These two functions provide a one-to-one correspondence between the non-negative integers and all vectors composed of n non-negative integers.

The one-to-one correspondence is defined as follows.  View all vectors of n non-negative integers as exponent vectors on n variables. Therefore, for each vector, there is a corresponding monomial.  Collect all such monomials and order them by increasing total degree.  Resolve ties by ordering monomials of the same degree in lexicographic order. This gives a canonical ordering.

Given a vector l of n non-negative integers, the corresponding integer m is its index in this canonical ordering.  The function vectoint(l) computes and returns this integer m.

Given a non-negative integer m, the corresponding vector l is the m^th vector in this canonical ordering of vectors of length n.  The function inttovec(m, n) computes and returns this vector l.

Here is a sample canonical ordering where n is 3:

The command with(combinat,vectoint) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{combinat}\right)\:$

$\mathrm{vectoint}\left(\left[1\,0\,1\right]\right)$

$6$

$\mathrm{inttovec}\left(6\,3\right)$

$\left[1\,0\,1\right]$