Sign in to use this feature.

Years

Between: -

Subjects

remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline

Journals

Article Types

Countries / Regions

Search Results (2)

Search Parameters:
Keywords = labeled Dyck path

Order results
Result details
Results per page
Select all
Export citation of selected articles as:
21 pages, 1780 KiB  
Article
Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application
by Yuriy Shablya, Dmitry Kruchinin and Vladimir Kruchinin
Mathematics 2020, 8(6), 962; https://doi.org/10.3390/math8060962 - 12 Jun 2020
Cited by 13 | Viewed by 4539
Abstract
In this paper, we study the problem of developing new combinatorial generation algorithms. The main purpose of our research is to derive and improve general methods for developing combinatorial generation algorithms. We present basic general methods for solving this task and consider one [...] Read more.
In this paper, we study the problem of developing new combinatorial generation algorithms. The main purpose of our research is to derive and improve general methods for developing combinatorial generation algorithms. We present basic general methods for solving this task and consider one of these methods, which is based on AND/OR trees. This method is extended by using the mathematical apparatus of the theory of generating functions since it is one of the basic approaches in combinatorics (we propose to use the method of compositae for obtaining explicit expression of the coefficients of generating functions). As a result, we also apply this method and develop new ranking and unranking algorithms for the following combinatorial sets: permutations, permutations with ascents, combinations, Dyck paths with return steps, labeled Dyck paths with ascents on return steps. For each of them, we construct an AND/OR tree structure, find a bijection between the elements of the combinatorial set and the set of variants of the AND/OR tree, and develop algorithms for ranking and unranking the variants of the AND/OR tree. Full article
(This article belongs to the Special Issue Advances and Novel Approaches in Discrete Optimization)
Show Figures

Figure 1

9 pages, 877 KiB  
Article
Euler–Catalan’s Number Triangle and Its Application
by Yuriy Shablya and Dmitry Kruchinin
Symmetry 2020, 12(4), 600; https://doi.org/10.3390/sym12040600 - 10 Apr 2020
Cited by 6 | Viewed by 3814
Abstract
In this paper, we study such combinatorial objects as labeled binary trees of size n with m ascents on the left branch and labeled Dyck n-paths with m ascents on return steps. For these combinatorial objects, we present the relation of the [...] Read more.
In this paper, we study such combinatorial objects as labeled binary trees of size n with m ascents on the left branch and labeled Dyck n-paths with m ascents on return steps. For these combinatorial objects, we present the relation of the generated number triangle to Catalan’s and Euler’s triangles. On the basis of properties of Catalan’s and Euler’s triangles, we obtain an explicit formula that counts the total number of such combinatorial objects and a bivariate generating function. Combining the properties of these two number triangles allows us to obtain different combinatorial objects that may have a symmetry, for example, in their form or in their formulas. Full article
Show Figures

Figure 1

Back to TopTop