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