Special Permutations
Memoization will optimize the solution by avoiding redundant calculations. Here’s the improved algorithm using memoization:
 Use a recursive function
dfs
that takes two parameters: a bitmask representing which numbers in
nums
have already been used in the current permutation.  the last number added to the current permutation.
 a bitmask representing which numbers in
 If all numbers have been used, increment the count of special permutations.
 Otherwise, try adding each number from
nums
to the current permutation and calldfs
recursively.  If adding the number violates the special condition, cut off this branch and don’t make the recursive call.
 Use memoization to store results for a particular bitmask and last number pair to avoid redundant calculations.
Here’s the code:


Key Takeaways:
 Using a bitmask is a compact way to represent which elements are currently being used in the permutation.
 Memoization stores results for subproblems and avoids recomputation, speeding up the algorithm significantly.
 This solution now efficiently calculates the result by avoiding redundant calculations for the same states.
For this, the following are a good preparation:
“46. Permutations”  This problem is a base case for any problem that involves finding permutations.
“47. Permutations II”  This problem extends the previous one by adding a condition of duplicate numbers.
“31. Next Permutation”  The next permutation concept is useful for understanding how to manipulate permutations and sequences.
“60. Permutation Sequence”  This problem makes you find a specific permutation given its rank. It’s a helpful exercise for manipulation and understanding permutations.
“78. Subsets”  While not about permutations, this problem on subsets helps understand the combinatorial nature of such problems.
“48. Rotate Image”  While the problem context is different, the principle of moving numbers around in a given order is similar.
“62. Unique Paths”  A combinatorial problem that involves counting unique ways to reach a goal, which helps understand how to count possibilities.
“79. Word Search”  This problem involves traversing a 2D grid, which can improve your skills of going through permutations in a systematic way.
“377. Combination Sum IV”  This problem about finding combinations sums could help in understanding how to work with integers and their relations.
“526. Beautiful Arrangement”  This problem is about permutations under special conditions, similar to the problem at hand.
These cover handling permutations and combinations, manipulation of integer sequences, and systematic traversal of possibilities.


Problem Statement:You are given a 0indexed integer array nums containing n distinct positive integers. A permutation of nums is called special if:
For all indexes 0 <= i < n  1, either nums[i] % nums[i+1] == 0 or nums[i+1] % nums[i] == 0. Return the total number of special permutations. As the answer could be large, return it modulo 109 + 7.
Example 1:
Input: nums = [2,3,6] Output: 2 Explanation: [3,6,2] and [2,6,3] are the two special permutations of nums. Example 2:
Input: nums = [1,4,3] Output: 2 Explanation: [3,1,4] and [4,1,3] are the two special permutations of nums.
Constraints:
2 <= nums.length <= 14 1 <= nums[i] <= 109
Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.
Problem Classification
Problem Analysis:
Domain: The problem falls under the domain of Combinatorial Mathematics and Number Theory in Computer Science. The problem requires understanding of permutations, modularity, and some properties of integers.
What components:
Input  A 0indexed integer array nums containing n distinct positive integers.
Output  The total number of special permutations, modulo 10^9 + 7.
Definitions  A permutation of nums is special if for all indexes 0 <= i < n  1, either nums[i] % nums[i+1] == 0 or nums[i+1] % nums[i] == 0.
Problem Classification:
Based on the problem statement, this problem can be classified as a Counting Problem. The goal is to find all permutations of the given integer array that satisfy a certain condition.
Explanation of Categorization:
The problem involves finding all permutations of a given array that satisfy a certain condition related to the mod operation. This requires understanding of permutation and combination from Combinatorial Mathematics and properties of mod operation from Number Theory. Hence, it falls under the domain of Combinatorial Mathematics and Number Theory in Computer Science. As the task is to count the number of permutations that satisfy the given conditions, it is a Counting Problem.
Language Agnostic Coding Drills
 Dissect the code and identify each distinct concept it contains:
a. Understanding and defining variables: Here, the code begins with defining some variables that will be used throughout the function. This is a basic concept which is found in all programming languages.
b. Bit manipulation: The function uses bit manipulation to keep track of which elements of the nums list have been used in a permutation. This is a slightly advanced concept which is used in many programming languages for tasks that involve working with binary numbers or flags.
c. Recursive function calls: The function uses recursion to generate and check all possible permutations of the nums list. This is an intermediate level concept, commonly used in programming languages for solving complex problems in a divideandconquer manner.
d. Caching/memoization: The code uses Python’s builtin @cache decorator to implement memoization, which can significantly improve the efficiency of recursive functions by storing the results of previous function calls. This is an advanced concept that requires understanding of how function calls are stored and retrieved in memory.
e. Modulo operation: The function uses the modulo operation to ensure that the final result is within the required range. This is a basic concept, commonly used in programming languages to prevent integer overflow and to wrap values within a specific range.
 Order of increasing difficulty and description:
a. Understanding and defining variables: This is a basic concept. A thorough understanding of this concept is essential for writing any program.
b. Modulo operation: This is a basic concept that is often used in calculations with integers, especially when dealing with large numbers or when we want the result within a specific range.
c. Bit manipulation: This is a slightly advanced concept. It is not typically required for basic programs, but it is used extensively in certain areas such as lowlevel programming, cryptography, and for solving some types of algorithm problems.
d. Recursive function calls: This is an intermediate level concept. Understanding recursion is critical for solving problems that have a nested or recursive structure, which includes many algorithm problems.
e. Caching/memoization: This is an advanced concept. Understanding this requires a good grasp of how function calls work and how data is stored and retrieved in memory.
 Problemsolving approach:
a. Begin by understanding the problem and defining any necessary variables. In this case, we need to know the length of the nums list and the modulo value.
b. Use bit manipulation to keep track of which elements have been used in a permutation. In this case, we use a bitmask to represent which elements of the nums list have been used.
c. Use a recursive function to generate and check all possible permutations of the nums list. The recursion is done by calling the function with different values of the ‘prev’ and ‘mask’ variables.
d. Use memoization to store the results of previous function calls. This can significantly improve the efficiency of the function by avoiding repeated calculations.
e. Use the modulo operation to ensure that the final result is within the required range. This is necessary because the problem statement asks for the result modulo 10^9 + 7.
These drills contribute to the overall solution by breaking the problem down into smaller, more manageable parts, each of which can be solved using a specific programming concept. The final solution is then obtained by combining the results of these individual parts.
Targeted Drills in Python
 Write Python drills for each concept:
a. Understanding and defining variables: This drill could involve a series of exercises that help students practice declaring variables, initializing them with appropriate values, and using them in expressions.


b. Modulo operation: This drill could involve exercises that help students understand how to use the modulo operation in different contexts, especially when working with large numbers.


c. Bit manipulation: This drill could involve exercises that require students to set, clear, and toggle specific bits in a number.


d. Recursive function calls: This drill could involve writing recursive functions to solve problems such as calculating factorial or finding the nth Fibonacci number.


e. Caching/memoization: This drill could involve exercises where students are required to optimize their recursive functions by adding a caching mechanism.


 Add drills for specific needs of the problem:
This particular problem also requires an understanding of how to generate and check all permutations of a list. A drill for this could involve writing a function that generates all permutations of a list.


 Describe how to merge these drills for the final solution:
The final solution to the problem involves combining all these concepts and skills. The general approach would be as follows:
 Start by defining and initializing the necessary variables.
 Use bit manipulation to keep track of which elements of the nums list have been used in a permutation.
 Write a recursive function to generate and check all possible permutations of the nums list. This function should also take care of the special condition of the problem.
 Use memoization to optimize the recursive function and avoid repeated calculations.
 Use the modulo operation to ensure that the final result is within the required range.
Each concept or skill is used at the appropriate place in the solution and contributes to solving a part of the problem. In this way, they are all combined to form the final solution.