# Total ordering of sets of fixed size

I'm curious if there is a name for this way of ordering finite sets of natural numbers (shown here for the case 3 elements, but can be extended to any number of them):

{0, 1, 2} < {0, 1, 3}
< {0, 2, 3}
< {1, 2, 3}
< {0, 1, 4}
< {0, 2, 4}
< {1, 2, 4}
< {0, 3, 4}
< {1, 3, 4}
< {2, 3, 4}
< {0, 1, 5}
< ...


The sets are generated recursively: increase the highest number and reset all the other elements to the lowest possible numbers, then apply this algorithm recursively to the remaining numbers.

The position within this ordering is given by:

C(x[1], 1) + C(x[2], 2) + C(x[3], 3) + ...


where x[i] is the i-th element in the sorted set and C(n, k) is the binomial coefficient.

Does anyone know of a name for this kind of total ordering? Furthermore, what other common ways are there to order sets containing a fixed number of totally ordered elements?

• In recursion theory and Peano arithmetic there is an ordering of unordered pairs $(x,y) \mapsto ((x+y)^2+3x+y)/2$ (here all of $x<y$, $x=y$, $x>y$ could be possible). – Yuval Filmus Feb 13 '14 at 2:18