Talk:Ordered Bell number

A paper reviewed by...

Why does the article refer to "A paper reviewed by Ellison and Klein" and not "A paper by Archangeli and Langendoen" or whomever wrote the paper in this anthology? That would make more sense. --WiseWoman (talk) 23:04, 3 March 2015 (UTC)

Because that paragraph is about what Ellison & Klein say in their review, not so much about what Archangeli and Langendoen say. —David Eppstein (talk) 23:28, 3 March 2015 (UTC)

Oh, it's clear that the paragraph is about Ellison & Klein's review, and their contributions in sentences 1, 2, and 4 of the paragraph. I was just curious to know who wrote the paper which Ellison & Klein refer to in their review that you refer to in your third sentence. --WiseWoman (talk) 23:58, 3 March 2015 (UTC)

Where is this name from?

This article has a nice historical introduction, but gives no or few explanation about the (curious, indeed) choice of naming this sequence of numbers after E.T.Bell (apparently, it's from a 2005 paper). Isn't the case to go back to the 1974 introduction by Louis Comtet, or to look for some more dignified choice? (btw, "Bell numbers" for the Dobiński's numbers of set partitions seems already quite a local attribution). pma 08:57, 28 January 2021 (UTC)

GA Review

This review is transcluded from Talk:Ordered Bell number/GA1. The edit link for this section can be used to add comments to the review.

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

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

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)

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)

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)

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)

Tonight's updates:

Added log-convexity (in the first paragraph of the summation section, since that's how it was proved in the reference I used).

Added a definition and example section with more examples including the connection to ordered partitions that you wanted elaborated.

Expanded the lead with a brief summary of the history section, so now all sections are represented by summarized material in the lead and the lead is longer.

Spelled out in words rather than formulas how $P$ comes from Pascal's triangle.

Searched some more but so far no luck finding a source for the simple and easily-proved bound $a(n)\leq (n+1)^{n-1}$ (via Cayley's formula). —David Eppstein (talk) 06:47, 22 April 2024 (UTC)

Another daily update: I added an anecdote about Knuth and OEIS to the history section, and a paragraph about parking functions to the application section, after finding a source that used parking functions to prove the simple lower and upper bounds I wanted to include. There's also some new text giving some explanation for why you would want to determine the exponential generating function. —David Eppstein (talk) 06:18, 23 April 2024 (UTC)

Re "is the approximation good for large": I added an explanation that (like Stirling) it has multiplicative error close to 1. Re "can you mention the base of the log in prose": it really isn't ambiguous in mathematics articles, but ok, done. Re "outcomes of a multi-candidate election": I agree this is not a great example (not all voting systems produce a meaningful ranking of all candidates, for example); removed. I'm not actually familiar with the tripos outcome system so I didn't try to use that as a replacement.

Also, skipping ahead: "The short description says "Number of weak orderings", which doesn't tell a layperson anything. It should be something like "Topic in mathematics"." I very strongly disagree. There are two main purposes of short descriptions: (1) when you search on mobile you see both titles and short descriptions; (2) some articles' see also sections use short descriptions to gloss their entries. In both cases you don't need a precise rigorous definition of the topic (WP:SDNOTDEF) but the short description does need to add information, relative to the bare title, to help readers figure out which search hit is the one they want or how a see also link might relate to the article listing it. When an article has a title that already suggests it might be mathematical, like "ordered Bell number", a short description like "topic in mathematics" provides precisely zero information to readers. It is unhelpful. You might as well just change all short descriptions of all articles to "Wikipedia article" and make them totally useless everywhere. On the other hand, "number of weak orderings" is short enough to serve as a short description, continues to suggest that the article is mathematical to the uninformed, and also helps inform mathematically-literate readers who aren't familiar with the specific topic to get a rough idea what it is about. Put it another way: imagine the searches one might do that have ordered Bell numbers as one of their hits. Among the people who might make such a search, and the searches they perform, how likely is it that one of these short descriptions will make a meaningful distinction between this topic and the other search results? The short description more likely to make such a distinction (within the character limit) is the better short description. —David Eppstein (talk) 07:20, 24 April 2024 (UTC)

Only one update today: I tried tackling the Kemeny paragraph by removing the technical description and instead attempting to explain it by example. —David Eppstein (talk) 07:34, 25 April 2024 (UTC)

I have changed the first paragraph of the summation section in an attempt to more accessibly explain the first summation formula. I moved combination locks earlier in the applications section. As for the last specific comment, on OEIS: I think it's generally considered reliable, and Wikipedia:Reliable sources/Noticeboard/Archive 420#Is OEIS reliable for this use? agrees. It has a significant level of editorial control and review of submitted content, by the members of a small and selective editorial board; in my experience every addition must pass through two levels of review. As for "what triangle of numbers": when an OEIS entry like [10] describes itself as being a "triangle of numbers", that means that the sequence describes the row-by-row ordering of the triangle. See for instance Pascal's triangle [11] where maybe this pattern is more clear. I have rewritten the footnote in an attempt to clarify this. Next I'll try looking through your suggested additional sources to see whether there's an more expansion that would be appropriate. Re the first one by Qing Zou: we now have better sources both for log-convexity and for tighter simple bounds $n!\leq a(n)\leq (n+1)^{n-1}$ so although it was useful for suggesting those two directions of expansion I don't think it's needed as a source itself. —David Eppstein (talk) 06:33, 28 April 2024 (UTC)

Just a note that I have been following these changes/explanations and I'm happy with them. I think this review should be marked on hold (and probably should have been two weeks ago) but as long as progress is being made I've no desire to set a particular deadline. I may have lower availability in the next fortnight but will prioritise this over other Wikipedia work. Two points: (1) "As has already been mentioned, the ordered Bell numbers count ... ordered multiplicative partitions of squarefree numbers" – am I missing a mention or is this only covered with the same brevity in the lead? (2) The Kemeny paragraph is much simpler now but something feels off about By the "complexity" of a relation he means the number of other relations ... when that's the first mention of "complexity". Could it be: He describes the "complexity" of a relation—the number of other relations ...? — Bilorv (talk) 21:20, 28 April 2024 (UTC)

Thanks! Given the slow pace at which I've been finding time to work on this article I can hardly complain if you also have limited upcoming availability. And I'd rather the revisions be thorough than that they be fast. —David Eppstein (talk) 21:54, 28 April 2024 (UTC)

Re: does this article need further expansion from the many additional sources? Searching Google Scholar for likely phrases like "ordered bell", "fubini number", "number of preferential arrangements" etc finds far too many publications to cite (768, 116, and 126 respectively), many in preprint form, in low-quality journals, or with very few citations. As one of the better recent ones states, "there are many variants of Fubini numbers". As a way of being more selective, I tried looking for the ones with signed reviews on MathSciNet; this finds many fewer hits but did not turn up much more that I thought should be added. As a principled way of seeking missing topics, rather than trusting my own search skills and value judgements, I decided to scan the OEIS entry on these topics to make sure that the main claims for that entry were also repeated here. I don't think we should cover everything OEIS does or remove material it doesn't cover, but OEIS provides an up-to-date survey on the topic and can reasonably be expected to mention its most important aspects. Based on this, I made the following changes:

• Added names (or really descriptions) as "the number of X" (where X is one of the ways of naming weak orders) to the lead.
• Briefly discuss compositions (unlabeled weak orders) in the definition section.
• Added Coxeter complexes to applications. I vaguely recall thinking about adding this long ago and deciding against because it was too technical, but maybe it is important enough to include anyway.

Only partway through, more to come. —David Eppstein (talk) 07:08, 29 April 2024 (UTC)

Ordered multiplicative partitions expanded into its own paragraph in applications, since (as you noticed) we previously had only a call-forward and call-back with no substance. This also provided an opportunity to connect again with the unlabeled weak orders mentioned in definitions. —David Eppstein (talk) 05:31, 30 April 2024 (UTC)

Some more updates:

• The fuss over logs appears to have mostly died down. No significant change to content was in play, just equation formatting. The current version of that part is more concise than the previous one (despite the extra variable), doesn't combine a division with the exponentiation (the part I strenuously objected to because I found it confusing myself), doesn't make invalid mixtures of $\sim$ and $\approx$ (as one suggestion did), and doesn't need binary logarithms (not something I minded but others apparently did).
• IntGrah made some improvements to the links in the approximation formula paragraph. I think they are relevant and not too many.
• I found an explanation for the Eulerian number summation formula in the same source the formula comes from. It turns out to be quite close to the parking function material but explaining the connection might veer too far into original research so for now they're separate and not connected to each other.
• From OEIS (and then another source) I found Ramanujan's use of these numbers to approximate log 2, now at the end of the approximation section
• More additions in the application section involve the moments of a geometric distribution (from OEIS) and calculating resistance in a hypercube (from Pippenger, and Pippenger's main reason for discussing these numbers).
• I made various improvements to wording (especially removing "count the number of" in favor of "count"), footnote ordering (to put them back in numerical order) and reference author and journal linking (including a new article for Toka Diagana).

There might still be a few more points to add from OEIS; I haven't completed my scan of that. —David Eppstein (talk) 07:03, 3 May 2024 (UTC)

@Bilorv: Ok, I think I'm done checking OEIS references, incoming wikilinks, and scholar searches to find more not-already-covered materials. Since the last update, my changes include:

• Adding a conjecture from OEIS that there is a modular identity for every modulus
• Splitting the overlong applications section into subsections involving combinatorial enumeration and other applications
• Adding an application to load balancing in the management of factory workers
• Adding an application to computational origami
• Adding an application to non-commutative algebra
• Adding an application to spam filtering

Not added: a supposed application in certain high-energy physics calculations that I don't understand and can't evaluate the significance of [12] [13]. Anyway, I think I've reached a stable point again, so it's ready for you to look over again. —David Eppstein (talk) 20:45, 4 May 2024 (UTC)

Second round

Okay, thanks for the overhaul and I'm much happier with broadness now. On making technical articles understandable, I think all the right pieces are there in the lead and first section. All that remains are some wording nitpicks:

• "They may also be described, rather than named" – I'm not sure how much of this sentence is important (particularly for the lead). I understand you use "described" rather than "named" deliberately but it seems strange to mention that within the sentence and "number of weak orderings" is effectively repetition of the lead's first sentence. The other names for weak orderings seem worth keeping.
• "Along with the weak orderings, they count several other types of combinatorial objects that have a bijective correspondence to the weak orderings" – This also seems a bit repetitive. How about: "The numbers also count objects that are in bijective correspondence to weak orderings, such as ..."
• "a partition of its elements, together with" – I think there shouldn't be a comma here as "together with" is a necessary criterion rather than extra information (and "together with" could just be "with").
• "Linear order" and "total order" are each used a few times – could one be used consistently?
• "A permutohedron is a convex polyhedron" – can "-hedron" be used for dimensions other than 3D? I understand "permutohedron" may be the standard name but in "convex polyhedron" it seems the meaning is a polytope of any dimension.
• "For sequences with faster than exponential growth" – it has taken me a few readings but I think this sentence is intended to convey that (a) the ordered Bell numbers have faster than exponential growth, (b) the OGF of the ordered Bell numbers does not converge and (c) faster than exponential growth is a necessary but not sufficient condition for OGF divergence. However, with its current wording I think it is not explicit that the OGF for ordered Bell numbers does not (rather than may not) converge.
• "Numbers that are neither squarefree nor prime powers have a number of ordered multiplicative partitions that is intermediate between these two extreme cases" – Maybe this should be "numbers with $n$ prime factors", and "that is intermediate" is redundant to "between".
• "As they point out" – could repetition of "point out" in this paragraph be avoided?

— Bilorv (talk) 21:25, 9 May 2024 (UTC)

• Copyedited "the number of" sentence to focus on the fact that people describe these by the many names that weak orders have.
• Re "Along with the weak orderings": tightened
• "together with": replaced by "and", with no comma
• Linear order vs total order: unified on "total" for consistency with the article title and usage on the weak order article
• Polyhedron: changed to polytope for the first instance. The second is for a 3d shape, so polyhedron is ok there. I don't want this article to have to explain the distinction in meaning between polytopes and polyhedra in n dimensions (a polytope is a convex hull of finitely many points; a polyhedron is an intersection of finitely many halfspaces); polytope as you suggest is the better word here and we don't need to go into more detail than that.
• Tightened sentence about divergence of the ogf.
• Re "Numbers that are neither squarefree nor prime powers": this is OEIS:A126706, described there as "Positive integers which are neither squarefree integers nor prime powers". These must have at least two prime factors, and at least one of those factors has to have exponent greater than one. But really, it's just "all the numbers that are not in either of the two extreme cases". "Intermediate between": tightened.
• First of two "point out"s removed.

—David Eppstein (talk) 05:45, 11 May 2024 (UTC)

The article has been much improved over the last month, and I hope my comments were somewhat useful in that. I'm now happy that this is a pass for GA. — Bilorv (talk) 08:10, 12 May 2024 (UTC)

Extremely useful, thanks! This sort of improvement is exactly what I hope for in the GA process. —David Eppstein (talk) 16:34, 12 May 2024 (UTC)

Logarithm notation

David Eppstein has reverted my edits about the choice of logarithm notation. I would like to explain why I think $\ln$ is better than $\log$ for this article.

Firstly, I don't know anywhere where it says $\ln$ is not used in mathematical writing, unless you're Paul Halmos. I point out that the articles for Bell number and Stirling's approximation, use the notation $\ln$.

Secondly, David also says that articles should be written for the widest audience. The widest audience would therefore include non-expert mathematicians as well. In that case, $\ln$ is recognised more widely than $\log$ as the natural logarithm. The fact that you have to write "here, $\log$ denotes the natural logarithm" is evidence.

On the question about whether the base should be ${\frac {1}{\ln 2}}$ or $\log _{2}e$: you've already explained later that it is approximately 1.4427. In my opinion, $\log _{2}e$ is more confusing. It could easily be misconstrued as $\log _{e}2$, since $e$ is far more commonly the base than 2. It takes a few seconds to realise this number is greater than one, whereas $\ln 2$ is a well known to be about 0.693 (off the top of my head).

Could we come to some agreement about this? IntGrah (talk) 20:33, 30 April 2024 (UTC)

Also, ${\frac {1}{\ln 2}}$ follows from the previous formula much more clearer; otherwise, the reader has to use the change of base formula themselves as well as converting the negative exponent to a reciprocal. IntGrah (talk) 20:51, 30 April 2024 (UTC)

This is a mathematics articles. Mathematics articles use log for the natural logarithm. To do otherwise would be to violate the standard conventions of mathematics writing, both off-wiki and on.

As for $\log _{2}e={\tfrac {1}{\log 2}}$: Yes, obviously. BUT, when trying to read the formula $({\tfrac {1}{\log 2}})^{n}$ one has to first realize that the unwritten base e is larger than 2, then that the unusual backwards ordering of larger base and smaller argument means that the logarithm is less than one, then that taking one over that produces a number greater than one, to eventually maybe conclude that you're actually looking at an exponentially-growing thing rather than an exponentially-shrinking thing. Is it obvious at a glance to you, without working through the steps and carefully keeping track of signs, that $({\tfrac {1}{\log _{4}2}})^{n}$ is really just $2^{n}$? It isn't obvious to me.

When the formula is expressed in the form $(\log _{2}e)^{n}$, one doesn't have to do that confusing double negation to reach the same conclusion. The base and argument of the log are in the more commonly seen order of a smaller base and larger argment. Therefore the log gives a number greater than one. Therefore it's exponentially growing. I think it is MUCH MUCH CLEARER this way. Specifically, I found the older formulation to be very confusing, myself, to the point where I had to work through it step by step to understand the growth rate. The current formulation expresses the point I am trying to express (exponential growth) more clearly, in the sense that I don't personally find it confusing. I do find the version you want to reinstate confusing. The change of base is a a bigger step from the summation formula but I think that is less important. —David Eppstein (talk) 22:54, 30 April 2024 (UTC)

Thanks for your reply. However, I still don't agree; the exponential growth is already highlighted by the succeeding sentence, and frankly it's stupid to write $\log _{2}e$. And the change of base step is still an issue. I'll leave it for now until someone else agrees with me. IntGrah (talk) 23:31, 30 April 2024 (UTC)

The expression $\log _{2}e$ appears in multiple other articles. To pick one that has little or no computer science content, and little or no connection to this article, it appears in clique game. What makes you think writing this expression in this form, there or here, is "stupid"? It is more concise than ${\tfrac {1}{\log 2}}$, in its TeX coding, in the number of symbols in the rendered expression, and especially in the vertical extent of the rendered expression. It uses logs in a way that I find more natural (base smaller than argument). It is just a logarithm of two numbers that happens to produce a value seen in some mathematical topics. What is it about that pair of numbers that makes it "stupid"? Which other pairs of numbers produce logarithms that are "stupid"? Is there a unique non-stupid way of writing each possible logarithm that one might see? What is your method for canonicalizing a logarithm into a non-stupid form? —David Eppstein (talk) 00:24, 1 May 2024 (UTC)

• Writing $\ln x$ does not violate the conventions of the wiki; the contrary is true. Thousands of Wikipedia mathematical articles use $\ln x$. It is the ISO standard. The off-wiki conventions do not apply to this article, because the wiki is not a scientific journal, and the wiki is also written for readers who may not know these "conventions". It was already mentioned by Bilorv that this is ambiguous notation. There is quite literally zero gain by writing $\log$ and having to explain its meaning in prose.
• For one, the article clique game uses base 2 logarithms consistently, unlike this article. I expect an article to stick to one base, and not assume the reader can follow with the an arbitrary change of base formula.
• It's not "more concise" than ${\frac {1}{\ln 2}}$. They use exactly the same number of symbols (l o g 2 e and 1 / l n 2). Is vertical space really getting expensive nowadays? Should Stirling's formula be written as ${\sqrt {2\pi n}}n^{n}e^{-n}$? Should ${\frac {1}{2}}$ also be chopped down to 0.5?
• I could not find any style guide which says logarithm bases should be ideally smaller than their argument. How would you write $\ln 2$ then? Is there a unique 'natural' way of writing every possible logarithm that one might see?
• What is it about that pair of numbers that makes it "stupid"? As I already said, it could be misconstrued as $\log _{e}2$, which you failed to address.
• Which other pairs of numbers produce logarithms that are "stupid"?

This is about the numbers 2 and $e$, which are two of the most common bases.

• What is your method for canonicalizing a logarithm into a non-stupid form? I never said there was a canonical form for logarithms. The goal is readability, which entails consistency and familiarity but does not entail "golfing" the expression in a way which might suggest the number is greater than one, but is already fully explained in the sentence after anyway. IntGrah (talk) 11:09, 1 May 2024 (UTC)

  Ok, you lost all credibility with me when you invoked ISO standards on how to write logs. ISO does write standards for this, it is true. Nobody follows them. See Binary logarithm#Notation. —David Eppstein (talk) 18:26, 1 May 2024 (UTC)

  And how exactly is their recommendation for binary logarithm related to this? Do you completely ignore the thousands of Wikipedia articles that use $\ln$? Do you assume everyone who reads mathematics articles assumes base $e$? IntGrah (talk) 18:44, 1 May 2024 (UTC)

  @IntGrah No skin in this game, but a heads up that there's further discussion going on over at WT:WPM#Log vs ln. GalacticShoe (talk) 18:59, 1 May 2024 (UTC)

  No, I do not assume base e. The article explicitly states that it defaults to the natural logarithm, a statement you tried to remove. —David Eppstein (talk) 19:00, 1 May 2024 (UTC)

  ...and you wouldn't need that statement if you'd just used $\ln$ in the first place. IntGrah (talk) 19:03, 1 May 2024 (UTC)

  A literature search finds hundreds (speaking conservatively) of examples where the authors say where ln is the natural logarithm. So, color me doubtful that the notation is so ubiquitously understood that it would not also need just as much clarification. XOR'easter (talk) 19:11, 1 May 2024 (UTC)

  There are also examples where authors write "where log is the natural logarithm" too, so what's the advantage? Wikipedia isn't a research journal. Even if it was not understood by everyone, it is still understood by more people than "log" — the article was in the DYK page, and is of interest to amateur combinatorists too. IntGrah (talk) 19:37, 1 May 2024 (UTC)

  If an amateur combinatorist can't understand the plain text explanation "where log is the natural logarithm", they're probably not a very good amateur combinatorist. XOR'easter (talk) 20:19, 1 May 2024 (UTC)

  That's unrelated to the point that $\ln$ is more widely known. IntGrah (talk) 20:36, 1 May 2024 (UTC)

  It is more widely used by engineers and physical scientists. It is very infrequently used by pure mathematicians. This is not an article in engineering or physical science. We should follow the conventions of the fields our articles belong to. —David Eppstein (talk) 23:22, 1 May 2024 (UTC)

Using "log" for the natural logarithm is indeed standard for mathematicians, and writing ${\frac {1}{\ln 2}}$ instead of $\log _{2}e$ is a mite peculiar. The point about making clear that the final functional form is an exponential growth rather than an exponential decay is a good one. In this case, $\log _{2}e$ is the more clear way to express the meaning the formula is supposed to convey, while also fitting more nicely into the following text. On the other hand, ${\frac {1}{\ln 2}}$ is an ungainly fraction that disrupts the prose flow. Changing the base between one formula and the next isn't a problem in this style of writing, since the reader is told that steps are being skipped over ("This leads to an approximation for..."). XOR'easter (talk) 19:08, 1 May 2024 (UTC)

How does it disrupt the prose flow? I don't think $\log _{2}e$ is clear. It's base two, which is unexpected. It's a different base from the one used beforehand. And $e$ is the argument for some reason. As I said, the fact that it is 1.443 is already explained clearly; there is no need for such a convoluted expression. IntGrah (talk) 19:31, 1 May 2024 (UTC)

It disrupts the prose flow by being a big honkin' fraction sticking up and down in the middle of a sentence. It's more convoluted than $\log _{2}e$, because it requires taking a logarithm and then dividing by it, rather than just taking a logarithm. XOR'easter (talk) 20:22, 1 May 2024 (UTC)

What's so scary about a fraction? There is already a fraction in the expression. IntGrah (talk) 20:40, 1 May 2024 (UTC)

I don't understand the objections to $\log _{2}e$ or any other base-2 logarithm. I don't understand how the number 2 is "unexpected". And it seems simpler to me than $1/\log 2$, $1/\log _{e}2$, or $1/\ln 2$, for XOR'easter's reason.

In advanced math, $\log$ (unsubscripted, meaning the inverse of $\exp$) is more common than $\ln$. And this article is advanced math. So I prefer $\log$ here. Mgnbar (talk) 20:42, 1 May 2024 (UTC)

It's not 'advanced math'. It's the number of outcomes of a horse race. It's readable for a high school student and understandable by an undergrad. There is no point in defining the meaning of the logarithm for the [two] times it is mentioned, especially when you override it by making it base 2. IntGrah (talk) 20:50, 1 May 2024 (UTC)

When base 2 is used in mathematics, I expect it to be meaningful, suggesting a relationship with the number two. In this scenario, it is not being used to convey twoness. Rather, it's a pointless application of change of base. IntGrah (talk) 21:12, 1 May 2024 (UTC)

Whether you write it as ${\tfrac {1}{\log 2}}$ or $\log _{2}e$, the same number two is there both times. There is an inherent meaningful relation with the number two in this formula, regardless of whether you put it in the base or the argument. —David Eppstein (talk) 23:02, 1 May 2024 (UTC)

It's one more operation. Taking a log, which happens to be smaller than one, and then taking a reciprocal to get a number bigger than one, takes multiple steps of reasoning. Just taking a log and seeing that is bigger than one is fewer steps. It is precisely this issue that caused me to be confused by the earlier version of the approximation formula in the article, the one that you want to revert to, and it was my confusion over this formula that led me to the present form. —David Eppstein (talk) 20:43, 1 May 2024 (UTC)

Then what about the number of steps between the previous summation and the approximation? The working is omitted, making it even more important to highlight the parallels between the two expressions. I fail to see how the 'number of function calls' is more important than the readability of the expression. Taking the logarithm of an irrational number with an integer base raises more questions and is certainly more work.

With ${\frac {1}{\ln 2}}$, the reader sees $\ln$, then 2, and knows this is some transcendental constant, which the reader does not need to evaluate. The reader can simply glance over it as 'some random number'. And, it obviously comes from the previous expression. The following sentence immediately explains it is exponential growth.

Compare with $\log _{2}e$: the reader sees a base 2 logarithm. Does this imply some relationship with the number 2? Why e? They then realise that this probably came from an application of the change of base formula. They recall the formula to verify what they are seeing: $\log _{2}e={\frac {\ln e}{\ln 2}}={\frac {1}{\ln 2}}$. Only then is it clear where it comes from. Meanwhile, they have not even begun to realise this number is greater than one; it would not cross their mind as to why greater-than-one-ness is important. They are simply baffled at why someone would write that, instead of one-over-ln-2. IntGrah (talk) 21:09, 1 May 2024 (UTC)

You're overthinking it. $\log 2$ is the natural logarithm of some number, which happens to be 2. $\log _{2}e$ is the binary logarithm of some number, which happens to be e. They have exactly the same descriptive complexity. —David Eppstein (talk) 22:59, 1 May 2024 (UTC)

I don't have any issue with using log instead of ln, and in principle $\log _{2}e$ is totally fine, but I have to admit that in context I found $\log _{2}e$ in the following a little jarring, and it took me a moment to figure out where it came from:

Based on a contour integration of this generating function, the ordered Bell numbers can be expressed by the infinite sum

$$a(n)={\frac {n!}{2}}\sum _{k=-\infty }^{\infty }(\log 2+2\pi ik)^{-(n+1)},\qquad n\geq 1.$$

Here, $\log$ stands for the natural logarithm, whose base is $e\approx 2.718$. This leads to an approximation for the ordered Bell numbers, obtained by using only the term for $k=0$ in this sum and discarding the remaining terms:

$$a(n)\sim {\frac {1}{2}}{n!}(\log _{2}e)^{n+1}.$$

Thus, the ordered Bell numbers are larger than the factorials by an exponential factor, whose base $\log _{2}e$ is approximately $1.4427$.

Wouldn't it be clearer to just say instead " ... and discarding the remaining terms:

$$a(n)\sim {\frac {n!}{2}}(\log 2)^{-(n+1)}\approx 1.4427^{n+1}{\frac {n!}{2}}.$$

Thus, the ordered Bell numbers are larger than the factorials by an exponential factor."? It both makes the point of exponential growth immediately, and also makes comparison to the previous formula completely direct. Gumshoe2 (talk) 21:17, 1 May 2024 (UTC)

I am happy with this solution. I can concede $\ln$ versus $\log$ on the basis that $\log _{2}$ is removed.

Although, telling the reader that $e\approx 2.718$ is a minor insult to intelligence, and it feels pointless to define the base when it is only used twice. That was the reason I preferred $\ln$. IntGrah (talk) 21:23, 1 May 2024 (UTC)

I don't think it's necessary to say e is 2.718, it's enough just to say "Here log stands for the natural logarithm." Gumshoe2 (talk) 21:26, 1 May 2024 (UTC)

Okay, that is fine. IntGrah (talk) 21:27, 1 May 2024 (UTC)

The reason for saying what $e$ is was to explain its later explicit appearance in $\log _{2}e$. There are so many other uses of the letter e that, if it is to be used in a formula, it needs to be disambiguated somewhere, just like log without an explicit base always means natural log in mathematics but still could use disambiguation. —David Eppstein (talk) 21:45, 1 May 2024 (UTC)

Yes, but if we are not to write $\log _{2}e$, then there is no need to highlight that 2.718 > 2 anymore. IntGrah (talk) 21:47, 1 May 2024 (UTC)

I am unhappy with this solution. I considered it and discarded it. The reason is that I am unsure what a formula combining both $\sim$ and $\approx$ is supposed to mean. Here, $\sim$ has a precise technical meaning that works for formulas with a variable $n$ in them (the ratio goes to one as $n$ goes to infinity) and $\approx$ has a meaning that has not been formally defined but informally means something like "this specific number is numerically close to this other specific number", without dependence on $n$. It is not true that the left and right sides of the combined formula have ratios that go to 1 as $n$ goes to infinity. Instead, the ratio blows up to something that is itself exponentially large because of the numerical approximation. So we get a mismatch between what we are telling the readers (this approximation formula gets more accurate for larger $n$) and what actually happens in the formula (it gets much less accurate). It is also not even true that the middle formula and the right hand side are numerically close for all $n$ (they are exponentially far apart for large $n$). —David Eppstein (talk) 21:36, 1 May 2024 (UTC)

That hadn't occurred to me, it's a fair objection. I don't have a better suggestion.

So then my only input is that for me as a reader it's definitely clearer to stick with a single choice of log base. Gumshoe2 (talk) 21:49, 1 May 2024 (UTC)

It seems to me that using "ln" instead of "log" would be clearer here, and 1/ln 2 instead of log_2 e. Tito Omburo (talk) 21:58, 1 May 2024 (UTC)

To be clear, how obvious is it to you whether the expression ${\bigl (}{\tfrac {1}{\ln 2}}{\bigr )}^{n}$ produces a big value or a small value? Can you tell that instantly just looking at the formula, or do you have to think about it for a while? That is the main thing I want this part of the formula to convey. —David Eppstein (talk) 22:27, 1 May 2024 (UTC)

The fact that it isn't instantly clear (to me) can be addressed by immediately subsequent text: "Since $\textstyle {\frac {1}{\ln 2}}\approx 1.44,$ this shows that the ordered Bell numbers are larger than the factorials by an exponential factor." (Or analogous.) I think that would read really easily and clearly. To be honest, I don't see the issue with it. Gumshoe2 (talk) 22:36, 1 May 2024 (UTC)

Next question: Since your preferred version fails to make this point clear, what is your reason for formatting it in this way rather than $(\ln 2)^{-n-1}$ (putting the +1 back into the exponent) or the even more abbreviated but often-seen form $\ln ^{-n-1}2$? Why separate out the inversion and exponentiation when they can be jammed into a single operation? After all, if we no longer are attempting to make the formula understandable, maybe we should at least make it concise? And next readability question: in this form, can you quickly tell whether the +1 makes the formula bigger or smaller? Readability is important. In case you hadn't noticed, this article is up for a GA nomination (although I suspect IntGrah may have endangered that), and WP:TECHNICAL is an explicit part of the GA requirements. —David Eppstein (talk) 22:50, 1 May 2024 (UTC)

I think the following is completely clear and readable (and as I said, I don't see the issue with it):

... and discarding the remaining terms:

$a(n)\sim {\frac {n!}{2}}\left({\frac {1}{\log 2}}\right)^{n+1}.$

Since $\textstyle {\frac {1}{\log 2}}\approx 1.44,$ this shows that the ordered Bell numbers are larger than the factorials by an exponential factor.

Answers to your questions: if you use $(\log 2)^{-n-1}$ instead, the followup sentence would be clunkier. The expression $\log ^{-n-1}2$ is potentially confusing, since $f^{-n}$ sometimes represents n-fold iteration of $f^{-1}$. I don't see the benefit of trying to jam everything into a single operation. The premise of your second to last question is incorrect. The answer to the last question is yes (and immediately so), at least in the way I just wrote it, where the formula is immediately followed by saying $\textstyle {\frac {1}{\log 2}}\approx 1.44.$ Gumshoe2 (talk) 23:08, 1 May 2024 (UTC)

From a practical point of view, that formula is going to require huge amounts of long-term on-going policing to stop gnomes from jamming the two fractions back into one fraction.

It is also not clear to me how "since (some number in the middle of a complicated formula has a value)" provides any explanation for why "larger than the factorials by an exponential factor". How would your sentence be any more meaningful if you wrote "since $\pi \approx 3.1416$, ... are larger than the factorials by an exponential factor"? In what way does providing a numerical approximation to part of the formula explain the exponential behavior of a different part of the formula? —David Eppstein (talk) 23:19, 1 May 2024 (UTC)

What is this comment supposed to even mean? It clearly explains that the base is greater than one and is thus exponential growth. IntGrah (talk) 23:24, 1 May 2024 (UTC)

Unfortunately, this time I don't understand your questions (even a little), since I think it's crystal clear how $\textstyle {\frac {1}{\log 2}}\approx 1.44$ is relevant in a way which $\pi \approx 3.1416$ isn't. I also don't understand what you mean by a "different part of the formula." Gumshoe2 (talk) 23:26, 1 May 2024 (UTC)

The numerical value of the logarithm, buried two levels below the exponentiation under a division operation, is not relevant to the fact that the exponentiation is an exponentiation. Your sentence is worded in a way that makes it sound like, if the log had a different value, the exponentiation would not be an exponentiation, which is nonsensical. The actual explanation, that the base of the exponentiation being larger than one causes the factor to be exponentially large rather than exponentially small, has been almost entirely skipped over in your wording. The result is a sentence that contains the two things we want to connect (numeric value ... exponentially larger) but where the path from one to the other has been muddied. We're missing the intermediate step. Because 2 < e, $\ln 2<1$. Because $\ln 2<1$, ${\frac {1}{\ln 2}}>1$. Because the base of the exponential is > 1, the whole formula is exponentially larger. Look at all those steps when we put them in. I was hoping that writing $\log _{2}e$ would make this clearer because fewer steps but instead it's leading to all this "I don't understand binary logarithms! Let's do everything we possibly can to contort things to avoid binary logarithms!" fuss. What's wrong with binary logarithms? —David Eppstein (talk) 23:31, 1 May 2024 (UTC)

Ok, I understand now. That makes sense, but it's really easy to fix: change the sentence to "Since $\textstyle {\frac {1}{\log 2}}>1$, this shows that the ordered Bell numbers are larger than the factorials by an exponential factor" or "Since $\textstyle {\frac {1}{\log 2}}\approx 1.44$ is larger than one, this shows ..." or, etc. Gumshoe2 (talk) 23:37, 1 May 2024 (UTC)

This was essentially how my edit looked. IntGrah (talk) 23:55, 1 May 2024 (UTC)

Another possibility

The following variation avoids all the horrible confusion of using binary logarithms (oh the humanity) at the expense of introducing a completely unnecessary new variable. In exchange we also never need to define or use $e$. Maybe some might like it better:

This leads to an approximation for the ordered Bell numbers, obtained by using only the term for $k=0$ in this sum and discarding the remaining terms:

$\displaystyle a(n)\sim {\frac {1}{2}}{n!}\,t^{n+1},$

where $t={\tfrac {1}{\log 2}}\approx 1.4427$. Thus, the ordered Bell numbers are larger than the factorials by an exponential factor.

What think you? —David Eppstein (talk) 23:44, 1 May 2024 (UTC)

I had this in mind as a possible solution, but thought it was unnecessary to introduce a single-use variable, so probably not. IntGrah (talk) 23:53, 1 May 2024 (UTC)

I don't understand why you think not using $\log _{2}$ means you have to introduce new variables? As I asked above, what's wrong with

... and discarding the remaining terms:

$a(n)\sim {\frac {n!}{2}}\left({\frac {1}{\log 2}}\right)^{n+1}.$

Since $\textstyle {\frac {1}{\log 2}}\approx 1.44$ is larger than one, this shows that the ordered Bell numbers are larger than the factorials by an exponential factor.

? Exactly the same content but no extra variables. You said "that formula is going to require huge amounts of long-term on-going policing to stop gnomes from jamming the two fractions back into one fraction" (which I don't really follow), but is there any other reason? Gumshoe2 (talk) 23:54, 1 May 2024 (UTC)

To me, a formula that is only an exponential, $t^{n+1}$ (or whatever other variable; I have no particular preference for t) is more obviously exponential than a formula that combines a logarithm, a reciprocal, and an exponentiation and leaves it to the reader to figure out which of those operations to focus on. Especially so when the notation for the reciprocal and the parens around it are so much more heavyweight and visually in your face than the notation for the other operations. Writing it with only an exponential, and no other operations, emphasizes that the exponential is the important operation in the formula, and all the rest is just a number whose precise calculation isn't so important. —David Eppstein (talk) 00:00, 2 May 2024 (UTC)

... and leaves it to the reader to figure out which of those operations to focus on ...

It's not left to the reader, because it's explained in the following sentence. IntGrah (talk) 00:06, 2 May 2024 (UTC)

To take Stirling's approximation as an example, do you see any issues with

${\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}$?

There's a fraction, which you despise for some reason. It doesn't take away from the fact that there is exponentiation going on. IntGrah (talk) 00:08, 2 May 2024 (UTC)

There's a "divided by a number" and an "exponentiation" from which I infer "exponentially smaller". That's exactly why putting both a division and an exponentiation into a formula that has the opposite outcome, "exponentially larger" confuses me so much. —David Eppstein (talk) 00:10, 2 May 2024 (UTC)

I see. It seems like you're looking at the formula (to some extent) entirely in and of itself, which is not how IntGrah and I (and, I believe, most readers) are viewing it. I think there's not too much more to discuss on it (at least on my part), maybe we can wait and see what other people think. Gumshoe2 (talk) 00:10, 2 May 2024 (UTC)

Well, for those readers who look at a big formula and think "big formula! let's just skip over it and read the text" rather than hoping for the formula to be readable and informative on its own, and not merely a calculation recipe, I wonder why the details of the formula even matter. Those readers are just going to see the "exponentially larger" part and not need any explanation of how that relates to the formula that they skipped over. —David Eppstein (talk) 00:12, 2 May 2024 (UTC)

I think it's kind of silly of you to describe looking for context or explanation in a followup sentence as "big formula! let's just skip over it and read the text." For further discussion I'll wait for some others to chime in. I should say though that I think the version you've proposed here is fine despite being (to my eyes) needlessly strange. Gumshoe2 (talk) 00:23, 2 May 2024 (UTC)

I think that version is permissible but not ideal, in the sense that I wouldn't edit it if I saw it. IntGrah (talk) 00:28, 2 May 2024 (UTC)

Writing it with only an exponential, and no other operations, emphasizes that the exponential is the important operation in the formula, and all the rest is just a number whose precise calculation isn't so important. I tend to agree with this. It's a subtle effect, but a noticeable one. XOR'easter (talk) 02:28, 2 May 2024 (UTC)

I'm not really sure what all the fuss is about, but it strikes me as reasonable to use $\log$ with the base spelled out, reasonable to use $\ln$ with or without the base spelled out, reasonable to use $(\log _{2}e)^{n+1}$, passable to use $(\log 2)^{-(n+1)}$ and unnatural to use $(1/\ln 2)^{n+1}$ or $t^{n+1}$. (On the other hand, ${\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}$ is ghastly in merging $n^{1/2}$ with the constant ${\sqrt {2\pi }}$ rather than the $n^{n}$ term, but there's no nice way to write Stirling's approximation.) — Bilorv (talk) 20:56, 2 May 2024 (UTC)