A108973

[Andrew Weimholt]

Tried to send this to the list, but it didn't get through.
Anyone else having trouble with emails not getting through?

-----Original Message-----
From:	Andrew Weimholt [SMTP:andrew at weimholt.com]
Sent:	Tuesday, December 19, 2006 3:20 AM
To:	'njas at research.att.com'; seqfan at ext.jussieu.fr; seqfans at seqfan.net
Subject:	RE: A108973

> no, don't wait - please send updates right away!
> Neil

Following Joshua's earlier advice, I submitted the 5th term...

a(4) = 1480206036768915456000 = 2^30 * 3^6 * 5^3 * 7^2 * 11 * 13 * 17 * 127

%I A108973
%S A108973 2,480,10321920,64561751654400,1480206036768915456000
%C A108973 New term added by Andrew Weimholt - Dec 19, 2006
%O A108973 0
%K A108973 ,more,nonn,
%A A108973 Andrew Weimholt (andrew at weimholt.com), Dec 19 2006

I actually found it back in early November, and was surprised to
see that the result contained a strange prime factor, 127.
I had expected that the largest prime factor would be less than
or equal to the dimension, 4n+3.
The factor, 127, can only be explained by the existence of multiple
sets of simplexes (such that no symmetry operation of the 19-cube can
transform a simplex from one set into a simplex of another).
Indeed, it turns out that there are 3 sets of 19-simplexes 
which can be inscribed on the 19-cube, occurring in the following numbers:
a(4)*32/127, a(4)*57/127, a(4)*38/127

Set 1:

  The symmetry group acting on the superposition of 1 of these simplexes
  with the 19-cube has order 171

  From this, the number of simplexes in this set is computed to be
  2^35 * 3^6 * 5^3 * 7^2 * 11 * 13 * 17 = 372965300603191296000

Set 2:

  The symmetry group acting on the superposition of 1 of these simplexes
  with the 19-cube has order 96

  From this, the number of simplexes in this set is computed to be
  2^30 * 3^7 * 5^3 * 7^2 * 11 * 13 * 17 * 19 = 664344441699434496000

Set 3:

  The symmetry group acting on the superposition of 1 of these simplexes
  with the 19-cube has order 144

  From this, the number of simplexes in this set is computed to be
  2^31 * 3^6 * 5^3 * 7^2 * 11 * 13 * 17 * 19 = 442896294466289664000
  
These results prompted me to go back and look at the 15 dimensional case.
It turns out there are 5 sets of 15-simplexes which can be inscribed on
the 15-cube. The group orders and counts are as follows...

  15d-set-1: ord =   2688,  count = 15941173248000
  15d-set-2: ord =   2688,  count = 15941173248000
  15d-set-3: ord =   1536,  count = 27897053184000
  15d-set-4: ord =   9216,  count =  4649508864000
  15d-set-5: ord = 322560,  count =   132843110400
  
For 11 and 7 dimensions, there's only 1 set each (unless you
only allow pure rotations in which case they can be divided
into left and right handed sets).

Andrew