In addition to the OP's 2011 paper with Ž. Jurić:

A New Formula for the Number of Combinations of Permutations of Multisets
Applied Mathematical Sciences, Vol. 5, 2011, no. 18, 875-881

there is a paper by Thomas Wieder, also in 2011:

Generation of All Possible Multiselections from a Multiset
CSCanada Progress in Applied Mathematics, Vol. 2, No. 1, 2011, pp. 61-66

which approaches counting the $k$-combinations of a multiset (sub-multisets of a multiset) in terms of "selection matrices" (similar to contingency tables).

In addition to the references cited in these two papers, Frank Ruskey's 2003 work-in-progress Combinatorial Generation has Sec. 4.5.1 with algorithms based on representing $k$-combinations as weak compositions with restricted parts.

In addition to the OP's 2011 paper with Ž. Jurić:

A New Formula for the Number of Combinations of Permutations of Multisets
Applied Mathematical Sciences, Vol. 5, 2011, no. 18, 875-881

there is a paper by Thomas Wieder, also in 2011:

Generation of All Possible Multiselections from a Multiset
CSCanada Progress in Applied Mathematics, Vol. 2, No. 1, 2011, pp. 61-66

which approaches counting the $k$-combinations of a multiset (sub-multisets of a multiset) in terms of "selection matrices" (similar to contingency tables).

In addition to the references cited in these two papers, Frank Ruskey's 2003 work-in-progress Combinatorial Generation has Sec. 4.5.1 with algorithms based on representing $k$-combinations as weak compositions with restricted parts.