Permutations/ Same Sequence Of Signs As Inverses

[Leroy Quet]

--- Richard Mathar <mathar at strw.leidenuniv.nl>
wrote:
>...
> I get
>
1,2,4,10,26,80,272,1076,4848,24832,142340,902440
> 

Thanks to Richard Mathar for the calculation.

Taking the nth term of this sequence (a(n)) and
subtracting A000085(n), then dividing by 2, we
get the number of pairs of non-self-inverse
permutations and their inverses that conform to
the conditions of having the same sequence of
signs for the differences between consecutive
elements.

0, 0, 0, 0, 0, 2, 20, 156, 1114, 7668, 53322,
381144,...

So, for example, for n=6, we have the pair of
permutations I wrote of in my original email on
this topic:

4,2,6,3,5,1
6,2,4,1,5,3

Then there is one, and only one, other pair of
permutations for n=6 that are inverses of each
other, are not self-inverse, and conform to the
conditions spelled out in the original post on
this topic. (Hmmm. I will try now to find that
pair of permutations by hand now. No matter how
easy it is to find that pair I will not post back
here that the problem is much too easy to make a
good puzzle. You have been warned.)
:)

Thanks,
Leroy Quet




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jp> From seqfan-owner at ext.jussieu.fr  Mon Dec 10 00:46:54 2007
jp> Date: Sun, 9 Dec 2007 15:46:39 -0800
jp> From: "Jonathan Post" <jvospost3 at gmail.com>
jp> To: seqfan <seqfan at ext.jussieu.fr>, "jonathan post" <jvospost2 at yahoo.com>
jp> Subject: Answering njas: definition of POLYPON
jp> 
jp> A057785  Blunt polypons (no 30 deg. angles) with n cells.
jp> COMMENT 	
jp> It would be nice to have a definition of "polypon"! - njas, May 09 2007
jp> 
jp> After looking at over 170 web pages, I have one likely positive hit, a
jp> vast number of typographical errors, and two weird false positive.
jp> 
jp> The probable correct coinage:
jp> 
jp> "If we use the grid below then we obtain sets of shapes which we could
jp> call polypons from the isosceles triangle (pons asinorum) which is the
jp> basis of the tessellation."
jp> http://geocities.com/alclarke0/PolyPages/PolyX/PolyX.htm
jp> 
jp> The two weirdities:
jp> 
jp> polypon: J 1 [where J = Liber Juratus (Sworn Book of Honorius)]
jp> Index of Angel names, magical words, and names of God
jp> http://www.esotericarchives.com/solomon/nameindx.htm
jp> Copyright 1998 by Joseph H. Peterson. All rights reserved.
jp> 
jp> and, stranger still, as I took courses from this professor:
jp> 
jp> Collected Plays, Volumes I and II by Oscar Mandel
jp> Professor Snaffle's Polypon, and Of Angels and Eskimos
jp> Santa Barbara, California:
jp> Unicorn Press
jp> Distributed by Spectrum Productions
jp> 979 Casiano Road
jp> Los Angeles CA 90049
jp> Volume One ISBN 0-87775-000-0
jp> Volume Two ISBN 0-87775-001-7
jp> 211 + 204 pages
jp> The hard cover set: $24.00
jp> 
jp> I disregard the Swedish Jazz group: "Helges Polypon-Boys" because we
jp> don't know the etymology: angel, typo, or variant of "polyphone"?

This looks like a racial mix of the Latin pons (=bridge) and the Greek
poly = many/multi. The unit element is a isosceles triangle with the
basis the longest side which spans a river or anything else of this kind:


By looking at the Clarke pictures, I guess, the unit element is a triangle with
internal angles of 120 degrees and two times 30 degrees. The polypons are
connected, non-overlapping assemblies of these, where connectivity is defined
via common sides; a common point is not enough. Only non-congruential
assemblies are counted, those which cannot be mapped onto each other by
rotations, translations or mirrors along a line or point. However, the
polypons are not all of these, because some of the free-form assemblies of
this kind would need placement of the unit that violates the format by the
grid. (The first case where this happens is with assemblies of 3 units: the
picture shows 2 examples with assemblies of 3 units, but I can imagine at
least 1 more where the unit would need to hide/cover one of the grid's edges.)

Richard