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Date: Mon, 13 Feb 95 00:59 WET
From: "Martin Schoenert"
To: cube-lovers@life.ai.mit.edu
In-Reply-To: "Martin Schoenert"'s message of Tue, 31 Jan 95 11:43 WET
Subject: Re: Re: Skewb thoughts
I wrote in my e-mail message about the SKEWB of 1995/01/15
Here is the table for H. The first column contains the lenght. The
second column contains the number of positions of that length in H.
The third column contains the number of positions of that length that are
local maxima, i.e., the number of positions such that for no
generator the position * is longer. The fourth column
contains the number of positions such that for one generator the
position * is longer. And so on. So the eleventh column
contains the number of positions such that for all eight generators
* is longer (this happens of course only for the 2 solutions).
length #pos #loc max
0 2 0 0 0 0 0 0 0 0 2
1 16 0 0 0 0 0 0 16 0 0
2 96 0 0 0 0 0 0 96 0 0
3 576 0 0 0 0 0 0 576 0 0
4 3456 0 0 0 0 0 240 3216 0 0
5 20496 0 0 0 48 729 2766 16953 0 0
6 118608 48 161 1231 4228 14779 32993 65168 0 0
7 630396 8266 33358 76349 121363 153892 137755 99413 0 0
8 2450966 1025322 621763 381098 234661 128570 47822 11730 0 0
9 2911712 2768641 126056 15344 1422 199 50 0 0 0
10 162056 161876 180 0 0 0 0 0 0 0
11 180 180 0 0 0 0 0 0 0 0
... note that this is the H = < RUF, RUB, LUF, LUB > ...
And in my e-mail message of 1995/01/31
I rerun the computation using the new subgroup H as a model for the
essential SKEWB. Here is the output.
0 1 0 0 0 0 0 0 0 0 1
1 8 0 0 0 0 0 0 8 0 0
2 48 0 0 0 0 0 0 48 0 0
3 288 0 0 0 0 0 0 288 0 0
4 1728 0 0 0 0 0 120 1608 0 0
5 10248 0 0 0 36 377 1322 8513 0 0
6 59304 12 87 662 2217 7561 15698 33067 0 0
7 315198 4331 16897 37723 61161 76931 66997 51158 0 0
8 1225483 515249 311594 186221 115830 65096 25012 6481 0 0
9 1455856 1384909 61839 8280 708 95 25 0 0 0
10 81028 80938 90 0 0 0 0 0 0 0
11 90 90 0 0 0 0 0 0 0 0
As you can see, the numbers in the first column are exactely half of the
corresponding numbers in my previous message (as was expected). The
numbers in the other columns are close to half of the corresponding
numbers in my previous message but not exactely identical. I have to
rethink what those numbers mean and how they relate to the corresponding
numbers for the basic SKEWB.
... note that this is now H = < RUF, LUB, RDB, LDF > ...
The reason that the numbers in the other columns of the second table are
not exactely half of the corresponding numbers in the first table is
rather simple. They are *both wrong*.
The correct numbers for H = < RUF, LUB, RDB, LDF > are as follows
0 1 0 0 0 0 0 0 0 0 1
1 8 0 0 0 0 0 0 8 0 0
2 48 0 0 0 0 0 0 48 0 0
3 288 0 0 0 0 0 0 288 0 0
4 1728 0 0 0 0 0 0 1728 0 0
5 10248 0 0 0 0 120 240 9888 0 0
6 59304 0 0 84 96 1740 6024 51360 0 0
7 315198 198 144 3600 9768 42900 94344 164244 0 0
8 1225483 15783 73016 199808 316776 343992 208584 67524 0 0
9 1455856 1001960 365792 74976 11760 1224 144 0 0 0
10 81028 80308 720 0 0 0 0 0 0 0
11 90 90 0 0 0 0 0 0 0 0
and the correct numbers for H = < RUF, LUB, RUB, LUF > are exactely twice
as large.
I figured out what those numbers mean. It is all rather simple.
Everybody who thought about them probably knows everything that follows.
I use the terms from my last few messages about models for the cube.
The basic states of cost 1 are exactely the elements in (F S F), where F
is the subgroup of essentially free elements, and S is the set of simple
elements (the set of generators) of G. Not all those elements need to be
different. Assume that there are basic states of cost 1. Each
basic state has neighbors, namely the elements (F S F).
The set of neighbors of each state is obviously a union of right cosets
of F. Furthermore if and are essentially equal, then there
sets of neighbors are equal. So we can map the whole concept to the
essential model G/F.
Recall that in the essential model G/F the set of elements of cost 1 was
exactely the set X = { ( F) | in F S }. Assume that there are
essential states of coset 1. Then each essential state ( F) has
essential neighbors, namely the essential elements ( F) X.
We can now count how many of the basic neighbors of a basic state
have smaller cost than . If all basic neighbors have smaller
cost than , then we call a basic local maximum.
Likewise we can count how many of the essential neighbors of the
essential state ( F) have smaller cost than ( F). If all
essential neighbors have smaller cost than ( F), then we call ( F)
an essential local maximum.
It is easy to see that a basic element is a basic local maximum if
and only if ( F) is an essential maximum.
In fact in most cases the number of basic states that have smaller cost
than is simply ( / ) times the number of essential states
that have smaller cost than ( F). One sufficient condition for this
to happen is, that S is invariant under conjugation by S and that all
classes have the same length.
This condition is met for the SKEWB, so the numbers in the first table
*had* to be twice the numbers in the second table. Sorry about any
confusion I caused.
Have a nice day.
Martin.
-- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .-
Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany