We define the set S as $S = \{(a_1, f_1), (a_2, f_2), ..., (a_i, f_i)\}$, where each fi is the frequency that ai is repeated in the multiset T.
For example, given $S = \{(a_1, 1), (a_2, 2), (a_3, 1)\}$ results in the multiset T={a1, a2, a2, a3}. Now, we can segmentize the multiset into six different cases in which each segment includes distinct ai as follows:
$1:\{\{a_1,a_2\},\{a_2,a_3\}\}$
$2:\{\{a_1,a_2,a_3\},\{a_2\}\}$
3:$\{\{a1,a2\},\{a2\},\{a3\}\}$
$4:\{\{a_1\},\{a_2\},\{a_2,a_3\}\}$
$5:\{\{a_1,a_3\},\{a_2\},\{a_2\}\}$
$6:\{\{a_1\},\{a_2\},\{a_2\},\{a_3\}\}$
Note that the order of elements in each segment does not matter, e.g., $\{a_1,a_2\} = \{a_2,a_1\}$
Note that each segment has distinct elements.
Now, I am looking for a formula that counts the number of cases (six cases in the example) where we can segmentize multiset T.
The code that creates these cases is in the below link, but I am looking for a formula that mathematically gives us the number of cases.