Talk:Pólya enumeration theorem

message: Wikipedia deserves only the best, and this example is extremely contrived. -Zahlentheorie 15:45, 29 April 2007 (UTC)
thread_title: Moved knights example to talk page

message: This is a good article; but I agree with many of the criticisms presented by Mikka. In the philosophy of "edit boldly", I'll try to make these changes, but what follows is my thinking:
Above all, we need a formal mathematical statement of the theorem: something like "Given a set A of cardinality n, a group G which acts on A, a set B, and a weighting function w on B such that w maps each element of B into some ring R[x,y,..], define f(x,y, ...) as sum(b in B, w(b)). Then if we define the generating function g(x,y, ...) by weight of the equivalence classes of B^A under the action of G, then g(x,y,...) = P_G(f(x,y,...), f(x^2,y^2, ...), ..., f(x^n, y^n, ...))".
Given this starting point, I found the "slots into which objects are distributed" metaphor generally confusing; it seems terribly backwards to me as it is used in the examples.
When I read that the "objects" in B (for which we have a counting function) are to be "distributed" into the "slots" of set A (which are to be permuted), my impulse is to imagine that no object from B will be in two slots at once: we seem to be talking about a function /from/ B /into/ A.
But the actual sets under consideration are the elements of B^A: mappings /from/ A /into/ B. In that sense, the "slots" are really being "placed" into "objects" in B. Given some configuration of "slots" from A placed into "objects" of B, we permute the "slots" in A according to the elements of G to form the set of all equivalent configurations (orbits).
Another metaphor that might be more generally used is coloring or labeling: we have a set A of "objects", which will be permuted by G. We have a set B of "colors" or "labels", for which we have a weighting function w. We color or label the elements of A with the elements of B.
In either case, the enumeration theorem is closely related to Burnside's lemma: we start with the set B^A; we form equivalence classes (orbits); and then count these orbits. (Some of the diagrams could be altered (particularly the 3-graph example) to show this more clearly: show all possible maps B^A; show the equivalence classes under the group action; finally, show a unique representative from each the equivalence class).
In Burnside's lemma, the method of counting is indirect: we instead count the elements left unchanged (fixed) by each element g in G. If we think of the "slots" in A as being placed into the various "buckets" of B (rather than the other way round) or "colored" by the elements of B, then it is easy to count these elements given some g in G: each cycle of g corresponds to some set of elements of A. For each cycle, all of those elements /must/ be in the same "bucket" of B (or have the same color of B) for g to leave the result unchanged; and vice versa. So if there are 3 cycles in g and 5 "buckets" in B (or colors in B), then there are 5^3 elements of B^A which are left unchanged by g. And thus follows the whole examination of G by looking at the number of cycles of the elements of G.
The main difference between Burnside's lemma and the enumeration theorem is that the theorem expresses the above in the language of generating functions. Springer Online at http://eom.springer.de/p/p073550.htm has a nice summary that made the whole formula much easier for me to understand (they use R^D instead of B^A, where n is the number of elements in D (the "number of slots" in the current article))
:If for the weights of the elements of one takes powers of an independent variable (or the product of powers of several variables), then for 
:\phi(x) = \sum_{k=0}^{\inf} {a_k x^k}
:(the so-called "series that enumerates figures", where a_k is the number of elements in R of weight x^k) and
:\psi(x) = \sum_{k=0}^{\inf} {b_k x^k}
:(the so-called "series that enumerates configurations", where b_k is the number of classes in R^D of weight x^k), the following applies, according to Pólya's theorem:
:\psi(x) = P_G(\phi(x), \phi(x^2), \ldots, \phi(x^n))
This makes much clearer what the "generating function f(x,y, ...) of the objects by weight" is: it's a statement about how we count elements in the labeling set (R or B). And the result is clearer: given a way of counting the elements in the labeling set R and some group G, we get a generating function that lets us count the equivalence classes we are interested in.
The whole issue of "generating functions" and "weights" and what they mean really needs to be addressed. There currently appears to be no place in Wikipedia where it is clearly stated how a generating function f(x,y, ...) is supposed to be understood as a way of talking about counting elements of some set. Should this be amplified upon here? In the generating function article? In the formal power series article? In a new article?
In the examples, let's be consistent with the statement of the theorem: always clearly identify what are the /elements/ of A, what are the /elements/ of B, and what is the weight function w (it should assign a value to each element of B in a non-arbitrary way). For example, the weight function in the first example is on the set B = {0,1}, where 0 means "0 edges present", 1 means "1 edge is present". The weight function is then w(b) = z^b. /That's/ why we have f(z) = a_0*z^0 + a_1*z^1 = 1 + z; in B, there is one instance of "0 edges present" and 1 instance of "1 edge present".
The "ternary tree" example is really good idea, in that it shows how you can go far beyond a simple application of Burnside's lemma. It also breaks us out of the idea that G always refers to some sort of Euclidean /spatial/ symmetry (which I think often accompanies the use of the term "symmetry"). And it really shows the strength of using generating functions.
But it needs to be edited a bit. I really have no idea what is being stated regarding whether ternary trees or binary trees are planar; but it doesn't matter: it's pretty clear what is meant by "isomorphic ternary trees", and what we want to know is /how to count them/. So really, the rest should just be removed. 
I intuitively understood what a ternary tree was by extension of the term binary tree: a rooted tree where every node has (outgoing) degree 0 or 3. But when I looked at the diagram, I thought I might be wrong. Yes, one can eventually figure out what is meant: blue is the root, red is a non-root node, nodes without children are not shown. But this requires a fair amount of consideration to figure out; simply using a larger diagram would make the result more immediate. And really, once I understand the concept, I can already imagine from the text what is meant by "isomorphic"; a diagram is almost unneeded.
It's also unclear how we get the result from the actual statement of the theorem using A, B, G, and w. In the text, we essentially equate both A and B with T3, and G with S_3; and no explicit value is given for w. There are 6 elements in S_3. Given an arbitrary element t of A = T3, what element of A = T3 does each of the elements of S_3 send t to? How should we understand the set of mappings B^A = T3^T3 here? What is w, as applied to the elements of B = T3? How do we show that T(z) = sum(b in B = T3, w(b)), as required by the theorem? A bit more detail would really help in reducing the above confusions.
As regards the proof moved to the Talk Page, all that is required of the weight function is that it map each element in B into "some" commutative ring. That ring could be Z, Z[x], Z[x,y], C, or any commutative ring. The choice of the ring and the mapping is obviously going to be meaningful in terms of understanding the result; but the proof follows in any case. It needs to be made consistent when the symbol "\omega" is being used to refer to the weighting /function/ on B, and some particular /value/ in the ring that that the weighting function maps B into. The assertion regarding f(x^{|c|}, y^{|c|}, ...) needs a bit more explanation. The final step needs to be made clearer than "the claim follows"; we haven't even stated a claim yet!
Anyway, I'll take a stab at these; please feel to comment, revert, add, modify, rant, etc. 71.198.111.245 02:12, 27 August 2007 (UTC)
thread_title: Some thoughts on improvement

message: "Suppose you have a set of n slots and a set of objects being distributed into these slots and a generating function f(x, y, ...) of the objects by weight."
The terminology 'weight' seems unclear to me... it is not explained what is meant by weight, nor does it have a link to follow. What is meant by weight, please? --Hirak 99 (talk) 10:44, 1 February 2008 (UTC)
replies: text: The weight of objects let you distinguish (groups of) them. The simplest "weight" is just constant 1 for every object, so that the weight of a group of objects is the number of elements in this group. Informally speaking, in this case counting of equivalence classes of a certain weight W corresponds to counting of equivalence classes that have exactly W elements. I will elaborate on this in the article. Maxal (talk) 15:13, 1 February 2008 (UTC)
thread_title:  Weights 

message: Does it really make sense to keep Spanish references in this English article? I do not see a point, especially since there is a lot of English references around. So, if there are no objections, I will move those references to Spanish wikipedia and create there a stub for Spanish version of this article. Maxal (talk) 15:02, 1 February 2008 (UTC)
thread_title:  Spanish references 

message: I am a mathematics professor, and I decided to use this article for my class.  However, the article was very confusing in its old form and came across as too technical.  I edited the article to make it more readable.  Greg Kuperberg (talk) 23:43, 26 May 2010 (UTC)
thread_title:  Many changes made for readability