# User talk:Watchduck

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## Empty graphs.

Hi!

Here, you wrote a definition in a separate section of the graph connectivity article, in order to provide an adequate anchor for a redirect. This is well done, I think; but you (incidently, or after careful consideration?) also included examples, which include make the empty graph among the connected ones. (The graph with no vertex also has no edge.) This, I find unusual, and not so good.

Now, many authors tacitly or explicitly demand that graphs be non-empty; i. e., they do not allow an empty graph. (If my memory is correct, e. g. Diestl demands this explicitly.) Now, personnally, I prefer to consider also the empty graph ${\displaystyle (\emptyset ,\emptyset )}$ as a (simple) graph, since this on some points makes the graph theory smoother. However, then, I do not consider the empty graph as connected. If I did, I would either have to make a numerous amount of exceptions in other definitions and calculations, or run into rather weird changes of how graphs behave.

Here is an example: A tree commonly is defined as a connected acyclic graph. It also is said to have the property (order = size+1). However, with your definitions, the empty graph would be both connected and acyclic; but with both order and size equal to zero.

Another example: A connected component of a graph often is defined as a maximal connected subgraph. Thus, with your definitions, the empty graph should have one connected component, namely itself. As a consequence, the theorem, that the number of components in a disjoint union of graphs equals the sums of the component numbers of these graphs, would have be to be modified, by subtracting the number of components who are empty graphs, but adding one, if they all are.

There is also the connectivity and connectedness connection, which breaks down for the empty graph.

So, for these and similar reasons, when I allow the empty graph, I consider it as "not connected". In this way, a tree is demanded to be non-empty, while I allow the empty forest - a forest in my opinion is best defined as any acyclic graph, i. e., as the disjoint union of zero or more trees (whence it also is characterised by the equation "order = size+#(components)").

I therefore would like to remove your zero vertex example. However, if you added it on purpose, and especially if you have references where the empty graph both is allowed and considered to be connected, then we could retain it, but with the warning that only some authors considers the empty graph as a valid connected graph.

(Another, rather minor point: The rest of that article uses the term "vertex" instead of "node". This should be easy and uncontroversial to fix, though, I think.)

Best regards, JoergenB (talk) 01:25, 9 April 2015 (UTC)

Hi. To me it was only important to add a section with a clear definition. Feel free to modify the details as you like. Watchduck (quack) 18:02, 9 April 2015 (UTC)
Done JoergenB (talk) 14:25, 12 April 2015 (UTC)

## Your Wikiversity article Permutation notation

Dear Duck,

Thanks very much for your quackage to me back in November 2016. I regret that I missed it until Wikipedia recently notified me. It appears that you, like me, have been troubled by the many unstated assumptions underlying practically every discussion of permutations and the process of permutation. I am very glad you have written what appears to be an excellent article on the topic. Have you published that piece, or a variant, in some journal or on ArXiv?

Maybe someday the many disciplines that make use of permutations could employ your framework as a guide, announce in advance what kind of notation is to be used in a paper or presentation, and thereby restore some peace of mind to those of us who become confused and uncomfortable when concepts and notation are used inconsistently from one moment to the next.

Hi @Dratman:
Thanks for your reply. I don't intend to publish my stuff in a more classical format. I am way too addicted to nested collapsible boxes, and I would rather have more interactivity than less.
What do you think about the terminology I use in the article (active/passive)? It makes sense to me, and I think it helps to understand the problem and the article. (And I guess it harmonizes with active and passive transformations, but I did not verify that yet.) But it's not a terminology I have found anywhere else - except in this article without sources. The simple reason is, that no one seems to use what I call passive permutations. Therefore I had the following statement in the article: The active interpretation of a permutation should usually be seen as correct, and the passive one as a misunderstanding. But Wcherowi has removed it, calling it an egregious POV statement. I wonder if "passive permutation" meant the same to him as it means in that article. Do you know any sources where the result of applying ${\displaystyle \pi }$ on a vector ${\displaystyle (x_{1},\dots ,x_{n})}$ actually is ${\displaystyle (x_{\pi (1)},\dots ,x_{\pi (n)})}$? I would be happy to include them. Greetings, Watchduck (quack) 21:15, 29 July 2017 (UTC)
Dear Duck,
I find this discussion quite interesting, and I would like to respond to your question about sources that use a certain type of notation, but I do not understand your subscripts. Could you please explain the subscripts and/or describe in more detail what you are referring to? Thanks Dratman (talk) 06:28, 30 July 2017 (UTC)
I don't understand what you don't understand.
The vector ${\displaystyle x=(x_{1},\dots ,x_{n})}$ and the permutation ${\displaystyle \pi ={\begin{pmatrix}1&\dots &n\\\pi (1)&\dots &\pi (n)\end{pmatrix}}}$.
Usually ${\displaystyle x}$ permuted by ${\displaystyle \pi }$ is ${\displaystyle (x_{\pi ^{-1}(1)},\dots ,x_{\pi ^{-1}(n)})}$, but with what I call passive permutations it would be ${\displaystyle (x_{\pi (1)},\dots ,x_{\pi (n)})}$.
Now I see what you mean. Thanks for explaining. Dratman (talk)
In Talk:Permutation#Major_Problem_with_Notation you wrote this as:
${\displaystyle \sigma .(x_{1},x_{2},\ldots ,x_{n})=(x_{\sigma ^{-1}(1)},x_{\sigma ^{-1}(2)},\ldots ,x_{\sigma ^{-1}(n)})~~~}$ and ${\displaystyle ~~~(x_{1},x_{2},\ldots ,x_{n}).\sigma =(x_{\sigma (1)},x_{\sigma (2)},\ldots ,x_{\sigma (n)})\,}$
I did not write those lines! That is part of someone's reply to what I wrote. Dratman (talk)
OMG! That is a very confusing way to answer. Maybe one day we will see a decent forum format for Wikimedia discussion pages. Watchduck (quack) 13:44, 30 July 2017 (UTC)
It seems that in your terminology this difference is somehow about left vs. right, rather than active vs. passive - which I don't understand.
Looking at that today, I think the whole left/right thing is something irrelevant which I should have omitted. Dratman (talk)
To me L/R is only a superficial difference, corresponding to flipping a matrix multiplication along the main diagonal. A/P is about a different matrix multiplication, no matter how it is flipped.
In other terms: A/P is about what the result of "${\displaystyle x}$ permuted by ${\displaystyle \pi }$" is. L/R is just about whether to write it as "${\displaystyle \pi .x}$" or "${\displaystyle x.\pi }$".
For A/P compare box 12 and box 16 in my article. For L/R compare left and right matrix multiplication in each box. Watchduck (quack) 09:38, 30 July 2017 (UTC)

## Edge enumeration

About this: the edge of the 1-variable Hasse diagram is Ax -> Ex. The edges of the 2-variable Hasse diagram are all of the following form: "Ax ... -> Ex ..." (with the same "...") or "Ex Ay -> Ay Ex". What is perhaps more interesting is that (if my poking around your data tables linked from OEIS is accurate), these already capture all edges for the 3-variable version: they are all of the form "Ax ... -> Ex ..." or "... Ex Ay ... -> ... Ay Ex ...". If this is also true for the 4- and 5-varaible versions, that would be promising. --JBL (talk) 01:32, 14 April 2018 (UTC)

I don't really understand what you are trying to say. But it sounds like the answer should already be in the data on GitHub for n up to 5: The edges are here. (The coordinates are here.) And details for n=4 are here. Given that predicate logic is so old and so important, I suppose that there is nothing new to be found here. I was surprised that and did not already exist, but I suppose the number of edges in these Hasse diagrams is long known, hidden is some dusty books. Numbers of edges don't seem to be prominent in the OEIS. I was also surprised that no sequence counts the edges in weak order diagrams like this. (After division by 2 that sequence would start 0, 1, 9, 79, 765, 9121. If there is no error in my manual calculation that is.) Watchduck (quack) 11:37, 14 April 2018 (UTC)