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My question is whether it is common for researchers in combinatorics to advertise their results, and then not to publish them? I am motivated by 1 particular example which I will state. But first I suggest an answer: they produce too many interesting results to publish them all. Posting slides and communicating to other mathematicians is sometimes more efficient.

The example is the generalization of Chung and Feller's theorem about the number of simple walks with a given number of points on the walk above the $x$-axis (for a point on the $x$-axis one considers it above if the previous point was, for consistency).

On Fluctuations in Coin Tossing Theorem 1.

At least one textbook, McKean's, stated the problem for odd numbers of steps.

Probability: The classical limit theorems Exercise 3.4.2 on p139.

This could possibly be interpreted not as a literal exact combinatorial problem but rather as an exercise for recovering Lévy's arcsine law in the continuum limit. Gessel stated the answer in one of his talks

Chung -Feller Theorems last few slides.

And finally Grünbaum gave an article with the result here

A Feynman-Kac approach to a paper of Chung and Feller on fluctuations in the coin-tossing game

published in Proceedings of the American Mathematical Society

By my count there was at least 1 and maybe 2 people who knew how to calculate this result but did not publish it. Incidentally, Feller seemed to revise his classic textbook once or twice as he obtained better and clearer pictures of this topic, the last using a graphical derivation of Ed Nelson. An excellent reference is the set of lecture notes online by Kim C Border

Lecture 16: Simple Random Walk (not to be confused as a replacement for looking at Feller, itself)

This all makes it difficult to know what has already been done. Of course the online slides and lecture notes make it easier for newcomers to learn those results and knowledge. How do professional combinatorics researcher navigate all this, if some of the best among them do not publish all the results they know?

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    This should probably be asked on mathoverflow, where there is a chance of having someone sufficiently knowledgeable about combinatorics to appropriately judge and comment on this. Probably the question -- My question is whether it is common for researchers in combinatorics to advertise their results -- should be slightly changed, however, because this sounds a bit accusatory as well as being a bit vague (common and advertise have varied meanings). Maybe ask (in mathoverflow) whether these results should perhaps have been formally published so as to be better known. Dec 29, 2021 at 18:42
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    David Renfro I did ask it at at Math Overflow , and was advised to move it here. I am asking a specific mathematics question acceptable for MO there, related to this question. Then I am pointing here for this question that seems more appropriate for Academia SE. Dec 29, 2021 at 19:06
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    One of the main learning resources in the representation theory of $p$-adic groups, Bill Casselman's $p$-adic book, has never been published.
    – LSpice
    Dec 29, 2021 at 23:52
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    @LSpice, I understand. Many of us learned from textbooks that were never published but circulated in notes. In Grünbaum's paper, he said Jim Pitman has a different proof, too. But he did not publish it, maybe for Anyon's and Anonymous_Physicist's reasons. Dec 30, 2021 at 0:14
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    Even prior to the internet, many important developments were not packaged as "published papers". After all, that is a constrictive vehicle. And, with the advent of the internet, communication is no longer restricted to mailing lists of friends and/or subscribers to expensive journals. People who already have tenure and are not worried about grant funding can afford, perhaps, to be more forthright in their communication. (Issues of correctness are not necessarily addressed effectively by "peer review", after all, but many (mathematically irrelevant) issues of "status" are raised...) Dec 30, 2021 at 2:23

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Elaborating on your "too many interesting results to publish" idea, I wonder how much of this is affected by personal webpages and the arXiv.

For well-known researchers, a notes section on their webpages might get as much attention as certain journals. An extreme case is the personal journal of Doron Zeilberger and his computer. See also his opinions section which includes commentary about refereed journals and whether they're worth the effort.

And while the arXiv doesn't post everything that is submitted, it does provide a place for material to be publicly available without the sometimes long hassle of journal publication. I know of people who post things there without any intent of further publication.

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Surely it is common for researchers in all fields to not publish all their results? I think many of us have mentioned something informally in private communication or in a talk or two, perhaps looking for feedback. There may be an intention to pursue those ideas further, and sometimes that gets delayed - in some cases indefinitely. Some results are not "ready" yet, others are seen (rightly or wrongly) as not significant or interesting enough to put in the effort to turn into papers. And, of course, some results were written up, but dropped after the submission was rejected.

Note that the significance bar is highly personal. The best in a field can be the most likely to find some important result and then drop it (at least for the time being) or delay publishing because they are busy, have other more interesting ideas to work on, are less worried about having to publish for status or grant money, have exceptional self-imposed standards for what they choose to publish, etc.

Maybe your impression that this happens more often in combinatorics than in other fields is correct, or maybe you're experiencing a Baader-Meinhof phenomenon. I don't know. I'm no mathematician myself, and I doubt there are good statistics on this. Either way your observation is a good reminder that a field is more than the sum of its published papers and books, and that there is a lot that can (only?) be learned from speaking to experts.

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    Well put. Good! :) Jan 1, 2022 at 22:36
  • This is good. It is more than the sum of its published papers and books. But the published material is sometimes what lives on. Jan 2, 2022 at 17:38

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