Let $\Sigma$ be a finite set of symbols with total order. Let $C_k$ be the set of all $k$-multiset (unordered collection of $k$ elements from $S$, with repetition allowed). We can order all the elements of $C_k$ in lexicographic order and index each element in that order i.e a bijective function $f: C_k \to [ 1, |C_k| ]$ can be defined where $f(x)$ will be the position/rank of $x$ in such ordering (Here $|C_k| = \binom{k + |\Sigma| -1}{|\Sigma|-1})$. I am looking for such bijective function which can be computed for any $x \in C_k$ without having sort the elements or to refer to any other element of $C_k$ in any way. Can we build such function?

**UPDATE**:
When repetition is not allowed, i.e. indexing $k$-subsets of a set is solved as follows: arrange $k$ elements in decreasing order. Let such an ordering be $c= c_k, ..., c_1$ ($c_k > c_{k-1}>...> c_1$), then
$f(c) = \sum_{i = 1}^{k} \binom{c_i}{i} $ gives such a bijection. (Combinatorial Number System)

**ADDITION:**
My original question is answered. This gives lexicographical ordering when k-combinations are taken as decreasing sequence. How do we define a function which gives ordering as that of lexicographic order when k-combinations are
taken as increasing. Mind that these two ordering are not 'reverse' of each others. Eg. When $S=\{0,1,2,3\}$ and we want ordering of 2-sets then the increasing ordering would give the ordering as: 01, 02,03, 12, 13, 23; and the decreasing ordering would give then ordering as: 01, 02, 12, 03, 13, 23 .

1more comment