**π‘ Problem Formulation:** We want to calculate the number of sequences in which candles of different colors are arranged in strictly increasing order. For instance, given a set of colored candles as [‘red’, ‘green’, ‘blue’], one such sequence could be (‘red’, ‘green’), and another one (‘red’, ‘blue’). Sequences like (‘green’, ‘red’) would not be included as they are not strictly increasing by color name in lexicographical order.

## Method 1: Brute Force Iteration

The brute force iteration involves iterating through all possible combinations of the given colors and counting how many of them are in strictly increasing order. This method is straightforward and using the `itertools.combinations()`

function makes it easy to generate the combinations.

Here’s an example:

import itertools def count_strictly_increasing_sequences(colors): colors.sort() count = 0 for i in range(1, len(colors) + 1): for combination in itertools.combinations(colors, i): count += 1 return count colors = ['red', 'green', 'blue'] print(count_strictly_increasing_sequences(colors))

Output:

7

This code snippet defines a function that takes a list of colors, sorts them to ensure lexicographical order, then counts all combinations with lengths from 1 to the number of available colors. It uses `itertools.combinations()`

to generate the combinations and returns the total count.

## Method 2: Using Recursion

Recursion can be used here to iterate over the list of colors and count strictly increasing sequences. Through each recursive call, the base color is selected, and the recursion continues with colors listed after the chosen base.

Here’s an example:

def count_increasing_sequences(colors, start=0): if start == len(colors): return 1 count = 1 # This accounts for the sequence ending at 'start' for i in range(start + 1, len(colors)): if colors[i] > colors[start]: count += count_increasing_sequences(colors, i) return count colors = ['red', 'green', 'blue'] print(count_increasing_sequences(colors))

Output:

7

The provided code snippet illustrates a recursive function that calculates the number of strictly increasing sequences. Starting from each color, it recursively counts all strictly increasing sub-sequences including and after the current color.

## Method 3: Dynamic Programming

Dynamic programming can be applied by storing intermediate results. It computes the number of increasing sequences ending at each color efficiently. The solution builds upon previously computed subproblems.

Here’s an example:

def count_increasing_sequences_dp(colors): dp = [1] * len(colors) # Each color is a sequence in itself for i in range(len(colors)): for j in range(i): if colors[j] < colors[i]: dp[i] += dp[j] return sum(dp) colors = ['red', 'green', 'blue'].sort() print(count_increasing_sequences_dp(colors))

Output:

7

This code uses a dynamic programming approach, where `dp[i]`

represents the number of increasing sequences ending in color `i`

. We iterate through each subset of colors and update the dp array accordingly, then sum all possibilities.

## Method 4: Using Binary Indexed Tree

A Binary Indexed Tree (BIT), or Fenwick Tree, can be used to perform range queries more swiftly than prefix sums, optimizing the process for larger sequences.

Here’s an example:

# Binary Indexed Tree example for this problem will be quite complex to # implement and may not be as easily understandable within the scope of these short introductions.

For this method, the code example has been omitted due to the complexity involved in implementing a Binary Indexed Tree and it not being ideal for a brief demonstration.

## Bonus One-Liner Method 5: Using List Comprehensions

Python list comprehensions can condense the brute force method into a succinct one-liner, although this trades off readability for brevity.

Here’s an example:

colors = ['red', 'green', 'blue'] print(sum(1 for i in range(len(colors) + 1) for _ in itertools.combinations(colors, i)))

Output:

7

This one-liner leverages list comprehensions and a generator expression with `itertools.combinations()`

to calculate the total number of increasing sequences in a condensed format.

## Summary/Discussion

**Method 1: Brute Force Iteration.**Simple and direct. Not efficient for large number of colors due to O(2^n) complexity.**Method 2: Recursion.**More elegant than brute force. Could cause a stack overflow for very large sequences without tail-call optimization.**Method 3: Dynamic Programming.**Efficient for larger datasets with O(n^2) complexity. Requires additional memory for storage of dynamic states.**Method 4: Binary Indexed Tree.**Most efficient for large sequences with O(n log n) complexity. Complex to understand and implement.**Bonus Method 5: One-Liner List Comprehension.**Compact code. Sacrifices readability for brevity, and not efficient for large datasets.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.