Talk:Ordered Bell number/GA1

GA Review

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Nominator: David Eppstein (talk · contribs) 06:50, 23 November 2023 (UTC)[reply]

Reviewer: Bilorv (talk · contribs) 20:47, 11 April 2024 (UTC)[reply]

I gather the bulk of the article was written in 2013, but it actually looks like there's been some developments since then. I like this paper (assuming it's reliable) because it has some ideas that can be understood by readers with a very low level of maths, like $2^{n}<a(n)<(n+1)^{n}$ (though $n!$ is a better lower bound that might be worth mentioning in the lead) and that the numbers are odd. It also shows log-convexity.

It looks like the numbers can be generalized to "higher-order Fubini numbers" and "Fubini polynomials", which might be the basis of a section.

Other possible sources (reliability not evaluated): [1], [2], [3], [4], [5]. This and this might not be worth including as they're only passing mentions but it's good to see that the numbers are natural enough to arise practically.

Some other thoughts on the existing content:

It may be useful to illustrate ties with more than three objects, without enumerating 75 or 541 of them. For instance: $b<f<a,c,d<e,g$ . I think in the same example you could show how this is equivalent to partitioning ( $\{a,c,d\},\{b\},\{e,g\},\{f\}$ and then ordering (or vice versa, partitioning bfacdeg).
"which count the number of permutations of n items with k + 1 runs of increasing items" – This makes it sound like $6,3,1,2,4,5$ has a single run of length 4 (so it counts towards $n=6,k=0$ ), but Eulerian number makes it seem like this is three runs ( $n=6,k=2$ ). If it's the latter, would "pairs of ascending numbers" be clearer? Or "pairs of ordered items", since you talk about "items" rather than "numbers" and we could play this game with descending pairs or letters in alphabetical order etc.
For clarity I think it would help to follow the numbers by their notation: "Stirling numbers of the second kind $\left\{{\begin{matrix}n\\k\end{matrix}}\right\}$ " and "Eulerian numbers $\left\langle {\begin{matrix}n\\k\end{matrix}}\right\rangle$ ".
I don't get the denominator $1-(e^{x}-1)$ : is that supposed to illustrate some step of calculation or that exponential generating functions are typically written in the form $1/(1-f(x))$ ?
Can you spell out what P is – e.g. the ith row and jth column is $i \choose j$ ?
Is the approximation good for large n (rather than n close to 1)?
Since it's ambiguous notation (used for e or 10), can you mention the base of the log in prose?
"For this reason, the ordered Bell numbers count ... the possible outcomes of a multi-candidate election" – I know combinatorics authors like elections, but I feel this muddies the water. It only counts the "outcomes" in the sense of throwing away vote counts and where all orderings matter. I feel that some version of "rankings where ties are allowed" would be clearer even if more abstract (I'm picturing Senior Wrangler to wooden spoon but again this complicates things with marks).
"by which he means the number of other relations one can form from it by permuting and repeating its arguments (lowering the arity with every repetition)" – On one hand I understood this without having to look it up, but on the other I feel it's not the clearest wording. I wrote down $\{(a,b,c)\in \mathbb {N} ^{3}:{\textrm {gcd}}(a,b)=c\}$ and I understand I can permute it and repeat arguments like ${\textrm {gcd}}(a,c)=b$ and ${\textrm {gcd}}(a,a)=c\}$ (on $\mathbb {N} ^{2}$ ) (though several of these generate the same relations because of properties like symmetry). I think the parenthetical could be "(though repetition decreases the arity of the relation)" or "other relations (of arity at most n)". I must admit to not understanding the next sentence: by changing $a,b,c$ to $a,a,c$ , what weak ordering do I induce?
I think combination locks belong in or just after the first paragraph of the section as a straightforward interpretation of the original definition.
I don't know if WikiProject Mathematics has a consensus on reliability of OEIS but I'm not sure I follow citation 13 (as of Special:Permalink/1213893425) at all: what triangle are we talking about and are you relying on the sequence or the comments as the source? One of the comments mentions Wikipedia, which is an issue, but others mention references listed on that page.
(Not a GA requirement.) The short description says "Number of weak orderings", which doesn't tell a layperson anything. It should be something like "Topic in mathematics".

A spotcheck of inline citations shows no issues. Great work so far but let me know what you think about expanding with further sources. — Bilorv (talk) 20:47, 11 April 2024 (UTC)[reply]

Thanks! I'll go through these points one at a time (not necessarily in order) as I find time. Starting with your unbulleted first paragraph: I think the simple bounds $n!\leq a(n)\leq (n+1)^{n-1}$ (tighter than $(n+1)^{n}$ ) may be worth mentioning somewhere, but we shouldn't put things in the lead that aren't summaries of later material. On the other hand we need a published source and the proof in the paper you link is too ugly for my taste. There's a much nicer proof of the tighter upper bound that follows from Cayley's formula; I've asked here for published references and may add it if I find one. I think the paper you link is reliable; at least, it's listed in MathSciNet and zbMATH. But we don't need to cite every possible paper in this topic; there would be too many. Its other main result is that these numbers are log-convex but I haven't seen much evidence that log-convexity is considered significant. We do have an article on logarithmically concave sequences but it doesn't mention log-convexity and we don't have a separate article on that. —David Eppstein (talk) 15:56, 16 April 2024 (UTC)[reply]

Another batch of replies:

Re the other potential sources: I couldn't evaluate most of them because ebscohost login needed. I tried logging into the Wikipedia Library and connecting to the ebsco database first, but still the links didn't work.

Re "may be useful to illustrate ties with more than three objects": this appears to be referring to the lead illustration, showing all weak orders on three objects. This article is about counting all lead orderings, not about the concept of a weak order itself, for which our other article does lead with an image of a single weak order (though maybe not a great image).

Re the gloss of Eulerian number: (6, 3, 1, 2, 4, 5) has three runs of increasing items: (6), (3), and (1, 2, 4, 5). But I can see that the one-element runs are confusing. I changed it to refer to the number of items with a larger successor, and added an inline copy of the notation for the Stirling and Eulerian numbers as you suggested.

Re the denominator $1-(e^{x}-1)$ , I don't know why that is there either, which suggests that it's not very informative. It was added last December by another user, AndriusKulikauskas, with the explanation "write out more explicitly". I don't think it was an improvement. Removed.

—David Eppstein (talk) 07:21, 18 April 2024 (UTC)[reply]

Huh, I did try testing the links by logging out but I think it's a timeout issue. I think the ScienceDirect works should link [6][7][8] and I was also trying to point to (all on The Wikipedia Library):

[9]
"The log-convexity of the Fubini numbers", Transactions on Combinatorics
"On central fubini-like numbers and polynomials", Miskolc Mathematical Notes

Log-convexity strikes me as a very natural condition so I thought it was of interest, while if I'm reading it right Fubini polynomials are a generalisation that has received a lot of attention (I found lots of results when searching). Maybe the polynomials are notable in their own right but should still get some treatment here. The others seemed like they would fit in "Applications" as well as what is currently there.

On "may be useful to illustrate ties with more than three objects", I think the image is fine but "Weak orderings arrange their elements into a sequence allowing ties" might be clearer with an example (it could be inline). Explaining the connection to Bell numbers in the lead in elementary terms is also possible. With WP:ONEDOWN in mind I think a high schooler could get a lot out of use out of a well-written lead on this topic. I'm not suggesting a ton of wordiness glossing "weak order", "partition" etc., but an example like the one I gave above.

In GA criteria terms it's 1(a) (Wikipedia:Make technical articles understandable) and 3(a) (Broadness) that I'm directing my attention to as I think the other criteria are met. — Bilorv (talk) 10:33, 20 April 2024 (UTC)[reply]