When I slept soundly last night, my dream took me back to the times when I was still a senior school student. I was given this problem during my college entrance exam:

Let $$$n$$$ be a positive integer and $$$S$$$ be a multiset of $$$\{1, 2, ..., n\}$$$. Integer $$$i\,(1 \leq i \leq n)$$$ appears $$$a_i \geq 0$$$ times in $$$S$$$. Also, I was given a positive integer $$$k$$$. A partition of $$$S$$$, namely $$$\cup\limits_{i=1}^t S_i$$$ is valid iff all of the following three conditions hold:

(1) $\