Abstract
We yield bivariate generating function for the number of n-length partial skew Dyck paths with air pockets (DAPs) ending at a given ordinate. We also give an asymptotic approximation for the average ordinate of the endpoint in all partial skew DAPs of a given length. Similar studies are made for two subclasses of skew DAPs, namely valley-avoiding and zigzagging, valley-avoiding skew DAPs. We express these results as Riordan arrays. Finally, we present two one-to-one correspondences with binary words avoiding the patterns 00 and 0110, and palindromic compositions with parts in \(\{2,1,3,5,7,\ldots \}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baril, J.-L., Barry, P.: Two kinds of partial Motzkin paths with air pockets. Ars Math. Contemp. (2024). https://doi.org/10.26493/1855-3974.3035.6ac
Baril, J.-L., Kirgizov, S., Maréchal, R., Vajnovszki, V.: Enumeration of Dyck paths with air pockets. J. Integer Seq. 26, Article 23.3.2 (2023)
Baril, J.-L., Kirgizov, S., Maréchal, R., Vajnovszki, V.: Grand Dyck paths with air pockets. Art Discrete Appl. Math. 7, Article #P1.07 (2024)
Baril, J.-L., Ramirez, J.L.: Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises, Submitted (2023)
Barry, P.: Riordan Arrays: A Primer. Logic Press (2017)
Burstein, A., Shapiro, L.W.: Pseudo-involutions in the Riordan group. J. Integer Seq. 25, Article 22.3.6 (2022)
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Hoggat, V.E., Jr., Bicknell, M.: Palindromic compositions. Fibonacci Q. 13(4), 350–356 (1975)
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)
Orlov, A.G.: On asymptotic behavior of the Taylor coefficients of algebraic functions. Sib. Math. J. 25, 1002–1013 (1994)
Prodinger, H.: The Kernel method: A collection of examples, Sém. Lothar. Combin., B50f (2004)
Shapiro, L.W., Getu, S., Woan, W., Woodson, L.: The Riordan group. Discrete Appl. Math. 34, 229–239 (1991)
Sloane, N.J.A.: The On-line Encyclopedia of Integer Sequences, available electronically at http://oeis.org
Author information
Authors and Affiliations
Contributions
All authors wrote and reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interests
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
