# Functions That Enumerate Pairs

Functions That Enumerate Pairs

A function 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 , , that satisfy the relations , , , with =i. One such triple is defined by ; , where is the largest integer for which .

f

i

k

l

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

k(i(x,y))=x

l(i(x,y))=y

-1

f

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

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

u

u(u+1)/2≤z;l(z)=u-k(z)