rencontres numbers

[Creighton Dement]

Dear Seqfans, 

Laxly stated, it is possible to measure the amount of "power" a
floretion X possesses by asking the following (very imprecice) question:
Let
symseq-1, symseq-2, .... (= ves, tes, etc.) be the collection of rules
of summing up over basis vectors (i.e. 'i, 'j, 'k, i', j', ...). Now,
how likely is it that if Y is a second floretion, symseq-n(X*Y) will be
a power sequence of the form m, m^2, m^3, m^4, ... for some m in
naturals.   
 
One immediately sees the power of the floretion X = 0. However, there
are also nontrivial examples. The floretion Y = - .5'j + .5'k + .5j' -
.5k' - .5'ij' - .5'ik' + .5'ji' + .5'ki' (in FAMP, see "Collections menu
-> Chung-shu's Division Group (clockwise-counterclockwise)" ) with Y*Y =
0 is "extremely powerful". In fact, I have never been able to put it to
use until today because it turns everything it touches into power
sequences-specifically, powers of 4 or the zero-sequence.

The rok-symmetry, below, works as such:  Instead of forming sequences of
the form symseq[X] = (X, X^2, X^3, X^4... ), it forms sequences as
symrokseq[X] = (X, (X+1)(X-1), ((X+1)(X-1)+1)((X+1)(X-1)-1),  ... ) .
Apparently, this was enough to jar Y out of its deep sleep and put it to
action (By the way, I believe one of my first messages to seqfan was on
the topic: "it seems that, very often, if vesseq(X) is an integer
sequence, then vesseq(X+1) is its binomial transform)". 

Here are the symseqs for Y (as expectected, they are quite dull):
http://www.crowdog.de/RokSym/Chung-shu.html


Here are the symrokseqs for Y 
http://www.crowdog.de/RokSym/Chung-shu-Rok.html

vesrokseq is 
http://www.research.att.com/projects/OEIS?Anum=A000240  
Rencontres numbers: permutations with exactly one fixed point.  

everything else is the zero-sequence... except for one additional
(unlisted!) sequence further down the page: 0, 0, -1, 6, -39, 278,
-2211, 19542, -190711, 2040606, -23781627 (unsure if the initial term
needs to be disregarded - furthermore, higher terms cannot be trusted
due to FAMP's limited ability to handle large numbers). 

But we can also perform a force transform on the floretion at this
stage. Again, the "golden rule" says to transform A000004 (the
zero-sequence) first. 

Here are the roksym sequences after A000004 has been transfromed:
http://www.crowdog.de/RokSym/Chung-shu-For-Rok.html

On that page, the following sequence is found:
http://www.research.att.com/projects/OEIS?Anum=A000166
Subfactorial or rencontres numbers, or derangements: number of
permutations of n elements with no fixed points.


I will check other transformations after I get back from work. 
(and will also try to update FAMP on my websight at that time.)  

Sincerely, 
Creighton