Encyclopedia of Integer Sequences

It is weird folks...

If I want to keep some compatibility with egf-s, I got some strange arithmetic :

1+1 = 1.1 = w where 1 is the arity of X and w is a notation

It follows - disregarding the associativity of multiplication - :

1.1 = 1 (in our world )
1.1 = 1 (in the mirror)
1.w = 1 + w
1.w = 1 + w
w.w = w + w + w

... I did not get yet a "contradiction"...

nick wrote:

w.w = w + w + w

There are alternative systems :

System E
1.1 = 1 + 1
It has a model, the even numbers starting with 2.

System T
1.1 = 1
1.1 = 1 + 1
w.w = w + w + w
It has a model, three distinct worlds that start with 1, 2, and 3.

The above are associative under multiplication. Do you have a model for your non-associative wonder numbers ?

Yes I have. Take N×N with
(a, b) + (c, d) = (a+b, c+d) and
(a, b).(c, d) = (ad + bc +ac, ad + bc + bd) ; then
1 = (1, 0)
1 = (0, 1).

There is nothing unusual here...

The arity of X is (1,0) and the arity of X is (0,1). Tracking back,

The arity of F_n is (n, 0) and the arity of 1/F_n is (0, n).

Example :
The Wonder Species Pascal , the natural generalization of Ens.
I have a set-box , but only a part of the slots are visible in a mirror.
I stick a cube. The stuck cube is either visible in the mirror, or not.

Pascal' = Pascal'^R + Pascal'^M

nick wrote:

The Wonder Species Pascal , the natural generalization of Ens.
I have a set-box , but only a part of the slots are visible in a mirror.
I stick a cube. The stuck cube is either visible in the mirror, or not.

Pascal' = Pascal'^R + Pascal'^M

Bro ? do you believe that ?
Pascal' = Pascal'^R + Pascal'^M = Pascal

Anyway Nick, good work ! Here is a dedication to you :

G funk wrote:

Transposition = X.X = X + X
either you put an egg in the mirror cup, or you put it in the real cup.
Anyway, after placing one egg, the other cup is automatically fulled.

Par Lewis Carroll ! I will maintain the name of Wonder Species !
Here is how it works :

X(X) = X(X)= X.X = X + X

I have only one cube; I place it in some mirrored real slot or in a real mirrored slot. In both cases, I obtain a filled slot in the Real world and one in the Mirror world. The Wonder Table is :

0 1 0 0 0 0 0 ....
1 0 0 0 0 0 0 ....
0 0 0 0 0 0 0 ....
0 0 0 0 0 0 0 ....
....
....

The word "arity" stands for the wonder cardinality (n, m). (the old "arity" became useless by passing to species, that are independent of arity)

And here starts a new inquire,
- what known things fit to the Arithmetics of Arities and
- they get a new shorter formulation using the four basic operations plus the transposition :

1/F or, equivalently, F ?

=========
Let me note a good news,
At CERN have trapped 38 anti-atoms, atoms made of anti-matter.


The above suggests the notation
} a {
for an anti-individual.

I think some more explanations are needed, even at Math Club was already explained the difference between combinatorial groups and topological groups.

There is a logical explanation of the combinatorial fact that Groupe'=Lin.

As known, one of the secrets of the combinatorial identities is that they catch an object and its very next context.

Hence,
Elem = X.Ens
reflects the very next context of an element (X) inside a set contains the other elements forming a set (Ens).

Subset = Ens.Ens
says that the very next context of a subset (the first Ens) contains also the complementary subset (the second Ens).

Similarly, the very next context of a group contains the other permutations of the Symmetric group. A group of order n is also a transitive (Burnside) subgroup of Sym(n) (Cayley).

We have a one-to-one correspondence between
- groups
- combinatorial groups
- transitive subgroups of Sym(n) (of order n)
- actions of Sym on the cosets of the target group
- molecular species Groupe.

While deriving at the fourth hyphen level, one gets a Lin.
(And I am also working to the Cubes-and-Puzzles explanation)

Nick, sometimes the most obvious things are the hardest to explain.

Either you take your time and you draw an extended Cayley table, of n×n!, instead of a regular n×n table,

Or you take your time and explain with caution the sticking theory, because no one likes stuck pieces. In the real world, stuck pieces means broken mechanisms.

Both cases, it is a nice winter here... let's enjoy it !

A000248

Sets of dices.

TO DO

============
A000001 Number of groups of order n.
(Formerly M0098 N0035) 107
1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, 1, 10, 1, 4, 2 (list; graph; listen; history; internal format)
OFFSET
1,4
COMMENTS
Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Dec 16 2004
=============


The OEIS sequence A000001 missed the number of primitives of Lin Species.

Fellows, wish me luck, because I will try (sooner or later) to edit this entry.

Hi fellows, do you believe this ? http://oeis.org/A000001

Nice hit Nick !

How about A069513 ?

It looks like they are primitives of cycs !

You know, the so-called fields, as the tetrahedron, A4 acting on four vertices ?