Papers that originated on math.SE

Question

I was thinking it would be interesting to have a list here on meta of any published papers that originated on math.SE. I realize that most of the questions here aren't research-level, and that it's more likely that Math Overflow questions and answers would generate papers, but I suspect that occasionally someone does post something here that becomes the germ of research that ends up in print. After all, there have been a few requests on meta already for how to cite a math.SE post in a paper.

So perhaps each answer could link to a question or answer on the main site, along with a brief description of the result and where the paper was published.

(Inspired by a similar question on meta.cstheory.stackexchange.)

(Disclosure: I'm a co-author on a manuscript based on a math.SE question that has been submitted for publication. Who knows if or when it will be published. But I would still be interested to know what others have done based on math.SE posts - if there are any yet.)

Answer 1

My paper "Enumerating Lattice Paths Touching or Crossing the Diagonal at a Given Number of Lattice Points" was just published in the Electronic Journal of Combinatorics.

Abstract: "This paper gives bijective proofs that, when combined with one of the combinatorial proofs of the general ballot formula, constitute a combinatorial argument yielding the number of lattice paths from $(0, 0)$ to $(n, rn)$ that touch or cross the diagonal $y = rx$ at exactly $k$ lattice points. The enumeration partitions all lattice paths from $(0, 0)$ to $(n, rn)$. While the resulting formula can be derived using results from Niederhausen, the bijections and combinatorial proof are new."

The history of the paper involves two questions on math.SE. It started with Elliott's "What's the probability that a sequence of coin flips never has twice as many heads as tails?" from about a year ago. In my answer to the question I came up with a formula for the number of sequences that obtain exactly twice as many heads as tails for the first time after $n$ flips. To get this I used a recurrence and a combinatorial identity. As I was thinking more about the formula I came up with, though, I started wondering if there was a more direct combinatorial proof. I couldn't find one after a few days and so asked the question "Combinatorial proof of $\binom{3n}{n} \frac{2}{3n-1}$ as the answer to a coin-flipping problem" on the site. Qiaochu Yuan's partial answer to my question pointed me in the right direction, and I was able to answer the question fully myself after a few more weeks. I realized my answer could be generalized, and the generalization and an extension became the paper that was just published.

Answer 2

The American Mathematical Monthly has recently published Absolute Convergence in Ordered Fields by Pete L. Clark and Niels J. Diepeveen. (Vol. 121, No. 10 (December 2014), pp. 909-916) The paper characterizes all ordered fields in which every absolutely convergent series is convergent: from one of the standard proofs over $\mathbb{R}$ one sees that the convergence of all Cauchy sequences is a sufficient condition; much less evidently, this turns out to be necessary as well. It also characterizes all ordered fields in which every convergent series is absolutely convergent: this holds precisely in a non-Archimedean ordered field in which Cauchy sequences converge.

The pre-origin of this paper is an exercise in some notes on sequences and series. Although formally linked to an undergraduate course I taught a few years ago, they soon became an outlet for me to record some material on infinite series so that I would not be tempted to talk about it in my lectures. (I endorse this practice: it has proven to be effective -- i.e., I really did manage not to ask this question in my Math 3100 course; doing so would have been, at best, a poor use of valuable lecture time -- and also to have some auxiliary payoffs.) In my notes I stated as an exercise that convergence of all absolutely convergent series in an ordered field implies convergence of Cauchy sequences. Looking back at the notes later, I realized that I had no idea how to do it. In that it reminded me of an analogous but easier fact about convergence in normed commutative groups (see here for a precise formulation), I suspected that I had had that in mind instead and really had never known the answer.

I posted the question on this site expecting to get the answer within an hour or two. After two weeks went by I placed a bounty on the answer (I think this was the first bounty I set on any SE site). Two days after that I got a simply remarkable answer by Niels Diepeveen.

Then I waited around for someone to tell me where this result could be found in the literature. That didn't happen. However, later I found the same question that I had asked posed as an unsolved problem at the end of a September, 2012 "Classroom Capsule" by Kantrowitz and Schramm in the College Mathematics Journal. (They must have submitted their paper before I posted my math.SE question. However, I can find no prepublication version of their paper available anywhere, and in fact I do not know of a honorable freely available copy of their paper. If you would like a copy, I would suggest that you email them and possibly hint that they could post a copy on their webpage and/or on the arxiv.) When I saw this I thought, "I had better write this up and submit it to the CMJ right away." The CMJ chose not to publish the paper. I rewrote it substantially, adding a lot of extra material whose purpose was mostly to slow down and conceptualize Diepeveen's laconically brilliant solution, and after several rounds of revision it was accepted by the Monthly.

I am sorry to say that though I included a link to the math.SE question in an early version of the paper, I removed it in revision: journals are a bit old-fashioned about the type of references they want to include. So I'm glad to record the story here.

Answer 3

Listing, J. M., Peter Taylor, and I had "The Lah Numbers and the $n$th Derivative of $e^{1/x}$" published in the most recent issue of Mathematics Magazine (86 (1): 39-47, 2013). The paper is mostly expository and consists of five different proofs that the Lah numbers are the coefficients in the $n$th derivative of $e^{1/x}$. It is based on the discussion the four of us had surrounding Listing's question about finding those coefficients a couple of years ago.

(By the way, this is the paper I was referring to when I asked this question about papers that originated on math.SE.)

Answer 4

My paper "A Combinatorial Proof for the Alternating Convolution of the Central Binomial Coefficients" appears in the June/July 2014 issue of the American Mathematical Monthly.

The origin of this paper was this question by user Alex M on evaluating an alternating binomial sum. My answer uses a fairly standard generating function approach. I wondered, though, about a combinatorial proof. After thinking about the problem for a day or so I couldn't come up with a combinatorial proof, nor could I find one in the literature. I then posted a question about finding a combinatorial proof to the main site. The question generated a lot of interesting comments but no answer. User Srivatsan eventually put a bounty on the question, yet the bounty expired without generating an answer. In the meantime the question became, for a time, the highest-upvoted unanswered question in the combinatorics tag.

Two months later I was perusing Richard Stanley's "Bijective Proof Problems" and found a new (to me) interpretation of the non-alternating version of the identity. That gave me the ideas I needed to put together a combinatorial proof of the alternating version, which is the substance of the paper published by the Monthly.

With this, the last three papers I have written that have been published have all originated with math.SE. I am surprised and grateful that a Q&A site that I started participating on for fun has turned out to be that helpful for my career. Thank you, math.SE. Thank you also to the various users whose questions and answers have taken my mathematical thinking to places it never would have gone otherwise.

Answer 5

I wrote an extended answer to Random points in a rectangular grid defining a closed path which was published in Discrete Mathematics and Theoretical Computer Science (link).

Title: Computing the number of $h$-edge spanning forests in complete bipartite graphs

Abstract: Let $f_{m,n,h}$ be the number of spanning forests with $h$ edges in the complete bipartite graph $K_{m,n}$. Kirchhoff's Matrix Tree Theorem implies $f_{m,n,m+n-1}=m^{n-1} n^{m-1}$ when $m \geq 1$ and $n \geq 1$, since $f_{m,n,m+n-1}$ is the number of spanning trees in $K_{m,n}$. In this paper, we give an algorithm for computing $f_{m,n,h}$ for general $m,n,h$. We implement this algorithm and use it to compute all non-zero $f_{m,n,h}$ when $m \leq 50$ and $n \leq 50$ in under $2$ days.

Answer 6

My paper,

• J. D. Hamkins, Every countable model of set theory embeds into its own constructible universe, Journal of Mathematical Logic, vol. 13, iss. 2, p. 1350006, 27, 2013.

found its origin in Ewan Delanoy's extremely interesting question, Comparing countable models of ZFC, which I was pleased to have answered after some intense work. Perhaps of all my theorems, this is one of the few of which I am most proud, and it was born here on math.SE.

The project led naturally to several further questions, including my MathOverflow question, Can there be an embedding j:V to L, from the set-theoretic universe V to the constructible universe L, when V not= L?, on which I am currently engaged in work in progress in collaboration with several authors.

Answer 7

Claude Leibovici and I recently wrote the paper Exact and approximate solutions to the minimum of $1+x+\cdots+x^{2n}$, which was prompted by this question on the MSE. The OP was able to find the minimum of this polynomial, denoted $f_{2n}(x)$, for the special cases $n=1,2$ but sought a general solution for $n\in\Bbb N$. We were able to accomplish this by deriving the relationship $$ \inf_x f_{2n}(x)=\frac{1+2n}{1+2n(1-x_{2n})}, $$ with $x_{2n}$ being the argument of the minimum and then applying Lagrange inversion to find an exact expression for $x_{2n}$. For the purposes of numerical computation we also derived a faster converging perturbation series for $x_{2n}$.

Abstract: The polynomial $f_{2n}(x)=1+x+\cdots+x^{2n}$ and its minimizer on the real line $x_{2n}=\operatorname{arg\,inf} f_{2n}(x)$ for $n\in\Bbb N$ are studied. Results show that $x_{2n}$ exists, is unique, corresponds to $\partial_x f_{2n}(x)=0$, and resides on the interval $[-1,-1/2]$ for all $n$. It is further shown that $\inf f_{2n}(x)=(1+2n)/(1+2n(1-x_{2n}))$ and $\inf f_{2n}(x)\in[1/2,3/4]$ for all $n$ with an exact solution for $x_{2n}$ given in the form of a finite sum of hypergeometric functions of unity argument. Perturbation theory is applied to generate rapidly converging and asymptotically exact approximations to $x_{2n}$. Numerical studies are carried out to show how many terms of the perturbation expansion for $x_{2n}$ are needed to obtain suitably accurate approximations to the exact value.