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Abstract
Using techniques from the theories of convex polytopes, lattice paths, and indirect influences on directed manifolds, we construct continuous analogues for the binomial coefficients and the Catalan numbers. Our approach for constructing these analogues can be applied to a wide variety of combinatorial sequences. As an application we develop a continuous analogue for the binomial distribution.
Highlights
In this work we construct continuous analogues for the binomial coefficients and the Catalan numbers
Our constructions are based on the theory of convex polytopes, the theory of lattice paths, and the theory of indirect influences on directed manifolds
We introduce our methodology for finding continuous analogues – applicable to many kinds of combinatorial objects – through the following table: Combinatorial Object
Summary
In this work we construct continuous analogues for the binomial coefficients and the Catalan numbers. Solutions to problem II lead naturally to the construction of continuous analogues for the sequence of natural numbers an as follows: the numbers an count the integral points in the interior of the polytopes Pn ⊆ Rdn , and we can think of the volume vol(Pn) as counting – measuring – points in Pn after the integrality restrictions are lifted. In both cases we begin by decomposing the given sequence of numbers as finite sums over time and patterns, where each summand counts the interior points of a lattice polytope. Once we have an interpretation of each summand as counting interior points of convex polytopes, we define our continuous analogous by removing the integrality restrictions, i.e. we compute volume of polytopes and replace finite sums by countable sums. This work takes part in our program aimed to bring geometric methods to the study of problems arising from the theory of complex networks [9, 11, 14, 15, 18]
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