Talk:Combinatorial species/Archive 1

It would help to know in what areas the term is used. It's linked to from Species - is "combinatorial species" a biological term also? Mathematical? Or both? Tannin

Actually, it didn't occur to me that I had not made that clear by using the word "combinatorial". But now I've made it more explicit. (For now, it's still only a stub article.) -- Mike Hardy


 * The description of the cycle index series seems slightly ambiguous. Namely, where is the dependence of $$\left( \sum_{\sigma \in S_A} |\mathrm{Fix}(F[\sigma])| x_1^{\sigma_1} x_2^{\sigma_2} \cdots \right)$$ on n? - Gauge 02:15, 10 January 2006 (UTC)

PS. It would be nice to indicate if the theory of combinatorial species has solved some interesting new problems. Thanks. CW 16 February 2006

This article refers to diagrams which are not present on the page. —Preceding unsigned comment added by 24.22.98.131 (talk) 07:41, 28 November 2007 (UTC)

Under "Basic Operations" the text refers to graphics which aren't there...? — Preceding unsigned comment added by 141.89.53.115 (talk) 08:25, 12 July 2012 (UTC)
 * For whatever reason, it was commented out with a completely unrelated edit summary: . Now all images are shown again. --Daniel5Ko (talk) 09:16, 12 July 2012 (UTC)

Note on the differentiation of virtual species
Start with two atomic species,

e.g. let P5 be the Frobenius group acting on 5 elements (or the field F5), and K4 the Klein group acting on itself.

P5 - K4, as virtual species, is nothing else than the class (P5 + A, K4 + same A), where A is any species.

By differentiating, one gets (Cyc4 + A', X.X.X + same A').

The trouble here is that A' is no more any species, but only those species that are derivatives.

For differentiation to be well defined, one should have Any' = Any. The completion comes by Cancellation property. And this put me "out of bussines"; nothing to differentiate. There are virtual cycle indices that have no x1 terms.

That is why I am trying to rotate the labels. A (-X) given by a Group ring of variables/labels would allow me to differentiate like in good old times.Nicolae-boicu (talk) 13:56, 4 August 2012 (UTC)


 * This comment is completely wrong. Differentiation is defined on virtual species as follows; in your example;

P5 - K4 is nothing else than the class (P5 + A, K4 + same A), where A is any species.
 * The derivative of (Cyc4 + A', X.X.X + same A'), which are all examples of (Cyc4 + A, X.X.X + same A), so it's equal to Cyc4 &minus; X.X.X . — Arthur Rubin  (talk) 02:03, 7 August 2012 (UTC)

Let me verify...
 * (F + A) - (G + A) ~ F - G ~ (F + B) - (G + B) is clear since A and B are given,
 * (F'+ A') - (G' + A') ~ F' - G' ~ (F' + B') - (G' + B') is also clear and differentiation is compatible with ~.

This would avoid (F + all species) for the sake of "completion". (Neither the set of al sets is not complete; it does not contain all its elements) Why not ? Good ideea, thanks ! Nicolae-boicu (talk) 14:58, 7 August 2012 (UTC)

Note on the Lin species
The combinatorial species Lin does not necessarily stands for a Linear Order, in the very same way that Ens does not stands for a Boolean Algebra.

A word xyzt has the very same combinatorial structure as { βx, 3y, Az, Πt } where β, 3, A, Π are specific "slots". Nicolae-boicu (talk) 20:21, 4 August 2012 (UTC)

Example of concrete use of a cyc
I have an example of actually using a cyc.

McKay's proof of Cauchy's theorem (1959, AMM) uses a p-tuples of elements of G

( x1, x2,....,xp ) and x1.x2......xp = 1

Well, this is a cyc ! (cyclic order does not matter) as usual, when species are present it turns spectaculary.Nicolae-boicu (talk) 05:54, 5 August 2012 (UTC)

Note on integration
In terms of permutation groups, primitives of molecular species are called transitive extensions. This does not close the issue, since (UV)'= U'V+UV'Nicolae-boicu (talk) 21:44, 5 August 2012 (UTC)

The spectrum of species
Every single sequence in OEIS has a species correspondent. Take an.Ens[n] instead of an, make the sum for all n and here is the species. Why then a new list ? Who decides that one sequence is more important then other to place it in a sublist ?

There are sequences in OEIS that have simple e.g.f.-s. An exercised eye simply extracts the species just looking to formula. It is some kind a reverse engineering here.

Me, I am waiting for the TBS, the third book on species, the dictionary, that would contain 100 ? 200 ? species. (no book, no article) Nicolae-boicu (talk) 12:36, 7 August 2012 (UTC)


 * Applies to almost any power series of enumeration, which is a more basic concept than this one. Even so, I don't think the table adds anything.  There are three examples above (sets, permutations, pairs), which seem adequate.
 * gee Arthur, like always your intervention is encouraging. So I do not have to wait for TBS, but for FBS, the first book on species !Nicolae-boicu (talk) 19:36, 10 August 2012 (UTC)

Note on the transport of structures
Since three basic operations are defined by splitting the set A with n elements like n = (n-1) + 1 (differentiation), n = m1 + m2 (product) or n = m1 + m2 +...+ mk (partitional composition) one should understand the transport of structure as a transport of partitional stuctures. Nicolae-boicu (talk) 17:37, 15 August 2012 (UTC)

Note on pointing
Pointing is about introducing coordinates. One cannot do too much math with non-coordinated structures. In a combinatorial set, all elements are 'total' conjugates without any other determination. For example, if |A|=3 and |B|=5, all we can write about A and B inside Species Theory may be expressed as 5 ' ' = 3 (Sym5 is a two point extension of Sym3).

To define a (mathematical) function on the (mathematical) set A, first one must point three times A toward •••A, thus obtaining an X.X.X; and then define the function on X.X.X, only after the elements of A got their own identity and they become distinct within respect to any conjugation.

X.X.X...X may be seen as a 'solid' structure, a 'zero-entropy' structure, with no possibility to make confusions between conjugate elements.

In Geometry, the coordination is implicitly introduced when choosing points or quadrangles or whatever. After choosing two points, all other points of an affine line are well determined (modulo the Galois conjugation)(unlike sets, the geometrical structures are efficiently coordinated with less pointing operations; after several pointing operations they became X.X.X...X's). In the corresponding algebraic structures, 0 and 1 are technologically introduced by the very first axioms, and all other numbers become well determined : one element-one sign, without confusions.

The pointing verb, the key to coordination, is 'let it be' (French 'soit'). For example, to totally coordinate the complex field and to eliminate the Galois confusion, one must point the i : let i be such that i.i = 1. Nicolae-boicu (talk) 12:00, 15 January 2013 (UTC)

note on sum and product rules
based on a Wikipedia tabel. I apologize, I still search a scholastic example.

1) US Drone Strike Statistics estimate according to the New America Foundation.

(As of 17 April 2013)

Let X be the sort of attacks, Y be the sort of casualties. Each line will be encoded in one species e.g. Line2007(X,Y). On has

Line2004 ( X, Y ) = X × [ Ens5 ( Y ) + Ens6 ( Y ) + Ens7 ( Y ) + Ens8 ( Y ) ]

Line2005 ( X, Y ) = Ens3 ( X ) × [ Ens12 ( Y ) + Ens13 ( Y ) ]

...

Line2013 ( X, Y ) = Ens12 ( X ) × [ Ens62 ( Y ) +...+ Ens73 ( Y ) ]

The unknown number between min and max entered the machinery as a sum of possibilities.

The whole table may be encoded in species terms as a product of its lines:

Table ( X, Y ) = Line2004 ( X, Y ) × Line2005 ( X, Y ) ×...× Line2013 ( X, Y ) /Nicolae-boicu (talk) 10:02, 14 July 2013 (UTC)

note on electrical interpretation
By interrupting one of L1 and L2 signals the output COM gets L2 or L1.

Nicolae-boicu (talk) 12:20, 16 February 2014 (UTC)

note on Cuantum Mechanics interpretation


Let's take the Schrödinger's cat. Then one has :

( alive and dead )' = dead or alive

the and/or conjunction
the And/or conjunction fits to the combinatorial multisorting. "He will eat cake, pie, and/or brownies" easily becomes cake + pie + brownies, Plate(X+Y+Z) Nicolae-boicu (talk) 09:30, 8 February 2015 (UTC)

the generalized differentiation
In one word, the generalized differentiation means sampling. Let's say that John has 3 chickens and 2 ducks and he wants to remove 2 birds.

$$ { C.C.C.D.D\over 2(C+D)   } = { C.C.C.D.D\over { 2(C) + C.D + 2(D) } } $$

$$ = { C.C.C.D.D\over { 2(C)} } + {C.C.C.D.D\over {C.D}} + { C.C.C.D.D\over {2(D)}} $$

$$ = {3\choose 2} .C.D.D + 3.2.C.C.D + {2\choose 2} .C.C.C  $$

$$ = 3.C.D.D + 6.C.C.D + 1.C.C.C $$

One may conclude, for example, that the probability for John to remove two birds of different kind is 6/10.

in the above I have noted the differentiation by a ratio line and a set of 3 chickens by 3(C).

Eventually, we have reached in our research the formula of binomial coefficient.

{ X.X.....X.X \over \{ X,X,...,X \}   }={   X^n \over k(X)  } = {n\choose k}.X^{n-k} Undersum (talk) 00:59, 8 July 2016 (UTC)

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Introduction
The "transport of structures"is introduced in the very first page of the very first article on species.

An example of transport is given using endofunctions. Let A be a set of three elements. Then there are 27 endofunctions of A, grouped in seven types:


 * 27 = 1 + 2 + 3 + 6 + 6 + 6 + 3

The associated species is:


 * 3.X3 + 2.X·E2 + C3   + E3

and the cycle index for 3-endofunctins is


 * (27.x13 + 9.x1.x2 + 6.x3 )/6

The question
If we look at the above species formula we can read ( in the 2.X·E2 term) that bijections like:

may be transported in functions like:

because they have the very same subjacent species, namely X·E2. The question is: What the transport of structures is transporting if the endofunctions are NOT transported ?

The answer
Normally bijections transport cardinality. A chain of bijections A->B->C-> ... ensures that all A, B, C,... have the same cardinality.

If the chain eventually hits back the set A like A->B->C->...->A, the chain of bijections act as a permutation on A. This permutation on A is then copied on every other set B, C, ...

The "transport of structure" transports permutations and nothing else.

Note on generalized differentiation
Suppose we have to investigate claims like the ones in Fano plane article :
 * There are 7 lines, and 24 symmetries fixing any line.
 * There are 28 triangles, which correspond one-for-one with the 28 bitangents of a quartic (Manivel 2006). For each triangle there are six symmetries fixing it, one for each permutation of the points within the triangle.

There are two ways of doing this; one is to use the generalized differentiation of a Fano species within respect to Ens3, by applying to ZFano the differentiator

{d_1.d_1.d_1 \over 6} + d_1.d_2 + d_3 that comes from   Z_Ens3 (d_1, 2.d_2,..., i.d_1,...)

The result is

$$ {Fano \over { \{ X,X,X \} } } = {1 \over 24}( a_1^4 + 3a_2^2 + 6 a_1^2a_2    + 8 a_1 a_3 + 6 a_4) + a_1.{1 \over 6}(a_1^3 + 3 a_1a_2  + 2 a_3)  = Ens_4 + X.Ens_3 $$

A faster way is to use directly the cycle index definition. We consider the action of Fano group on 3-sets and we slightly modify the definition by taking the Fano group insted of Sn

$$ {1 \over |Fano|} \sum_{\sigma in Fano} fix(\sigma). a^\sigma  $$ then we evaluate fix(σ) for each type; the result is the very same.

Nboyku (talk) 20:49, 31 May 2018 (UTC)