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1. Manes, Konstantinos. Απαρίθμηση προτύπων σε μονοπάτια Dyck και Grand-Dyck.

Degree: 2014, University of Piraeus (UNIPI); Πανεπιστήμιο Πειραιώς

The Catalan numbers are considered to be the second most significant numbers in Combinatorics, after the binomial coefficients, because they appear frequently in various combinatorial problems. Professor R. Stanley maintains a record including more than 200 different combinatorial objects which are enumerated by the Catalan numbers, therefore are structurally equivalent. Perhaps, the most popular among these are Dyck paths (or words) and binary trees.Dyck paths are the main object studied in this dissertation. They have a very simple geometrical representation and for that reason they are suitable for studying properties which are then translated into properties of other objects in the Catalan family.Moreover, by introducing various restrictions (parameters), we obtain special categories of Dyck paths which are often equivalent to other known objects, so that any results are also extended to these objects.In this dissertation, we mainly study the parameter “number of occurrences of the string t”, where a string is considered to be any binary word. In the first chapter, the basic definitions and necessary mathematical tools are extensively presented.In the second chapter, we study the parameter “number of occurrences of t at height j”, that is we enumerate Dyck paths, with respect to their length and number of occurrences of t at height j. The result is expressed via the corresponding generating function for any binary word t.In the third chapter, we study the parameters “number of occurrences of t” and “number of occurrences of t at height at least j” in Dyck paths. We again obtain the results via the corresponding generating function for the cases where t is a Dyck prefix or a Dyck suffix and for some other general cases as well. In the fourth chapter, we study the parameter “number of occurrences of t” in Grand-Dyck paths, where t has length 3. In addition, by considering the auxiliary parameter “number of up-steps below zero level”, we obtain in some cases refinements of the Chung-Feller theorem. In the fifth chapter, three new parameters of Dyck paths, not related to strings, are studied and complete enumerative results are obtained. These parameters are defined by refining the well known parameter “number of peaks”. In the sixth chapter, exact as well as asymptotic formulas are presented, for the mean value and variance of the parameters studied in previous chapters.

Οι αριθμοί Catalan θεωρούνται ως οι πιο σημαντικοί αριθμοί της Συνδυαστικής, μετά τους διωνυμικούς συντελεστές, λόγω της εντυπωσιακά συχνής εμφάνισής τους σε διάφορα προβλήματα. Ενδεικτικά, ο R. Stanley διατηρεί αρχείο με περισσότερα από 200 διαφορετικά σύνολα συνδυαστικών αντικείμενων που απαριθμούνται από τους αριθμούς Catalan και άρα είναι πληθικά αλλά και δομικά ισοδύναμα. Τα πιο διαδεδομένα από αυτά είναι ίσως τα μονοπάτια (λέξεις) Dyck και τα δυαδικά δένδρα.Το κεντρικό αντικείμενο μελέτης της διατριβής αυτής είναι τα μονοπάτια Dyck, τα οποία αποτελούν απλά μια αναπαράσταση στο επίπεδο των λέξεων Dyck. Λόγω της απλής και εύληπτης γεωμετρικής…

Subjects/Keywords: Μονοπάτια Dyck; Μονοπάτια Grand-Dyck; Αριθμοί Catalan; ΓΕΝΝΗΤΡΙΕΣ ΣΥΝΑΡΤΗΣΕΙΣ; Συνδυαστική απαρίθμηση; Τύπος αντιστροφής Lagrange; Dyck paths; Grand-Dyck paths; Catalan numbers; GENERATING FUNCTIONS; Combinatorial enumeration; Lagrange inversion formula

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APA (6th Edition):

Manes, K. (2014). Απαρίθμηση προτύπων σε μονοπάτια Dyck και Grand-Dyck. (Thesis). University of Piraeus (UNIPI); Πανεπιστήμιο Πειραιώς. Retrieved from http://hdl.handle.net/10442/hedi/34603

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Manes, Konstantinos. “Απαρίθμηση προτύπων σε μονοπάτια Dyck και Grand-Dyck.” 2014. Thesis, University of Piraeus (UNIPI); Πανεπιστήμιο Πειραιώς. Accessed January 15, 2021. http://hdl.handle.net/10442/hedi/34603.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Manes, Konstantinos. “Απαρίθμηση προτύπων σε μονοπάτια Dyck και Grand-Dyck.” 2014. Web. 15 Jan 2021.

Vancouver:

Manes K. Απαρίθμηση προτύπων σε μονοπάτια Dyck και Grand-Dyck. [Internet] [Thesis]. University of Piraeus (UNIPI); Πανεπιστήμιο Πειραιώς; 2014. [cited 2021 Jan 15]. Available from: http://hdl.handle.net/10442/hedi/34603.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Manes K. Απαρίθμηση προτύπων σε μονοπάτια Dyck και Grand-Dyck. [Thesis]. University of Piraeus (UNIPI); Πανεπιστήμιο Πειραιώς; 2014. Available from: http://hdl.handle.net/10442/hedi/34603

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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