Given a valid combination, how to get its index in the sequence of integer partition
This is called ranking combinatiorial objects, in this case combinations. Ranking and unranking algorithms exist for various simple combinatorial objects and even a general ranking algorithm exists for arbitrary combinatorial objects given that computing the number of objects matching certain criteria is feasible and there is a total ordering (eg lexicographic ordering).

For combinations, a reference is Algorithms for Unranking Combinations and Other Related Choice Functions, Zbigniew Kokosinski 1995

Assuming a lexicographic ordering among combinations of $K$ from $N$ the following algorithm can be used (which runs in $O(K)$ time assuming binomial computations are cheap):

Assuming the combination is given as an array (item) of length $K$ and where 

$$item[0]&lt;item[1]&lt;\dots&lt;item[K-1]$$

and each $0 \leq item[i] \leq N-1$

(If you want the combination in reverse order simply change the algorithm below to match in reverse order and skip last step. Also since your combinations take values between $1$ and $N$ you might want to subtract $1$ from below algorithms, where $item[i]$ is used)

$index = 0$

for $i=1$ to $K$:

____ $c = N-1-item[i-1]$

____ $j = K+1-i$

____ if $j \leq c$ then $index \leftarrow index+\binom{c}{j}$

Finally: $index \leftarrow \binom{N}{K}-1-index$

Note: The algorithm above can be modified to rank combinations with repeated/duplicate elements. Ie

$$item[0] \leq item[1] \leq \dots \leq item[K-1]$$

One needs to replace the counts used to represent combinations with repeated elements instead of binomials which count combinations with unique elements.

$index = 0$

$N \leftarrow N+K-1$

for $i=1$ to $K$:

____ $c = N-1-item[i-1]-i+1$

____ $j = K+1-i$

____ if $j \leq c$ then $index \leftarrow index+\binom{c}{j}$

Finally: $index \leftarrow \binom{N}{K}-1-index$

In essense ranking combinations is a bijection from $N,K$ combinations to natural numbers $0 \dots \binom{N}{K}-1$. This defines a combinatorial number system.

Algorithms above are modified to match my use case (for my combinatorics library Abacus) where combinations take values $0 \dots N-1$ and are non-decreasing. For your use case you can modify them to match wikipedia article.