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 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.
Highlights
Combinatorial objects such as permutations, combinations, partitions, graphs, trees, paths, etc.play an important role in mathematics and computer science and have many applications in practice
Sometimes combinatorial sets can be described by using number triangles, for example, for their enumerating
For the presented combinatorial objects, we have obtained an explicit formula for their enumerating
Summary
Combinatorial objects such as permutations, combinations, partitions, graphs, trees, paths, etc. Knuth [1] gave an overview of the formation and development of the direction related to designing combinatorial algorithms In this field of mathematics, the following tasks are distinguished: enumerating, listing, and generating combinatorial objects. To define a number triangle, it is necessary to specify the rules for generating elements Tn,m of this triangle It can be some expression for Tn,m in the form of an explicit formula or a recurrence relation. We study labeled binary trees with ascents on the left branch and labeled Dyck paths with ascents on return steps For these combinatorial objects, we obtain an explicit formula that counts the total number of such objects and a bivariate generating function. We present the relation of the obtained number triangle to Catalan’s and Euler’s triangles and call the obtained number triangle as Euler–Catalan’s triangle and the elements of this triangle as the Euler–Catalan numbers
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