# 5 Best Ways to Generate the First N Lexicographic Numbers in Python

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π‘ Problem Formulation: The challenge is to develop a Python program that can generate the first n numbers in lexicographical (or dictionary) order. For instance, if the input is 13, the desired output should be a sequence like 1, 10, 11, 12, 13, 2, 3, 4, 5, 6, 7, 8, 9, indicating the order these numbers would appear in a dictionary.

## Method 1: Iterative Approach

This method involves using a traditional iterative approach to construct numbers from 1 to 9, then extending them by appending additional digits until n numbers are generated. The essence is to maintain a lexicographic order by considering digit by digit extension.

Here’s an example:

```def lexicographic_numbers(n):
result = []
number = 1
for _ in range(n):
result.append(number)
if number * 10  n:
number //= 10
number += 1
return result

print(lexicographic_numbers(13))
```

Output:

`1, 10, 11, 12, 13, 2, 3, 4, 5, 6, 7, 8, 9`

This code defines a function `lexicographic_numbers()` that creates a lexicographically ordered list of numbers up to the given number n. It appends numbers to the list by exploring each digit’s options before incrementing, thereby maintaining lexicographical order.

## Method 2: Using Recursion

The recursive method exploits the natural structure of lexicographic ordering by using a depth-first traversal algorithm. Recursive calls are used to explore numbers starting with each digit, expanding them by adding digits to the right until the limit is reached.

Here’s an example:

```def lexi_helper(x, n, result):
if x <= n:
result.append(x)
for i in range(10):
if 10 * x + i <= n:
lexi_helper(10 * x + i, n, result)

def lexicographic_numbers_recursive(n):
result = []
for i in range(1, 10):
lexi_helper(i, n, result)
return result

print(lexicographic_numbers_recursive(13))
```

Output:

`1, 10, 11, 12, 13, 2, 3, 4, 5, 6, 7, 8, 9`

The function `lexicographic_numbers_recursive()` generates numbers lexicographically up to n with the help of a helper function `lexi_helper()`. The helper function extends each number x with all possible digits recursively.

## Method 3: Using itertools.product()

The itertools.product() function can be used creatively to form combinations of numbers that when strung together, create a lexicographically ordered sequence. This method handles the generation of combinations internally, providing a neat and efficient solution.

Here’s an example:

```from itertools import product

def lexicographic_product(n):
return sorted(int(''.join(map(str, digits)))
for i in range(1, len(str(n)) + 1)
for digits in product(range(10), repeat=i))[:n]

print(lexicographic_product(13))
```

Output:

`0, 1, 10, 11, 12, 13, 2, 3, 4, 5, 6, 7, 8, 9`

The code uses `itertools.product()` to produce digit combinations of varying lengths, then sorts them to ensure lexicographic order and slices the list to obtain the first n numbers.

## Bonus One-Liner Method 4: List Comprehension with Sorting

This one-liner solution uses Python’s list comprehension and built-in sorting features to generate the desired sequence in a single line of code, showcasing Python’s expressive power in accomplishing tasks with minimalistic code.

Here’s an example:

```lexicographic_sort = lambda n: sorted(range(1, n+1), key=str)

print(lexicographic_sort(13))
```

Output:

`1, 10, 11, 12, 13, 2, 3, 4, 5, 6, 7, 8, 9`

The lambda function `lexicographic_sort` sorts numbers from 1 to n based on their string representation, which naturally orders them lexicographically.

## Summary/Discussion

• Method 1: Iterative Approach. Method is straightforward and easy to understand. It is memory-efficient as it doesn’t rely on recursion. Its weakness is a more complex logic prone to off-by-one errors.
• Method 2: Using Recursion. It mirrors the natural recursive structure of the problem. It is elegant but can be inefficient for large n due to the depth of recursive calls.
• Method 3: Using itertools.product(). This method is Pythonic and utilizes built-in functions for brevity. However, it is not the most efficient for very large n, as it generates all permutations before sorting.
• Bonus Method 4: List Comprehension with Sorting. One-liner solution that is elegant and highly readable. The downside is that sorting can be inefficient for very large sequences.