Functions That Enumerate Pairs

z

0

red point 0 ↦ {0,0}

blue point {0,0} ↦ 0

A function

f

that enumerates pairs is a bijective mapping of the set of non-negative integers to the set of all ordered pairs of non-negative integers. Thus there are three functions

i

,

k

,

l

that satisfy the relations

i(k(z),l(z))=z

,

k(i(x,y))=x

,

l(i(x,y))=y

, with

-1

f

=i

. One such triple is defined by

i(x,y)=(x+y)(x+y+1)/2+x

;

k(z)=z-u(u+1)/2

, where

u

is the largest integer for which

u(u+1)/2≤z;l(z)=u-k(z)

.