Talk:Pólya enumeration theorem

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[edit] Expert required

The section "Example computation: enumerating rooted ternary trees" is strange:

• the definiton of ternary tree doesn't match the image
• what the heck is "flat tree"?
• funny language: "a leaf of size zero"

Sorry have to run now, but there is more. `'mikka 02:36, 14 March 2007 (UTC)

Hello there, did you read the caption on the image, which says that leaves (green) are not shown. If you add the leaves, you will see that the definition matches the image (outdegree three, including leaves). I was not able to find the word "flat" in the article. Do you mean planar? You will find the definition of planar trees in any introductory textbook on graph theory. Furthermore, the generating function counts the number of nodes, excluding leaves. A node contributes one to the total and a leaf, zero, hence the phrase "size zero." The use of the word "size" or "size function" is standard terminology from generating functions, again to be found in any introductory textbook on the subject. -Zahlentheorie 14:04, 14 March 2007 (UTC)

Hey, please don't refer me to "intgroductory books". I already know I am an ignoramus. We are talking about the wikipedia article, right? That said, we have to restrict the talk to the article at hand and wikipedia in general, right? I am not asking you to educate me in graph theory, I am too old and lazy to learn something new, that's why I am writing wikipedia :-) That said, back to the problems.

1. Image: Last time I've read a book in graphs, a leaf is a vertex. On the pic I see graphs on 4 vertices. The blurb about "leaves not shown" I noted as a yet another strange piece.
2. yes, "planar" (like I said above, I was in a hurry). Please direct me to wikipedia article about "planar trees" Last time I read a book in graph theory about planar graphs I was surprized to learn that all trees are planar, who would believe this?
3. "leaf of size zero": sorry to insist, but this language is sloppy and good in a chat among experts, but not in an encyclopedia article. There are at least 3 different problems with this phase in our context.

In conclusion, judging your attitude, I would kindly ask to refrain from further answers to me on the topic, if you are going to continue in the same defensive way, and let other experts say. `'mikka 18:21, 14 March 2007 (UTC)

Here is another puzzle:

1. "The set T3 of rooted ternary trees"
2. "an element of T3 is either a leaf of size zero, or a node with three children"

And another one: "The slots in this problem are the three slots where the children are attached to their parent node". All my life I thought that "children are attached to their parent node" by edges or arcs, but not by slots.

Now, looking into the intro:

• "and a generating function f(x, y, ...) of the objects by weight." What is weight and how it is related to generating function? The article "generating function" says nothing about weights.
• "permutation group" better be linked (done)
• back to ternary trees: how/what "weight" is to do with them?

I am inclined to insist that this article reads like Turboencabulator. Probably I was mistaken to ask for expert's attention. For an expert all may be OK: a couple of "evident" (to expert) links in a train of logic omitted, no big deal. `'mikka 18:21, 14 March 2007 (UTC)

P.S. The Russian wikipedia version is in a sharp contrast with this one, but don't tell me that it is probably my problem with English language comprehension. `'mikka 18:24, 14 March 2007 (UTC)

Hello there, thanks for the many useful observations, which will lead to a better article. The word slot is used in all of the examples. It's the common term that links them. PET says that if you have a generating function by weight of some objects, and distribute them into some slots, with a group permuting the slots, then you get the generating function of the new class by substituting into the cycle index. That's why a common term is used rather than a more specific one. It's to illustrate the common structure of these problems. E.g. for the tree example the slots are the three locations where subtrees are attached to the parent node. For the colored cube, it's the faces of the cube etc. As for generating functions by weight, these are sums of monomials in some variables, with the power of the respective variable indicating the value of the parameter, and the value of the parameter of the derived class being the sum of the values for the constituents (the objects in the slots). -Zahlentheorie 00:46, 15 March 2007 (UTC)