Recently we were solving problems from past Indian ICPC regional . We weren't able to solve these 2 problems , and I couldn't find any editorial either . It would be really helpful if you guys can give me some hints to these problems.

Chemicals

Description

There are N bottles each having a different chemical. For each chemical i, you have determined C[i], which means that mixing chemicals i and C[i] causes an explosion. You have K distinct boxes. In how many ways can you divide the N chemicals into those boxes such that no two chemicals in the same box can cause an explosion together?

Constraints

• T ≤ 50
• 2 ≤ N ≤ 100
• 2 ≤ K ≤ 1000

My Ideas

I thought of modelling the given dependencies as a graph and we are asked to find the number of ways to partition the graph into independent sets . But I realised that counting independent sets is intractable , so there must be a much more efficient or different way to solve this problem .

Dividing Stones [Solved]

Description

There are N stones, which can be divided into some piles arbitrarily. Let the value of each division be equal to the product of the number of stones in all the piles modulo P. How many possible distinct values are possible for a given N and P?

Constraints

• T ≤ 20
• 2 ≤ N ≤ 70
• 2 ≤ P ≤ 10⁹

Idea (thanks to ABalobanov)

We represent every partition as p₁^a₁ * p₂^a₂ * ... * p_k^a_k, where p₁, p₂, ..., p_k are primes up to 70. We can achieve this value for a given n iff a₁ * p₁ + a₂ * p₂ + ... + a_k * p_k ≤ n . So the partition looks like p₁ + p₁...a₁ times + ... + p_k + p_k...a_k times , if the value is less than n we can add extra 1 .

My solution