The Encoding of Natural Numbers as Nested Parentheses Strings with Associated Probability Distributions

TL;DR

This paper introduces an efficient method to encode natural numbers as nested parentheses strings using Catalan numbers, exploring associated probability distributions and potential applications in data encoding and recursive functions.

Contribution

It presents a novel encoding scheme for natural numbers based on nested parentheses and Catalan numbers, with analysis of resulting probability distributions.

Findings

Encoding is efficient and recursive.

Probability distributions are characterized and analyzed.

Potential applications in data encoding and recursive function theory.

Abstract

We provide an efficient encoding of the natural numbers {0,1,2,3,...} as strings of nested parentheses {(),(()),(()()),((())),...}, or considered inversely, an efficient enumeration of such strings. The technique is based on the recursive definition of the Catalan numbers. The probability distributions arising from this encoding are explored. Applications of this encoding to prefix-free data encoding and recursive function theory are briefly considered.