A108973
Andrew Weimholt
andrew at weimholt.com
Tue Dec 19 12:20:04 CET 2006
> 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
More information about the SeqFan
mailing list