Testing For Simple Permutations

Having explained what a simple permutation is in my previous post, I thought that testing the number associated with my diurnal age would be a good idea. Here is the output for today's number \( \textbf{28146} \) (permalink):

Simple Permutation Analyzer initialized...

--- Analyzing Number: 28146 ---

Intermediate Stage:

  Digits ranked descending: [8, 6, 4, 2, 1]

  Rank mapping (1=highest): {8: 1, 6: 2, 4: 3, 2: 4, 1: 5}

  Normalized permutation:   [4, 1, 5, 3, 2]

Result: False

Reason: Failed: Sub-block [3, 2] from digits [4, 6] forms an interval.

Alternative: Try 28416 instead. It is a simple permutation using your digits.

Having found that the digits 28146 do not form a simple permutation, the program finds a permutation that is simple, namely 28416. I tried entering a number with repeated digits and the number was regarded as invalid. I asked Gemini why:

The Definition of a Strict Permutation

The primary reason duplicate digits are rejected is rooted in the mathematical definition of a permutation. In combinatorics, a standard permutation is strictly defined as a rearrangement of a set of distinct, distinguishable items. Every item in the set must be unique.

When you introduce duplicate items (like the number 88234), you are no longer working with a standard set. You are working with what mathematicians call a multiset. While you can arrange the elements of a multiset, the rules, formulas, and structural properties change entirely.

The Problem of Ambiguity

Permutation theory relies on knowing exactly how many unique arrangements can be made. If all elements are distinct, a sequence of length n will always have n! (n factorial) possible arrangements.

However, duplicates introduce ambiguity. Imagine the number 122. Let's label the identical twos to tell them apart: 1, 2_a, 2_b. If we swap the two 2s, we get a structurally identical number:

• Arrangement A: 1, 2_a, 2_b (Displays as 122)
• Arrangement B: 1, 2_b, 2_a (Displays as 122)

Because swapping identical digits does not create a new, distinct sequence, the total number of unique arrangements drops. The mathematical framework built to analyze the structure of permutations assumes that every single shift in position creates a fundamentally new sequence. Duplicates break this assumption.

How Duplicates Break the "Simple" Math

Beyond the theoretical definition, duplicate digits physically break the mathematical logic used to define a simple permutation.

Recall the formula we used in the algorithm to test for intervals: max(block) - min(block) == length - 1. This formula only works because it assumes that an interval of distinct integers will have a predictable span. For example, a block of three distinct, consecutive numbers like {4, 5, 6} will always have a span of 2 (because 6 - 4 = 2, and the length 3 minus 1 is 2).

If we allow duplicates, this logic collapses. Consider a block of two identical numbers: {5, 5}.

• The length of the block is 2.
• The maximum value is 5.
• The minimum value is 5.
• max - min is 0.

The formula expects a span of 1 for a block of length 2. Because the span is 0, the math fails to recognize what is happening. Furthermore, the entire concept of a "continuous range of numbers" becomes nonsensical when numbers repeat. Does {5, 5} constitute an interval? What about {4, 5, 5, 6}?

Because the definition of simple permutations was explicitly designed to map the structural scattering of distinct integers, attempting to apply it to repeating digits forces the math to evaluate scenarios it was never designed to handle.

Fair enough. I've incorporated this algorithm into my daily number analysis. The no repeated digits restriction means that only xenodromes are acceptable input.