Advances and Applications in Discrete Mathematics
Volume 20, Issue 2, Pages 219 - 236
(March 2019) http://dx.doi.org/10.17654/DM020020219 |
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PARTIALLY ORDERED SETS AND COMBINATORY OBJECTS OF THE PYRAMIDAL STRUCTURE
O. V. Kuzmin, A. A. Balagura, V. V. Kuzmina and I. A. Khudonogov
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Abstract: An important direction in the study of complex systems that one has to deal with in the most diverse fields of science and technology is to consider them as multi-level systems, or systems with a hierarchical structure. The method of analyzing hierarchies seems more substantiated by solving multicriteria problems with hierarchical structures than the approach based on linear logic. In this case, the process of step-by-step construction of a management solution in a complex hierarchical system can often be interpreted as a trajectory on a lattice describing the corresponding partially ordered set.
In this paper, relating to the development of methods for set-theory analysis of complex systems, combinatorial objects of the pyramidal structure are studied using partially ordered sets.
Basic concepts and basic relations for the studied combinatorial objects are given.
Generalizations of Pascal triangle and Pascal pyramid are considered, as well as important special cases of the generalized Pascal pyramid are A- and B-pyramids, formed by generalized trinomial coefficients of the second kind and the first kind, respectively.
In the study of combinatorial numbers, the pyramidal properties of a function of a partially ordered set are used - the number of extensions to complete (linear) ordering. An algorithm for constructing partially ordered sets and geometric lattices corresponding to the combinatorial objects under study is found.
Relations are obtained that connect generalized trinomial coefficients of the first kind and second kind and the elements of important special cases of the generalized Pascal triangle, generalized Stirling numbers of the second kind and first kind, respectively, constructed on different bases.
This article provides interpretations of the studied combinatorial objects in terms of lattice paths in two-dimensional and three-dimensional Euclidean spaces. Interpretations of the obtained formulas in terms of partially ordered sets are given. |
Keywords and phrases: partially ordered sets, lattice, generalized Pascal pyramids, combinatorial algorithms, generalized trinomial coefficients, generalized Stirling numbers.
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