   Date: Mon, 13 Feb 95 00:59:00 WET   From: Martin Schoenert <Martin.Schoenert@math.rwth-aachen.de >
~~~  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 <pos> such that for no
generator <gen> the position <pos>*<gen> is longer. The fourth column
contains the number of positions such that for one generator <gen> the
position <pos>*<gen> is longer. And so on. So the eleventh column
contains the number of positions <pos> such that for all eight generators
<pos>*<gen> 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 <n_b> basic states of cost 1. Each
basic state <g> has <n_b> neighbors, namely the elements <g> (F S F).

The set of neighbors of each state is obviously a union of right cosets
of F. Furthermore if <g_1> and <g_2> 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 = { (<x> F) | <x> in F S }. Assume that there are
<n_e> essential states of coset 1. Then each essential state (<g> F) has
<n_e> essential neighbors, namely the essential elements (<g> F) X.

We can now count how many of the basic neighbors of a basic state <g>
have smaller cost than <g>. If all <n_b> basic neighbors have smaller
cost than <g>, then we call <g> a basic local maximum.

Likewise we can count how many of the essential neighbors of the
essential state (<g> F) have smaller cost than (<g> F). If all <n_e>
essential neighbors have smaller cost than (<g> F), then we call (<g> F)
an essential local maximum.

It is easy to see that a basic element <g> is a basic local maximum if
and only if (<g> F) is an essential maximum.

In fact in most cases the number of basic states that have smaller cost
than <g> is simply (<n_b> / <n_e>) times the number of essential states
that have smaller cost than (<g> 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
```     