On Two Infinite Families of Pairing Bijections

TL;DR

This paper introduces two general methods for generating infinite families of pairing bijections between natural numbers, using n-adic valuations and characteristic functions, with executable Haskell code provided.

Contribution

It presents novel mechanisms for creating parameterized and diverse pairing bijections, expanding the toolkit for bijective functions between N x N and N.

Findings

Two mechanisms for generating pairing bijections are described.

A large family of pairing bijections can be constructed using these methods.

Executable Haskell code is provided for all functions.

Abstract

We describe two general mechanisms for producing pairing bijections (bijective functions defined from N x N to N). The first mechanism, using n-adic valuations results in parameterized algorithms generating a countable family of distinct pairing bijections. The second mechanism, using characteristic functions of subsets of N provides 2^N distinct pairing bijections. Mechanisms to combine such pairing functions and their application to generate families of permutations of N are also described. The paper uses a small subset of the functional language Haskell to provide type checked executable specifications of all the functions defined in a literate programming style. The self-contained Haskell code extracted from the paper is available at http://logic.cse.unt.edu/tarau/research/2012/infpair.hs .