I once suggested here (jokingly) the definition of mathematics as the study of SL_2(R) and related structures. With all due respect, taxonomy is not mathematics, even if you prefer a much broader definition. As I wrote some time ago, you are classifying ordered trees with respect to symmetry, which can be measured by the number of different labelling of the same tree.

To make a theory out of all this, you are welcome to pose and attempt to solve some real problems, such as:
- how fast does the number of those trees grows with the number n you are presenting? (does it get bigger than any polynomial in n? bigger than 2^n (2-to-the-power n)? is there a number c such that the number of trees, when divided by c^n, approaches 1?)
- what is the connection between the number of trees and the partition function p(), which counts the number of ways a number can be written additively (for example p(4)=5 since 4=3+1=2+2=2+1+1=1+1+1+1; this function was seriously studied first by Ramanujan).
- are there any natural operations that can defined on the set of trees, which will have interesting properties (such as x+y=y+x or x+(y+z)=(x+y)+z ) and transform this set into an object of some interest?
- can your notions of information and clarity degree be (first of all, defined, then) quantified beyond the level of yes-no? Can they be extended to infinite ordinals?

hi uzi

t(n is much bigger then p(n
And therefore any polynomial
And much more complicate so you cannot expect to have any simple formula

But there is written algorithm to calculate it

we are talking here
only about mathematics
hebrew english
have bo different

a note written today by robert Ihnot

According to the Russian paper Pravda, "Fermat's theorem is the unproved theorem indicating that the equation xn + yn = zn has no solution for x,y,z nonzero integers when n is greater than 2."

However, the matter has changed: "Doctor of Technical Sciences Alexander Ilyin will present his proof of Fermat's theorem at a meeting to be held at the Academy of Aviation and Aeronautics. His colleagues in Omsk believe Alexander's proof is flawless and simple."

http://english.pravda.ru

Remember you heard it here first!

"I have a friend in Minsk, Who has a friend in Pinsk, Whose friend in Omsk Has friend in Tomsk With friend in Akmolinsk."

There is a general proof by means of the Taniyama-Shimura theorem but it is subject to certain conditions. )

Emptiness {} and Fullness {__}, are the natural limits of any language, formal or informal.

Let Emptiness be represented by {}

Let a point be represented by {.}

Let a segment be represented by {._.}

Let Fullness be represented by {__}

By {} ← x(={.}) we mean that {.} is a potential {}.

By x(={._.}) → {__} we mean that {._.} is a potential {__}.

Monadic Mathematics is:

({},{__}):={x|{} ← x(={.}) AND x(={._.}) → {__}}

A set is only a framework to explore our ideas.

The concept of an oredered set does not depend on the quantity concept as shown here:

By Complementary Logic multiplication is noncommutative,
but another interesting result is the fact that multiplication
and addition are complementary opreations that can be ordered
by different internal symmetrical degrees where the quantity
remains unchanged, for example:

A Number is anything that exists between ({},{__})

Or in more formal definition:

({},{__}):={x|{} <-- x(={.}) AND x(={._.})--> {__}}

Where -->(or <--) is ASPIRATING(= approaching, but cannot become closer to).

If x=4 then number 4 example is:

Number 4 is a fading transition between multiplication 1*4 and
addition ((((+1)+1)+1)+1) ,and vice versa.

This fading transition can be represented as:

(1*4)..............={1,1,1,1}.<-------------.Maximum symmetry-degree,
((1*2)+1*2)........={{1,1},1,1}..............Minimum information's
(((+1)+1)+1*2).....={{{1},1},1,1}............clarity-degree
((1*2)+(1*2))......={{1,1},{1,1}}............(no uniqueness)
(((+1)+1)+(1*2))...={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1)..........={{1,1,1},1}
(((1*2)+1)+1)......={{{1,1},1},1}
((((+1)+1)+1)+1)...={{{{1},1},1},1}.<------ Minimum symmetry-degree,
..............................................Maximum information's
..............................................clarity-degree
..............................................(uniqueness)

First let us examine a well-known relation between mathematical objects and their representations.

=>> is ‘represented by’

|{}|=>>0 ; |{{}}|=>>|{0}|=>>1 ; |{{},{{}}}|=>>|{0,{0}}|=>>|{0,1}|=>>2 ;

|{{},{{},{{}}}}|=>>|{0,{0,{0}}}|=>>|{0,1,2}|=>>3 ; …

Let us see how we use my method to construct a collection of R members.

R members are constructed like this:

1) First let us examine how we represent a number by my system:

=>> is ‘represented by’

a) |{}|=>>0

b) There is 1-1 and onto between ‘0’ and the left edge of {._.} and we get {‘0’_.}

c) |{{}}|=>>|{0}|=>>1

e) There is 1-1 and onto between ‘1’ and the right edge of {._.} and we get {‘0’_’1’}

In short, {.} is the initial place of R collection, which is represented by ‘0’, where {‘0’_.} is the initial place of the second place of R collection, which is represented by ‘1’, and we get our first two must-have building-blocks of R collection.

2) When we get {‘0’_’1’} we have our two must-have numbers, which are ‘0’
and _’1’. Be aware that ‘0’ is the representation of {.} where ‘1’ is the representation of {._.}.

3) If we get {.}_AND_{._.}, then and only then we have the minimal must-have information to construct the entire R collection because:

a) We have ‘0’ AND _’1’ that give us the to basic scale factors 0 and _1.

b) We also have our initial domain _1, which stands at the basis of any arbitrary scale factor that is determined by the ratio between the initial domain _1 and another segment, which is smaller or greater than the initial domain _1 , for example:

0 = .

1 = 0__1

2 = 0____2

3 = 0______3

.5 = 0_.5

0_________pi

The negative numbers are the left mirror image of the above numbers.

There is no division in my number system because both {.} and {._.} are indivisible by definition.

In short, any segment is an independent element that clearly can be shown in the above 2-D representation.

If we use a 1-D representation, we get the standard Real-line representation, but then we can understand that division is only an illusion of an overlap of independent elements when they are put on top of each other in a 1-D representation,
for example:

0_.5__1_____2_____3__pi

Since in my system nothing is divisible, then '/' stands for a ratio between at least any given two (indivisible) numbers.