[chordbug]

We truly live in an age where facts that are worth "maybe one sentence of space on Wikipedia" can be expanded into full-blown AI-coded interactive websites. I'm not sure how to feel about this. I think in this case it ascribes an inappropriate sense of grandeur: making a mathematical curiosity (and is the result even that surprising?) seem like some deep truth has been unveiled, or we finally found God's Number.

[Someone]

> and is the result even that surprising?

Not as far as I can tell from skimming https://en.wikipedia.org/wiki/Random_permutation_statistics.

[thaumasiotes]

> and is the result even that surprising?

No.

The exposition has its problems too. Consider:

>> Zero fixed points — not a single hexagram occupies the same position in both orderings. The structural difference is total.

As a mathematical matter, the expected number of fixed points for any permutation is 1. Some have more. For some to have more, others must have less, and all of those will have 0.

But as a logical matter, "the structural difference is total" is pure gibberish. Consider these two permutations on 5 elements:

    1. [2, 3, 4, 5, 1]
    2. [5, 1, 2, 3, 4]

"Not a single element occupies the same position in both orderings."

But of course these two permutations have a nearly identical structure (they are rotations in opposite directions, and are each other's inverses); they are far more closely related to each other than either is to

    3. [4, 3, 2, 1, 5]

even though permutation 3 shares the assigned position of "3" with permutation 1, and the assigned position of "2" with permutation 2.

Then:

>> We reframe the question:

>> Transform the question "what is the structural distance between two orderings"

>> into the mathematical problem "what is the cycle structure of a specific permutation in S₆₄?"

This is nonsense. The 'question' cannot be transformed into the 'problem', because they are completely unrelated ideas. It's like transforming the question 'what is the Levenshtein distance between two strings?' into the problem 'if a specific string were in alphabetical order, how would it be pronounced?'.

[gezhengwen]

You are right, zero fixed points does not mean total structural difference. Your counterexample is good. My wording was wrong, I will fix it. What interests me is not the statistical rarity, but that 81% of elements are in one orbit — this means the reordering is highly coupled, not a bunch of small local swaps.

[mcphage]

> What interests me is not the statistical rarity, but that 81% of elements are in one orbit — this means the reordering is highly coupled, not a bunch of small local swaps.

But what is the significance of the reordering being highly coupled?

[gezhengwen]

The observation itself is the value — it tells you the King Wen sequence is not a bunch of small local adjustments, but a holistic rearrangement. But it cannot tell you why King Wen arranged it this way.

[Bimos]

You didn't explain or prove there is a reason for it.

[gezhengwen]

You are right, the presentation may be overdone. The result itself is a small mathematical fact. I made the interactive page so people can verify it themselves, not to make it look grand. Thank you for the criticism, I will adjust.