<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; charset=iso-8859-1">
<META content="MSHTML 6.00.2800.1106" name=GENERATOR>
<STYLE></STYLE>
</HEAD>
<BODY bgColor=#ffffff>
<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></BODY></HTML>