Euler–Catalan’s Number Triangle and Its Application

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.

Combinatorial Generation Algorithms for Directed Lattice Paths

“…Another generalization of Dyck paths related to catastrophes was considered in [10], where generating functions for enumerating such lattice paths and asymptotic approximations for their coefficients were obtained. The enumeration of labeled Dyck paths with ascents on return steps can be found in [11]. Moreover, there are a large number of other special cases of lattice paths, such as Delannoy paths [12], Schroder paths [13], Motzkin paths [14], Riordan paths [15], Lukasiewicz paths [16], etc.…”