New sequence: crossing number of the Aboulafia/Zaks permutation-reversal chord diagram

[Dario Clavijo]

Hi all,

I'd like some feedback before formally submitting this through the
OEIS web form, in case anyone recognizes it or spots an issue.

Background: a recent blog series (blog.klipse.tech, "Aboulafia"
series, July 2026)[0] that I came across on HN revisits a 13th-century
enumeration rule by the Kabbalist Abraham Aboulafia[1] for listing all
permutations of a word by repeatedly rotating the first letter to the
back and recursing. The series shows this order is identical to the
one produced by Zaks' 1984 suffix-reversal algorithm (itself the
mirror image of the prefix-reversal sequence from Bill Gates' only
paper, with Papadimitriou, on pancake sorting)[2][3]. I verified this
equivalence computationally up to n=10, and separately confirmed that
the suffix/prefix reversal-length sequence for this order is exactly
A055881(m)+1, and that its total length for n symbols is n!-1 =
A033312(n).

The series also observes that if you place the n! permutations on a
circle in this order and draw a chord from each permutation to its
reversal, the chords visually organize into a dense caustic curve
starting around n=7. I wanted a numeric handle on that rather than
just an eyeballed image, so I computed the number of pairs of chords
that cross, for n=2..10:

n : 2, 3, 4, 5, 6, 7, 8, 9, 10
a(n): 0, 0, 18, 620, 24300, 1216320, 78015840, 6295423680, 625975560000

(Offset 2; for n=1 there's nothing to pair, and no permutation of n>=2
distinct symbols equals its own reversal, so there are no fixed
points/loops -- every point gets matched, giving exactly n!/2 chords.)

I searched OEIS, Wolfram|Alpha, and Google using both the individual
terms and the combinations 18,620,24300,1216320 and
1216320,78015840,6295423680, and found no matches, including under
obvious transformations. I don't have a closed form for a(n). One
thing I did notice: the fraction of possible chord-pairs that actually
cross, a(n) / C(n!/2, 2) (total number of chord-pairs), rises to a
peak around n=7-8 (~0.384) and then decreases (0.382 at n=9, 0.380 at
n=10) -- which lines up with the blog's qualitative observation
visually form a dense caustic-like curve right around n=7.

Generator (Python, verified against a slower O(chords^2) brute-force
check for n<=7, and consistent with a Fenwick tree[5] O(N log N) sweep
up to n=10):

def g(n):
if n == 1: return []
p = g(n - 1)
return (p + [n]) * (n - 1) + p

# generate permutations by repeated suffix reversal, pair each with
# its reversal's position in the sequence, then count crossing chords
# via a Fenwick tree sweep (O(N log N)).

I'm happy to post the full script if useful. Proposed
cross-references: A000142 (n!), A033312 (n!-1, move count), A055881
(per-move reversal lengths), A235748 (companion ruler sequence for
rotation-generation of S_n).

Any thoughts on whether this is a reasonable addition, and whether the
offset/keyword choices above (nonn,more) look right, would be
appreciated before I submit.

Thanks,
Dario

PS: links:
[0] https://blog.klipse.tech/aboulafia/2026/07/06/a-13th-century-enumeration-algorithm-ignored-for-700-years.html
[1] https://es.wikipedia.org/wiki/Abraham_Abulafia
[2] https://www.sciencedirect.com/science/article/pii/0012365X79900682
[3] https://en.wikipedia.org/wiki/Pancake_sorting
[4] https://link.springer.com/article/10.1007/BF01937486
[5] https://en.wikipedia.org/wiki/Fenwick_tree

[brad klee]

This proposal is similar to https://oeis.org/A187781, and personally I find an
advantage to having a good recipe for filtering diagonals.

As for the caustic claim, and then the claim of observing thousandths of a percent,
I would take some care not be misled by eyesight or by small numbers.

Remember, if we are really playing with points and lines, we don't see any of
this because points and lines have a vanishing width.

Most of what we're seeing in the blog image is the computer adding a finite width
to the lines, which gets integrated and summed over by our eyes.

What happens with a caustic is that it sweeps a boundary curve out in space and you
can see it even in platonic geometry where that sweeping out happens continuously
and tangents fill a solid area.

If I remember correctly, Bill Harter used to teach in his classical mechanics class
that caustics can also be obtained classically by taking a baton, throwing it
through space, and observing the envelope it traces out. I'm almost sure this
experiment has to do with the famous "cycloid" curve.

What do we get if we do such an experiment and the camera is only sampling at a
finite rate? Is it really a caustic or is it more like a Moire pattern?

If you'd like to keep investigating this question, my suggestion is to try and
obtain a continuous limit where a curve is increasingly well defined from the
tangent manifold provided by the construction. This might be possible by relating
diagrams in the series.

More practically, for a submission in the near future, I would suggest making a
picture without the chords and just give the points as disks with a fixed and
very small radius so that we can actually see them.

If any caustics exist, or anything similar, I think they should be defined more
sparsely in terms of these intersection points. If it then happens that some
hull of the intersection points creates a division of space where tangents are
and are not found, then a "discrete caustic" would begin to make more sense.

Comparing two discrete caustics from two separate diagrams you might then be
able to give point density per caustic length, and that would likely have
observable statistics (but I'm guessing length integrals would need to be
taken numerically, at least at first). A "true continuous caustic" as observed
classically would need a dense limit with a nice spacing property.

There's potentially some room for growth here. Counting pairwise intersections
seems like a good starting place before putting them into subsets. Be careful
if there are places where more than two chords cross at one point not to let
that anomaly confuse definitions or mess up calculations.


Harm.On.ica one-shot agreed with your terms, and claims the terms * did not *
come from a hidden source, lol. There's some code in cpp somewhere now.

<<
n = 1..11
0, 0, 0, 18, 620, 24300, 1216320, 78015840,
6295423680, 625975560000, 75277323475200
>>

Harm.On.ica also found https://oeis.org/A002866 might be related to chords
through the centroid, and sometimes found multi-points off center. That's
another interesting question with its own difficulties.


All the best,







--Brad