connections among distant sequences

[Wouter Meeussen]

in some of the stuff below, it can be argued that
a single "strategy" connects distant sequences.
Maybe arbitrary, maybe original & productive.
(cum grano salis)

Consider all possible combinations of two-argument
functions P,Q and R, wraped around their arguments
a,b,c and d,  as in

P[a, Q[b, R[c, d]]]
P[a, Q[R[b, c], d]]
P[Q[a, b], R[c, d]]
P[Q[a, R[b, c]], d]
P[Q[R[a, b], c], d]

the 5 results represent the five "bracketings"
with 3 sets of brackets (= 3 functions).
(bracketings are counted by Catalans, or binary trees).

Same thing can be done for w=1 ... 6 or more functions
with (w+1) arguments, but calculations get nasty & slow.

And now for some counting.
You'll notice that both functions and arguments remain
"in sequence" (please pardon this cheap pun).

What if we replace the functions with functions A or B,
in all 2^w combinations, 
PQR -> AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB.
and, to top that, we present the arguments in any order,
effectively generating all permutations of them.

What can this result be called?
"
All possible outcomes of applying the functions A and\or B
warped w times on the w+1 arguments a[1] ... a[w+1]
"

Now, imagine that functions A and B have no special nice properties,
Then every result is different, and we easily count the *different*
results as function of w:     2^w cat[w] (w+1)! 
or {4,48,960,26880,967680, 42577920}=A052714=(2w)!/w! 2^w 

BUT
the functions A and B can be given the properties of associativity
and commutativity, or "flat-ness" and "orderless-ness",
the first implying    A[b,A[b,a]]  equals  A[b, b, a]
the second means A[b,a] equals A[a,b] as you all know.

RESULTS: 
{function A}    {function B}   w={1, 2,3  ,    4}
{nil}           {nil}            {4,48,960,26880}  A052714
{Orderless}     {nil}            {3,27,405, 8505,229635}  A011781
{Orderless}     {Orderless}      {2,12,120, 1680,30240}  A001813 

{Flat}          {nil}            {4,42,744,18480}  do not match anything 
{Flat}          {Orderless}      {3,21,249, 4155}  do not match anything  
{Flat,Orderless}{nil}            {3,25,351, 6901}  do not match anything 
{Flat,Orderless}{Orderless}      {2,10, 86, 1036}  do not match anything 

{Flat}          {Flat}           {4,36,528,10800}  A052716 
{Flat,Orderless}{Flat}           {3,19,195, 2791}  A053554  A048172 
{Flat,Orderless}{Flat,Orderless} {2, 8, 52,  472}  A006351 


A052714 by encyclopedia at pommard.inria.fr, Jan 25 2000
A011781 by killough at wagner.convex.com (Lee D. Killough)
A001813 by njas & James A. Sellers (sellersj at math.psu.edu), May 01 2000
A052716 by encyclopedia at pommard.inria.fr, Jan 25 2000
A053554 by njas, Jan 16 2000
A048172 by njas
A006351 by njas
---------------------------------------------------

If we drop the (w+1)! permutations of the arguments,
and only keep the cat[w] parentesizations of the functions,
w instances of A and b, then we get
{nil}           {nil}
{Orderless}     {nil}
{Orderless}     {Orderless}
are all                          {28,40,224,1344,8448,54912} A052701
                                 =cat[w]2^w 

{Flat}          {nil}
{Flat}          {Orderless}
{Flat,Orderless}{nil}
are all                          {2,7,31,154,820,4575,26398} A007863 

{Flat}          {Flat}           
{Flat,Orderless}{Flat}           
are both                         {2,6,22,90,394,1806,8558} A006318
                                 =2* A001003
---------------------------------------------------
needless to say that fitting to a sequence, using a 4 term match,
should be called "poor & unsafe".
Maybe someone with brains & training could have a peek at this?


"in obscuritate finio"       ;-)
wouter.meeussen at pandora.be
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