3 Make my question more specific.

The definition of inversion vector for permutation is very well defined. Each permutation can be mapped to a unique inversion vector. So is there a well defined inversion vector for each multiset permutation, which maintains such one-to-one correspondence? I'm particularly interested in the multiset permutations where the multiplicity of each element is equal.

For permutation, an inversion vector corresponds to a factoradic number, i.e., $i_k∈[0,i−1]$. The decimal number converted from the factoradic number could be used for ranking a permutation. Seems the inversion vector of a multiset permutation looses such good properties as the range of $i_k$ no longer only depends on i ? Or is there a numeral system similar to factoradic numeral system which corresponds to the inversion vector of a multiset permutation?