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 Encyclopedia of Integer Sequences

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nick
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PostSubject: Encyclopedia of Integer Sequences   Fri Oct 09, 2009 10:01 pm

Some of the sequences listed in Encyclopedia have explanations in terms of species identities. I think it could be useful to list some identities. Caution, a sequence does not uniquely define a combinatorial species.

=======
A001787
Number of edges in n-dimensional hypercube.


X . ENS . ENS

I fix a vertex, and a choose one of n adjacent edges. (here comes X)
Once edge is fixed, the adjacent faces, cubes, and so on are also fixed.
By cutting the n-hyper-cube with a median hyper-plane of my initial edge I got a (n-1) hypercube whose vertices lays one-to-one in the middles of the parallel edges of my initial edge. So each subset of the (n-1) remaining directions define uniquely the path to a parallel edge. (SUBSET = ENS . ENS)

=======
A003465
Number of ways to cover an n-set.

FAM = COVER . ENS

Given a family of non-empty sets FAM, there are individuals covered, so they form a COVER and there are non-covered individuals, which form a set (ENS).
WARNING ! The exponential generating series displayed in Encyclopedia could be divergent, having no function to represent.

=======
A000110
Number of partitions of a set of n labeled elements.

PART = ENS ( ENS*)

A partition is a set of non-empty sets.

=======
A000079
Number of subsets of an n-set.

SUBSET = ENS . ENS = LIN2 ( ENS )

To give a subset means to cut into two sets, the chosen ones and the others, or to establish an oriented pair of sets.

========
A000012
The simplest sequence of positive numbers: the all 1's sequence.

ENS

Sets, the only structures having this sequence that can be defined on n-sets . Also known as cardinal numbers.

=========
A000142
Factorial numbers

a) LIN' = LIN . LIN = LIN2(LIN)

Linear orders of an n-set. Also known as ordinal numbers.

b) PERM = ENS ( CYC )

A permutation is a set of cycles.

c) LIN = CYC'

A linear order is a stuck cycle . Also known as cyclic numbers.


=== here are some books.pdf on species and EGF (SGE)===

http://web.mac.com/xgviennot/Xavier_Viennot/cours_files/Ch3.pdf
http://algo.inria.fr/flajolet/Publications/book.pdf
http://bergeron.math.uqam.ca/Site/bergeron_anglais_files/livre_combinatoire.pdf
http://www.math.uwaterloo.ca/~dgwagner/CO220/co220.pdf

=== a personal definition of these objects ===


The two structures above have something in common, an abstract object, a mathematical object. First view points to the number of vertices that is the same.
A second common thing occurs when one wants to label the two structures. Whatever is there, I love it.


Last edited by nick on Sun Oct 10, 2010 4:26 pm; edited 4 times in total
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PostSubject: Re: Encyclopedia of Integer Sequences   Mon Oct 12, 2009 12:30 am

A055882
1, 2, 8, 40, 240, 1664, 12992, 112256, 1059840, 10827264, 118758400,...


ENS ( SUBSET - 1 ) = PART × SUBSET

sets of objects labeled with subsets or, defined simultaneously on the same set, a partition and a subset.

There are twenty six labels for n = 3
[ _ | a ], [ _ | b ], [ _ | c ], [ a | _ ], [ b | _ ], [ c | _ ],
[ b | a ], [ c | b ], [ a | c ], [ a | b ], [ b | c ], [ c | a ],
[ _ | a, b ], [ _ | b, c ], ], [ _ | c, a ], [ a, b | _ ], [ b, c | _ ], [ c, a | _ ],
[ c | a, b ], [ a | b, c ], ], [ b | c, a ], [ a, b | c ], [ b, c | a ], [ c, a | b ],
[ _ | a, b, c ], [ a, b, c | _ ] ;

The subset [ _ | _ ] of the empty set is not allowed ( there is link with the -1 in the formula) and every labeling contains exactly one time the symbols a, b, and c.

I'd say, if a representation does not works with sets try with boxes.

there are
8 labellings with labels in fourth and fifth rows,
8 labellings with labels in the first row
12 labellings with labels in the first row and second row
12 labellings with labels in the first and third row

==========
A000166
1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961,...
derangements, permutations with no fixed points


DER . ENS = PERM

A permutation has two kinds of point, the fixed ones that form a set, and the ones that loose the original position by permutation.
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PostSubject: Re: Encyclopedia of Integer Sequences   Mon Jan 25, 2010 10:36 am



FUNC (X, Y) = ENS ( X. ENS (Y) )

A function is a set of couples of (sets of y's and x's) ; func (x,y) = exp (x. exp(y))


SURJ (X, Y) = ENS ( X. ENS* (Y) )

A surjection is a set of couples of (non-empty sets of y's and x's) ; func (x,y) = exp (x. (exp(y) - 1))

We get ENS (X) . SURJ (X, Y) = FONC (X,Y)

INJ (X, Y) = ENS (X). ENS (X.Y)

An injection is a couple of (x's and a set of couples of x's and y's); inj (x,y) = exp ( x + x.y )

How it works ? it is underarithmetics !
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PostSubject: Re: Encyclopedia of Integer Sequences   Sat Oct 09, 2010 5:55 am

A007395
2, 2, 2, 2,...
The Alternating Species

ALT


If one agrees that species of numbers may be seen as series of the symmetric group actions and that we may associate
- the Sn acting on 1-element set with cardinal numbers
- the Sn acting on itself with ordinal numbers

it naturally comes to ask about An, which is a normal subgroup of Sn (in addition to 1 and Sn).

If we may see a set (cardinal number) as a collection of individuals in a bag, or we may see a sequence (ordinal number) as a linear order of individuals, what is the case for alternating numbers ?

Take a look :


In such a box we can place individuals in precise two ways modulo legal moves.

Here, to take the derivative means to stick one cube.
Known as "ensemble orienté" , notation E±, we have
ALT' = ALT

Remark: If I remove a cube, the grid-box becomes a simple bag; (and I am still working on what I obtain if I add a cube) (I am a bit confused about placing individuals in named-slots)(if the names of slots are First, Second, Third..., ordering means naming. Nevertheless, an observer whom does not know English is around, he will conclude that I am naming objects ; Shocked naming means ordering ? ).

OK, one more try. Let say I have three little tomcats.
- I also have three good name, Jack, Tom and Huck, and I want to name them.
- The other thing I can do, is to align them from left to right.

There are precise six ways to do each of above. Moreover, the actions of Sym(3) on this are transitive, since permuting tomcats I can reach all configurations of names of positions. There is only one transitive action of Sym (3) on a 6-element set. Thus, naming means ordering and vice-versa. So,
- where vanish the Peano's rules ? how could be established something like NAMES' = NAMES.NAMES ?


---------------
It is the right time now to try my sticking technique, let's say, I will try to stick a permutation.

PERM' = CYC'. PERM

Yes, I have to stick all the cycle my stuck individual belongs to, while the rest of individuals freely permute.
Of course, if I stuck one individual in a bag, the rest of them will rest in a bag :

ENS' = ENS

A stuck subset is the sum of two other subsets, one when the stuck individual is inside the initial subset, one when it is outside :

SUBSET' = SUBSET + SUBSET (a table would be good) and

ENS*' = ENS Sticking a non-empty set, one obtains a set.

PART' = ENS.PART A partition with a stuck individual is formed by the class of the stuck one and the rest of partition.

Also, when sticking a proper cycle inside a permutation, one obtains a nonempty LIN and the rest of permutation:

DER'.ENS = LIN.PART

==============
and a nice definition for a cycle,
A cycle is a stick-one-stuck-all structure
(CYC' = NAMES)





There is a natural way to build stick-one-stuck-all species.

- Start with a group H of order n.
- represent H as a subgroup of Sym(n). It acts now transitively on its elements, and each symbol 1,a,b,c,... in H is transported exactly one time in any other symbol. If { 1, a, b, c,....} are the symbols of H, there is a unique permutation in H that transports x in y.

- Consider the (n-1)! classes H, sH, tH, sH....
- Such a class, let's say sH, will also permute transitively {1, a, b, c,...} because |sH| = n and sx(z) = sy(z) implies x = y (x, y are in H and z is a symbol in H).

In fact, for each z in H, we will find in every class exactly one permutation that fixes z.
This is good, because it means that each class contains one "pure” permutation of given (n-1) symbols, and classes may be represented without implying one specified element of H.

- Sym (n) acts transitively on classes, producing a species S.
- By construction, we have Stab(H) = H. The index of H in Sym(n) is, by construction, (n-1)!

By disregarding one of symbols, the remaining (n-1)! "pure" permutations will continue to act transitively on (n-1)! Classes, delivering a nice LIN species, which satisfies S' = LIN.

Abstract :
To give a subgroup H of Sym(n) means also to give a species, considering the multiplication on left cosets. The trick of this construction is that the group has a large index in Sym(n), of (n-1)! Greater subgroups of Sym(n) will not behave like this; they will act transitively on a smaller amount of cosets. For example, to Cayley-embed An one needs (n!/2)! symbols, that will produce the necessary amount of cosets. In conclusion, the n-subgroups of Sym(n) deliver species S that satisfy S’ = LIN(n-1).

Thus, groups are stick-one-stuck-all species, integrals of LIN.

Reversely, given a transitive action table with n! rows and (n-1)! columns, one should be able to uniquely recover a group by taking Stab(some element).


For example, the cube above IS Z2 × Z2 × Z2 as species and it has the derivative LIN7.


=======
let me note another ideea,
I also got the X - take an egg cup :

What is the issue here ? the issue is that with these recipients of various capacities, and the individuals, and my sticking technique, and a table hehe, is the time for me to read one more time some books bounce I like.
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PostSubject: Re: Encyclopedia of Integer Sequences   Thu Oct 14, 2010 11:53 am

nick wrote:

Obviously, the fourth basic operation, namely the composition is no more simple.
Could it be simple ?

=======
A067994 (almost)
2, 0, 12, 0, 120, 0, 1680, ....

E(X.X), labeling batteries before recycling.

A123023]
1, 0, 3, 0, 15, 0, 105, 0, 945, 0, 10395, 0, 135135,...

E(E2), labeling fuses before recycling.

========after recycling three bateries =======
My stick-one-stuck-all structures, integrals of LIN, seem to be more groups-acting-on-themselves, or Cayley-tables, than subgroups of order n in Sym(n).
My question is, now, could exist a group so abstract that it does not act on something ?
==================================

Here is a counter-example, E3(B), recycling three labeled batteries.
B{a,b} = {B1, B2} where B1 could be {a, {b}} and B2 the other one {b, {a}}.
Then
E3(B){a, b, c, d, e, f} =
{
{ B1{a,b}, B1{c,d}, B1{e,f}},
{ B1{a,b}, B1{c,d}, B2{e,f}}
{ B2{a,b}, B2{c,d}, B2{e,f}}
....
{ B1{a,c}, B1{b,d}, B1{e,f}},
......
{ B2{a,f}, B1{b,e}, B1{c,d}},
}
there are 8.15 = 120 E3(B) structures on my six labels a,b,c,d,e, and f.

The relabeling action is transitive, since B1{x,y} may be transported in B2{x,y} by the permutation (xy) and
{B1{xy}, B1{z,t}, B1{u,v}} may be transported in any other {B1, B1, B1}.

The stabilizator of { B1{a,b}, B1{c,d}, B1{e,f}} contains six permutations :
Stab = (), (a c e)(b d f), (a e c)(b f d), (a c) (b d), (a e)(b f), and (c e)(d f).
This molecular species is revealed as the transitive action of Sym(6) on the 120 cosets of Stab.
This is a counter-example, not every subgroup of index (n-1)! of Sym(n) delivers a LIN by derivation.

E3(B) ' = E2(B). (X+X) <> LIN5

Nevertheless, even it is isomorphic to Sym(3), Stab is here a non-transitive (two blocks) action on {a, c, e} and {b, d, f}.
To deliver a LIN by derivation, a species (subgroup of Sym(n)) must be the natural (canonical) embedding of an n-group, or the group-acting-on-itself, or the so called abstract group. Let me try :

ABSTRACT GROUP' = ORDINAL NUMBER

How does it look ? (I am thinking at a T-shirt Rolling Eyes)
...let me think ...let me think

something like : Ag' = On
... it looks too chemical... someone could believe that they have found silver in Ontario...
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PostSubject: Re: Encyclopedia of Integer Sequences   Sat Oct 23, 2010 9:38 pm

nick wrote:

something like : Ag' = On
... it looks too chemical... someone could believe that they have found silver in Ontario...

So what ? It looks nice. Why should under-arithmetics have to follow those scary math notations ?
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PostSubject: Re: Encyclopedia of Integer Sequences   Sun Oct 24, 2010 6:40 am

Hi Doc ! you again ? Take a look here :
http://math.ucr.edu/home/baez/qg-spring2004/octopus_toby.pdf

This fellow says :
Quote :
(I've also drawn and counted the first 18 octopi.)
Doesn't looks to you like a case of is some incomplete induction ? Anyway :

A029767
Labeled octupi with n nodes.

Within respect to the authors' notation,
Oct = C(L+)


=========
Doc, I got something new : I have just watch a demo at the club, about stick-one-stuck-one structures... of course, they are known in Mathematics as "Platonic Solids" bom
Yeah, I know you want a debriefing...

It was like that : after derivation, the speaker used the other hand for the stuck point; while the rest of the points remained with some liberty, following some synchronous cycles :

Object' = X.(sum of some cycles(lin)).

CUBE' = X. (6.CYC4(X6) + 8.CYC3(X8) + 12.CYC2(X12))

... labeling a cube is a mess, I got a factor 61 which comes from 6/4+8/3+12/2. I really thought that a cube is something more "spherical".

My stick-one-stuck-one species must be some SPECIES(X.X), or, structures labeled with "batteries".
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PostSubject: Re: Encyclopedia of Integer Sequences   Sun Oct 24, 2010 10:23 pm

nick wrote:
CUBE' = X. (6.CYC4(X6) + 8.CYC3(X8) + 12.CYC2(X12))

Very Happy You Shouldn't be disappointed.

I see above some d(1/4+1/3+1/2) = d + 1 + 1 where :
d is the order of the cubical group,
1 stands for the one-stick
1 stands for the one-stuck
1/4, 1/3 and 1/2 are coefficients in some e.g.f.

You should try to see what happens in four or more dimensions, even there are no more exclusively cycles there. Since you said that Ag' = Cyc', they must be equipotent. Just replace Cyc with Ag and see what happens.

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PostSubject: Re: Encyclopedia of Integer Sequences   Sun Oct 24, 2010 11:33 pm

Doesn't work, doc.

There are no more cycles there, but cubical groups, dihedral ones and so on, and these are not abstract groups, but groups acting on something else then themselves.

Sorry doc, there are no groups like abstract groups !

However, to do this requires to explain a group-action in terms of species- that are group actions for me.
Before this, I'd have to explain first n! as a subgroup of n!! :

A000197
1, 1, 2, 720, 620448401733239439360000
a(5) has 199 digits and is too large to include.

================= The definition of a (labeled )platonic solid / could be useful ==========

Ps' = X. (sum of something of LIN) - where LIN describes the solidity
Ps = something(X.X) where X.X describes the stick-one-stuck-one hypothesis

================== Before printing a T-shirt ===========

Fellows, there are three kinds of groups.

Type 1: The good old fashion groups of permutations - see for example the Burnside book.
Type 2: The Cayley groups, subsets of G × G × G,
Type 3: and the Abstract groups, stick-one-stuck-all puzzle-boxes, defined as the primitives of Lin species, for each n. Not for all, but for each.

To underline the relations among these types of groups there is a huge example in Math :
- take the morphisms V ----- >W of vectorial spaces. They are simply to describe, just formulate the element of a W base in terms of a V base.
- then add a new equation, to form a particular case : V=W. Now, to describe a morphism is no more simple, but is something about Jordanisation;

The example above clarifies the link between the first two types of groups : a Cayley group acts on itself, not on something else.
It is like adding to the definition of an action G × M ----> M a new equation, G = M, to obtain a very generous particular case.

Let's now take a type 3 group, a stick-one-stuck-all puzzle box of capacity n; let us note this puzzle box with PB.
By derivation, unlike the case of a lin-box where I obtain n distinct other boxes, in this case I obtain only one puzzle-box, a (n-1) lin-box.
The PB box has some liberty of relabeling (or re-placement of the cubes). The important thing is that by moving a cube from the slot A to the slot B, all other slots must be deranged; since if not, by sticking a cube in one unaffected slot C, the derivate of this machine would not more be a totally stuck one, or a lin-box. If you want, stick-one-stuck-all means derange-one-derange-all.

Let' now introduce another remark, after taking a look at a table of a type 2 group. Well, each permutation of the initial symbols acts by derangement.

In conclusion, Type 2 groups are Type 3 groups, primitives of LIN, acting by derangement.

How about reversely ?
Given a type 3 group,
- an abstract group,
- something equipotent to Cycn,
- sub-group of of index (n-1)!, in Sym (n),
- acting by deranging a n-box,,
- primitive of LIN (stick-one-stuck-all structure)

could I extract some Type 2 group ?

( or, equivalently, given an action of an n-group on an n-set,
knowing that is an action by derangement (excepting the 1), does this implies the action is transitive ? )

uf, seen like that is simple, YES. Take 1.A, a.A, b.A... the orbit of A. If it does not cover the whole n-set, there is some repetition, s.A=t.A in the action table. This means that another permutation than the identity does not derange A, contradiction !
Deranging implies transitivity, that implies the action is equivalent to the action on itself.

In conclusion, Type 3 groups are Type 2 groups as numberoidal structures.


Definitivelly, Cayley-groups are primitives and the only primitives of LIN.



A000001
Number of groups of order n.

.
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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Oct 26, 2010 6:29 am

nick wrote:
numberoidal

Shocked Numberoidal ? Never heard about this, google certified. What could a numberoid be ?
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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Oct 26, 2010 6:58 am

Doc, there are many kinds of numbers.

The most known of them are the cardinal numbers and the ordinal numbers.

At the same level of understanding, the other kinds of numbers are numberoids.

We have:
- cardinal numbers,
- ordinal numbers, (these two are numbers)
- 15-puzzles,
- cycles and groups, or stick-one-stuck-all numberoids,
- platonic solids, or stick-one-stuck-one numberoids,
- rubik cubes,
- trees,
- and many many other types of numberoids.

The glue that stick them at the same level of understanding is a small amount of combinatorial operations. Hence the name of "Combinatorial Species".

Doc, Combinatorics infiltrated Mathematics for centuries, just take for example (a+x)n, where, par Descartes, a is a constant, x is a variable, and n is an exponent.

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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Oct 26, 2010 9:41 am

Well, Nick, I got a suggestion for your T-shirt :

Ag' = Cyc'

where :
Ag stands for Abstract groups/ Cayley groups / s1sA puzzles etc.
Cyc stands for the combinatorial species Cyc.

You got two advantages :

1) You clearly caution about the isotopes of Cyc's, and
2) You do not have to warn about LIN = NAMED.
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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Oct 26, 2010 12:45 pm

Dr. Post wrote:
to warn about LIN = NAMED.

Why, will you print your smock ?
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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Oct 26, 2010 12:55 pm

oh, nope.

study It just happens that I am satisfied by the dr. Kroneker's conclusion, that's all. Even it is not about numberoids, but about integers. For example, let's say you do not have the concept of -4. Then, take a mirror and look through it to a "4" digit.

We see only half of a 3D space. That are mirrors good at, to take a look to the other part of our space. In the case of LIN and NAMED, we have a similar "distorsion" of perception, some kind of "temporal" distorsion. I'd say, Math is the most perfect human "imperfection".

Anyway, since we are a little off-topic, I'd suggest to go back to our books and papers. I bet you are eager to write down the combinatorial definition of the hecatonicosachoron.
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PostSubject: Re: Encyclopedia of Integer Sequences   Tue Oct 26, 2010 2:50 pm

Then Ag' = Cyc' will be.

I also see too advantages :

- Cyc avoids all that sophisticated stuff you have just explained about Lin, On and Names;
- Ag signifies that I have finally managed to understand some abstract things.


========

Doc, since these combinatorial exercises I developed some kind of "sterescopic perception".
Take a look to a Cayley table that has a normal sub-group.

Does't looks to you like a clear combinatorial proof that

A normal subropup is a group of group ?
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PostSubject: Re: Encyclopedia of Integer Sequences   Wed Oct 27, 2010 12:09 pm

nick wrote:
I developed some kind of "sterescopic perception".

Mathematics of the twentieth century are based on the linear double perception. For example, in the sequence

Pi(r.m) = r.m + N = r.(m +N) = r.Pi(m)

the second and the third terms are "double objects". Then, you explicitly write one of them in the left side , and the other in the right side. Hence the well-known acronyms of LHS and RHS.

This happened because of the democratization of the typewriters
wikipedia wrote:
By about 1910, the "manual" or "mechanical" typewriter had reached a somewhat standardized design.
The typewriters dramatically increased the speed of math papers writing.

Definitively, your proof by taking a look at a Cayley table could be a combinatorial proof, in the best mathematical meaning.

But, if someone accepts such non-linear-double-perception-based proofs, it will be also forced to accept pure synthetic geometry proofs, that are also 2D or more. This could not always be the case :
wikipedia wrote:
Dans l'enseignement secondaire en France (et aussi dans d'autres pays), la géométrie synthétique a eu tendance à être supplantée par la géométrie analytique dans les années 1960-1970, lors de la période dite des mathématiques modernes, avant d'effectuer un retour en tant que base de l'apprentissage du raisonnement.

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PostSubject: Re: Encyclopedia of Integer Sequences   Wed Oct 27, 2010 2:09 pm

Doc, this is another sophisticated stuff that beats me.

The analytic geometry is a particular case of synthetic geometry, that requires to draw two (or more) perpendicular axes (dites cartesiennes, par Descart), to project the points of the problem on these two (or more) axes, then to solve the problem. Sad I do not see the opposition they talk about.

Let's get back to the s1sA structures, and to the s1s1 structures, and reformulate :

Could describe an abstract normal sub-group defined by Ag(Ag) the numberoidal structure of a type 2 subgroup (a closed subset of a closed subset of G×G×G) ?

I mean, without introducing further operations. Has the notion of a (normal) subgroup a simple "synthetic" description in bold letters ?

Something like this :

Abstract normal subgroup = Ah(Ag)

where
- Ah is an abstract Holder group,
- Ag is an abstract group.

Of course, if Ah and Ag are simple groups, we have Ah(Ag) = Ag(Ah)


.
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PostSubject: Re: Encyclopedia of Integer Sequences   Wed Oct 27, 2010 10:21 pm

A000522 (nice small index !)
1, 2, 5, 16, 65, 326, 1957, ... (Gosh, all those guys never heard about Sokoban !

Sokoban' = Sokoban + Lin.Sokoban

By sticking a cube in a Sokoban puzzle,

Either I do not disconnect the box, obtaining a new Sokoban puzzle,
Or I disconnect the box : In the area without pusher all cubes are stuck while the pusher's area is a new Sokoban.

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PostSubject: Re: Encyclopedia of Integer Sequences   Thu Oct 28, 2010 9:39 am

nick wrote:
Then Ag' = Cyc' will be.

Hey Nick, I hope that you did'n print your T-shirt yet, since I got another suggestion.

Ag' = N

Ag meaning Abstract Groups, or stick1stuckAll structures.
' means to stick one cube
N stands for Named Numbers (ordinal numbers).

Then :
- you avoid the chemistry department, to not produce confusion;
- I will thing about one very important aspect:

What is that thing that allows:
- the construction of Lin over Named
- that allows to separate the combinatorial structure Cyc among the numberoidals Ag.

I see two advantages :
- the formula will engage only your reformulated terms/personal understanding
- it could be possible to isolate that special glue that links a combinatorial species(N+1) to the same combinatorial species(N), by comparing
Ag' = N to Cyc' = Lin

for example Z5 is linked to Z4 and not to Z2×Z2. Even you take the field Z5, its multiplicative group is [/b]Z4.

Once isolated the glue, this will allow us to talk about numberoidal structures when the context will require.

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PostSubject: Re: Encyclopedia of Integer Sequences   Thu Oct 28, 2010 10:04 am

Doc, how about:

Ng' = N

where
- Ng means Nick's personal understood groups, or Numberoidal groups,
- ' means Nick' personal understood derivation by sticking
- N stands for the ordinal numbers First, Second, Third,... read by a non-english speaker.

Thus :
- my T-shirt becomes a quite personalized T-shirt,
- I correctly avoid the G notation, that beats me,
- no chemical confusions
- and, also, it suggests that by derivation we got a tremendous loss of information !
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PostSubject: Re: Encyclopedia of Integer Sequences   Fri Oct 29, 2010 10:06 pm

nick wrote:
- I correctly avoid the G notation, that beats me,!

Whooa !
Bro, this fellow has a problem with the G-word, ain't you think ?
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PostSubject: Re: Encyclopedia of Integer Sequences   Fri Oct 29, 2010 11:32 pm

Bro, this fellow is quite confused. He does not know the difference between a definition and a proof.
nick wrote:
Does't looks to you like a clear combinatorial proof that
A normal subropup is a group of group ?
normal_sougroup is a word that occures only once, hence it must be a defined term, not a proofed one.

Nevertheles, since the G-word has no other combinatorial definition, it definitively stands for GROUP.
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PostSubject: Re: Encyclopedia of Integer Sequences   Fri Oct 29, 2010 11:52 pm

Yeah, G' = N looks better.

Nick, print your T-shirt G' = N.
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PostSubject: Re: Encyclopedia of Integer Sequences   Sun Oct 31, 2010 10:07 am

How about a college level proof that one cannot interchange two cubes in the 15-puzzle, other than my verification.
Par Rubik, I know how to cycle every three of them !

The Wikipedia's Parity_of_a_permutation#Proof_1
looks more like a proof by postulation; they postulate that the identity is an even permutation.

Ok. How can I proof the amount of nice puzzles equals the amount of impossibles ones, without postulating that identity is an even permutation ?.

????

Par Rubik !
( a b x ) ( a b y ) ( a b z ) ( a b x) (a b y) = ( x z y )

This means that I can do all the stuff above using tri-cycles that pass by two flagged points, a and b !
I have also ( a b x ) ( b a y ) ( a b x ) = (a b) (x y)

This is a good news, because if one mess my 15-puzzle by transposing cubes two at a time, and he makes an even number of transpositions, then, at each couple of steps, either it makes an XX that I recover with the second formula, or it makes a tri-cycle that I recover with the first formula.

As I intuited, the "impossible" puzzles are at a distance of one transposition of the "solvable" ones.

Anyway, one do not need to make the additional transposition (x y) since I know how to recover it by paying an ( a b ). My "solvable: puzzles are at a distance (a b) of the "impossible" ones.

Since the length of a chain of transposition either is odd either is even, my puzzles are either "solvable", or at a distance
of ( a b ) to a solvable one.

This shows that the number of "solvable" ones equals the number of "impossibles".
It remains to show that the "impossibles" exist - being really impossibles, and - consequently - that my "solvable" puzzles are the only ones solvable.

Doc, how about this ? Look at the Wiki proofs by postulation :

- Wiki EN 1 considers that I cannot obtain the identity by the mean of odd permutations,
- Wiki EN 2 involves some rational polynomials, defined as 1 for identity that is a proof "by definition",
- Wiki FR claims that the parity of identity is even : since that huge formula is not defined on identity. then N(0), the signature of identity, must be 1, not -1 or { 1, -1}.

I still have to show, without postulating, that there are unreachable configurations in the puzzles larger than 4×4.

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PostSubject: Re: Encyclopedia of Integer Sequences   Mon Nov 01, 2010 7:35 pm

I do not see a solution, Nick.

I think you should postulate :

There are, for each n, alternating puzzles that you cannot solve.

This would imply, as consequences :

- The Altern Species exists,
- Your solubles are the even permutations for each n, and since identity is soluble, it must be even for each n,
- The number of solubles are n!/2 = to the number of impossibles,
- The discriminant / signature function has indeed the value 1 on identity and it is well defined.

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