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<DIV><FONT face="Palatino Linotype" size=2>Christopher,</FONT></DIV>
<DIV><FONT face="Palatino Linotype" size=2>The seq you seek
is <STRONG><FONT face="Times New Roman" size=3>A002464.
</FONT></STRONG></FONT><FONT face="Palatino Linotype"><FONT size=2>There's quite
a bit of information on it in the Encyclopedia, including a simple recursive
formula.</DIV></FONT></FONT>
<DIV><FONT face="Palatino Linotype" size=2><STRONG><FONT face="Times New Roman"
size=3></FONT></STRONG></FONT><FONT face="Palatino Linotype"><FONT size=2>I
wrote a program to calculate the first few terms; it found the first 10 or 11 in
less than one minute.</DIV></FONT></FONT><FONT face="Palatino Linotype"
size=2><FONT size=2>
<P>Regards,<BR>Jack Kennedy</P></FONT></FONT>
<BLOCKQUOTE dir=ltr
style="PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px">
<DIV style="FONT: 10pt arial">----- Original Message ----- </DIV>
<DIV
style="BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: black"><B>From:</B>
<A title=cmt1288@comcast.net href="mailto:cmt1288@comcast.net">Christopher
Tomaszewski</A> </DIV>
<DIV style="FONT: 10pt arial"><B>To:</B> <A title=seqfan@ext.jussieu.fr
href="mailto:seqfan@ext.jussieu.fr">SeqFan</A> </DIV>
<DIV style="FONT: 10pt arial"><B>Sent:</B> Sunday, December 14, 2003 11:09
PM</DIV>
<DIV style="FONT: 10pt arial"><B>Subject:</B> Permutations</DIV>
<DIV><BR></DIV>
<DIV><FONT face=Arial size=2>SeqFan members,</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2> I am currently attempting to
derive a formula giving the total number of permutations of n elements such
that no two elements which were consecutive in the original set are
consecutive in any of the permutations.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2> In order to easily remember
the original positions of elements in empirical study, I simply use the set
{1,2,3,...,n}.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Here are all 24 permutations of {1,2,3,4}, with
those satisfying my criteria in red:</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2>{1,2,3,4} {1,2,4,3} {1,3,2,4}
{1,3,4,2} {1,4,2,3} {1,4,3,2}</FONT></DIV>
<DIV><FONT face=Arial size=2>{2,1,3,4} {2,1,4,3}
{2,3,1,4} {2,3,4,1}<FONT color=#ff0000> {2,4,1,3}</FONT><FONT
color=#000000> {2,4,3,1} {3,1,2,4} </FONT><FONT
color=#ff0000>{3,1,4,2}</FONT><FONT color=#000000> {3,2,1,4}
{3,2,4,1} {3,4,1,2} {3,4,2,1}</FONT></FONT></DIV>
<DIV><FONT face=Arial size=2>{4,1,2,3} {4,1,3,2} {4,2,1,3}
{4,2,3,1} {4,3,1,2} {4,3,2,1}</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>I have already searched the database for the
sequence resulting from finding this number for each of n, and it returned
nothing. I based the search on the first 5 terms I have derived
empirically thus far. I know 6, but I did not include the 6th because,
although I checked it aganist all the characteristics I am aware of this
sequence possessing, I can not be absolutely sure of its validity. Does
anybody know anything about this, or even heard of such a sequence
before? Or perhaps somebody is aware of a program that might be able to
generate futher terms without manually checking each of the
permutations? Manual search can not hope to generate even
several more terms when the exponential growth of the factorial function is
considered.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV align=right><FONT face=Arial size=2>Sincerely,</FONT></DIV>
<DIV align=right><FONT face=Arial size=2>Christopher M.
Tomaszewski</FONT></DIV></BLOCKQUOTE></BODY></HTML>