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<DIV>Thank you all for your kind answers!<BR>You got me thinking ;)</DIV>
<DIV> </DIV>
<DIV>"The combinations function is one of the most fundamental
in<BR>combinatorics, and while S_n is equally fundamental, looking<BR>at numbers
modulo 2 is a fairly specialized operation. (Franklin )"</DIV>
<DIV> </DIV>
<DIV>Sorry, I was not clear enough. My "point of view" is modulo i, not just
modulo 2.</DIV>
<DIV> </DIV>
<DIV>Please, let me defend (and I hope, explain) my modulo i point of
view:</DIV>
<DIV> </DIV>
<DIV>1. Index of permutation, [0..n!-1] <BR> Let Ix be integer index
of a permutation from Sn, Ix element of [0..n!-1].<BR> We can get
rid of base 10 integers expressing them in Factorial Number System,
FNS.<BR> That is modulo i number system, i element of [2..n], base
i!.<BR> Let R be index of a permutation expressed in FNS.</DIV>
<DIV> </DIV>
<DIV>2. Permutation Generator, PG<BR> Lets define PG as
transformation of R to permutation, P.<BR> I'm aware of two family
of PG's which converts R to P using modulo i arithmetic (+,-), i element of
[2..n].</DIV>
<DIV> </DIV>
<DIV>3. Permutation<BR> P obtained above can be viewed as own index
expressed in base n number system.<BR> So, Ix and R and P are all
the same thing, what differs is only used number system.<BR> That
allows me to view the whole Sn as modulo i thing, <BR> with i being
fixed, i.e. n, or i varying from 2 to n...</DIV>
<DIV> </DIV>
<DIV>Now back to the subject:<BR>Lets define n! as "number of different patterns
in the set of all permutations Sn taken modulo n".<BR>Generalized question now
becomes:<BR>What is the number of different patterns in the set of all
permutations Sn taken modulo i, i element of [2..n]?</DIV>
<DIV> </DIV>
<DIV>Sequences:<BR>SEQ CONSTRUCTION (FIXED i)<BR> i n= 1 2
3 4 5 6
7
8 OEIS<BR>-----------------------------------------------<BR> 2 1,2,3,
6, 10, 20, 35, 70,.. A001405<BR> 3 1,2,6,12,
30, 90, 210, 560,.. A022916<BR> 4 1,2,6,24, 60,180, 630,
2520,.. A022917<BR> 5 1,2,6,24,120,360,1260, 5040,..
-------<BR> 6 1,2,6,24,120,720,2520,10080,.. -------<BR> 7
1,2,6,24,120,720,5040,20160,.. -------<BR> ..<BR>-----------------------------------------------<BR> limes A000142
n!</DIV>
<DIV> </DIV>
<DIV>Solutions:<BR>----------<BR>i=2 A001405 Central binomial coefficients:
C(n,floor(n/2)). <BR>i=3 A022916 Multinomial coefficient
n!/([n/3]![(n+1)/3]![(n+2)/3]!)<BR>i=4 A022917 Multinomial coefficient n!/
([n/4]!, [(n+1)/4]!, [(n+2)/4]!, [(n+3)/4]!)<BR>...</DIV>
<DIV> </DIV>
<DIV>Thank you again,</DIV>
<DIV>--ivica<BR></FONT></DIV></BODY></HTML>