I.; Tsikouras, P. P., Counting strings in Dyck paths, Discrete Math., 307, 2909-2924 (2007) · Zbl 1127.05005 |
| [13] | N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, http://oeis.org/. · Zbl 1044.11108 |
| [14] | Stanley, R., Catalan Numbers (2015), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1317.05010 |
| [15] | Zhuang, Y., A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Math., 341, 358-379 (2018) · Zbl 1376.05009 |
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