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We provide an efficient encoding of the natural numbers {0,1,2,3,...} asstrings of nested parentheses {(),(()),(()()),((())),...}, or consideredinversely, an efficient enumeration of such strings. The technique is based onthe recursive definition of the Catalan numbers. The probability distributionsarising from this encoding are explored. Applications of this encoding toprefix-free data encoding and recursive function theory are briefly considered.

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