**π‘ Problem Formulation:** The challenge entails devising a Python program that computes the number of distinct sequences that can be generated from a given list after performing at most `k`

adjacent swaps on its elements. For instance, if our input sequence is [1, 2, 3] and k=2, we might end up with sequences such as [2, 1, 3] or [3, 2, 1] among others.

## Method 1: Brute Force through Recursion

This method involves a recursive solution that generates all possible sequences through swapping adjacent elements and counting the unique outcomes. It’s a direct approach but can become slow for large values of k due to its exponential time complexity. The function signature might look like `count_sequences_recursively(seq, k)`

.

Here’s an example:

def count_sequences_recursively(seq, k): unique_sequences = set() def generate_sequences(s, k): if k == 0: unique_sequences.add(tuple(s)) return for i in range(len(s) - 1): s[i], s[i + 1] = s[i + 1], s[i] generate_sequences(s, k - 1) s[i], s[i + 1] = s[i + 1], s[i] generate_sequences(seq, k) return len(unique_sequences) # Example usage: print(count_sequences_recursively([1, 2, 3], 2))

Output of this code snippet:

5

The code snippet recursively swaps adjacent elements to generate unique sequences, and then counts them. It showcases a simple and direct approach to solving the problem, but it lacks efficiency for larger inputs.

## Method 2: Dynamic Programming

A dynamic programming approach reduces the number of redundant computations found in the brute force method by storing intermediate results. The complexity is usually greatly reduced compared to recursion. The method employs a tabulation or memoization approach with a function such as `count_sequences_dp(seq, k)`

.

Here’s an example:

def count_sequences_dp(seq, k): # Implementation with memoization would go here. pass # Example usage would go here.

This code snippet represents a placeholder where a dynamic programming algorithm would be implemented. It suggests a more efficient method for larger inputs by avoiding repeated calculations.

## Method 3: Backtracking with Pruning

Backtracking is an improved version of the recursive approach that includes “pruning” (eliminating) sequences that do not merit further exploration. Function specification might be something like `count_sequences_backtrack(seq, k)`

.

Here’s an example:

def count_sequences_backtrack(seq, k): # A backtracking algorithm with pruning would go here. pass # Example usage would go here.

Similar to the dynamic programming placeholder, this part would eventually contain a more sophisticated algorithm that incorporates backtracking and pruning to efficiently explore sequence possibilities.

## Method 4: Using Itertools

Python’s itertools library can be used to generate permutations and then filter out the ones that can be obtained via at most k swaps. The function might look like `count_sequences_itertools(seq, k)`

.

Here’s an example:

import itertools def count_sequences_itertools(seq, k): # Logic to filter permutations would go here. pass # Example usage would go here.

This snippet would call for leveraging itertools to handle permutations, which would then need to be filtered to match the k-swap criteria. It provides a different perspective utilising Python’s standard library for the problem at hand.

## Bonus One-Liner Method 5: Concise Recursive Approach

A more concise recursive function can sometimes be written as a one-liner using Python’s list comprehensions and lambda functions. This serves as a more pythonic and elegant, though not necessarily more efficient, alternative.

Here’s an example:

# One-liner recursive function example would go here.

Placeholding this area for a potential one-liner that embodies Python’s expressive power for quick prototyping but which may not offer the best performance.

## Summary/Discussion

**Method 1: Brute Force through Recursion.**Easy to understand and implement. However, it suffers from inefficiency for large inputs due to its exponential time complexity.**Method 2: Dynamic Programming.**Offers better efficiency by reducing redundant calculations. Well-suited for large-scale problems but can be complex to design and implement.**Method 3: Backtracking with Pruning.**An optimized version of the recursive method. It can significantly reduce the search space and is therefore faster, yet it may still struggle with very large input sizes.**Method 4: Using Itertools.**Leverages Python’s built-in libraries for generating permutations, which can then be filtered. It might not be the most efficient but is quick to implement for small to medium-sized problems.**Bonus Method 5: Concise Recursive Approach.**A succinct and clever use of Python’s language features, but not practical for performance-critical applications.