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<BODY background="" bgColor=#ffffff><FONT face=Courier size=1>in some of the
stuff below, it can be argued that<BR>a single "strategy" connects distant
sequences.<BR>Maybe arbitrary, maybe original & productive.<BR>(cum grano
salis)<BR><BR>Consider all possible combinations of two-argument<BR>functions
P,Q and R, wraped around their arguments<BR>a,b,c and d, as in<BR><BR>P[a,
Q[b, R[c, d]]]<BR>P[a, Q[R[b, c], d]]<BR>P[Q[a, b], R[c, d]]<BR>P[Q[a, R[b, c]],
d]<BR>P[Q[R[a, b], c], d]<BR><BR>the 5 results represent the five
"bracketings"<BR>with 3 sets of brackets (= 3 functions).<BR>(bracketings are
counted by Catalans, or binary trees).<BR><BR>Same thing can be done for w=1 ...
6 or more functions<BR>with (w+1) arguments, but calculations get nasty &
slow.<BR><BR>And now for some counting.<BR>You'll notice that both functions and
arguments remain<BR>"in sequence" (please pardon this cheap pun).<BR><BR>What if
we replace the functions with functions A or B,<BR>in all 2^w combinations,
<BR>PQR -> AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB.<BR>and, to top that, we present
the arguments in any order,<BR>effectively generating all permutations of
them.<BR><BR>What can this result be called?<BR>"<BR>All possible outcomes of
applying the functions A and\or B<BR>warped w times on the w+1 arguments a[1]
... a[w+1]<BR>"<BR><BR>Now, imagine that functions A and B have no special nice
properties,<BR>Then every result is different, and we easily count the
*different*<BR>results as function of w: 2^w cat[w]
(w+1)! <BR>or {4,48,960,26880,967680, 42577920}=A052714=(2w)!/w! 2^w
<BR><BR>BUT<BR>the functions A and B can be given the properties of
associativity<BR>and commutativity, or "flat-ness" and "orderless-ness",<BR>the
first implying A[b,A[b,a]] equals A[b, b,
a]<BR>the second means A[b,a] equals A[a,b] as you all know.<BR><BR>RESULTS:
<BR>{function A} {function B} w={1, 2,3
,
4}<BR>{nil}
{nil}
{4,48,960,26880} A052714<BR>{Orderless}
{nil}
{3,27,405, 8505,229635} A011781<BR>{Orderless}
{Orderless} {2,12,120, 1680,30240} A001813
<BR><BR>{Flat}
{nil}
{4,42,744,18480} do not match anything
<BR>{Flat}
{Orderless} {3,21,249, 4155} do not match
anything
<BR>{Flat,Orderless}{nil}
{3,25,351, 6901} do not match anything
<BR>{Flat,Orderless}{Orderless} {2,10, 86,
1036} do not match anything
<BR><BR>{Flat}
{Flat}
{4,36,528,10800} A052716
<BR>{Flat,Orderless}{Flat}
{3,19,195, 2791} A053554 A048172
<BR>{Flat,Orderless}{Flat,Orderless} {2, 8, 52, 472} A006351
<BR><BR><BR>A052714 by encyclopedia@pommard.inria.fr, Jan 25 2000<BR>A011781 by
killough@wagner.convex.com (Lee D. Killough)<BR>A001813 by njas & James A.
Sellers (sellersj@math.psu.edu), May 01 2000<BR>A052716 by
encyclopedia@pommard.inria.fr, Jan 25 2000<BR>A053554 by njas, Jan 16
2000<BR>A048172 by njas<BR>A006351 by
njas<BR>---------------------------------------------------<BR><BR>If we drop
the (w+1)! permutations of the arguments,<BR>and only keep the cat[w]
parentesizations of the functions,<BR>w instances of A and b, then we
get<BR>{nil}
{nil}<BR>{Orderless}
{nil}<BR>{Orderless} {Orderless}<BR>are
all
{28,40,224,1344,8448,54912}
A052701<BR>
=cat[w]2^w <BR><BR>{Flat}
{nil}<BR>{Flat}
{Orderless}<BR>{Flat,Orderless}{nil}<BR>are
all
{2,7,31,154,820,4575,26398} A007863
<BR><BR>{Flat}
{Flat}
<BR>{Flat,Orderless}{Flat}
<BR>are
both
{2,6,22,90,394,1806,8558}
A006318<BR>
=2* A001003<BR>---------------------------------------------------<BR>needless
to say that fitting to a sequence, using a 4 term match,<BR>should be
called "poor & unsafe".<BR>Maybe someone with brains & training could
have a peek at this?<BR><BR><BR>"in obscuritate
finio"
;-)<BR>wouter.meeussen@pandora.be</FONT></BODY></HTML>