## A satisfying weekend

This past weekend was a satisfying one for me, mathematically: I had two papers accepted.

On Solutions to a General Combinatorial Recurrence” was accepted by The Journal of Integer Sequences after being rejected by the first two journals I submitted it to. I’ve probably had more rejections than acceptances over the years, so rejection is nothing new, but what had surprised me some about the rejections is that I thought this paper was particularly good.  The paper gives a partial, but substantial, answer to research problem 6.94 in Concrete Mathematics:

Develop a general theory of the solutions to the two-parameter recurrence $R(n,k) = (an + bk + c)R(n-1,k) + (a' n + b' k + c')R(n-1,k-1) + [n=k=0],$ for $n, k \geq 0$, where $R(n,k) = 0$ when $n<0$ or $k < 0$.

(Well, actually, they have fancier notation, but for some reason I can’t get it to parse properly using WordPress’s $\LaTeX$ capabilities.)  So after two rejections it was particularly nice to have it accepted.  And not only was it accepted, the referee report was one of the most positive I’ve ever received.  Satisfying, as I said.  (Aside: I use the last two identities in the paper in my answer to the math.SE question “Formula for $\sum_{k=0}^n S(n,k) k$, where $S(n,k)$ is a Stirling number of the second kind?.”)

The Lah Numbers and the nth Derivative of $e^{1/x}$” was accepted by Mathematics Magazine this weekend, too.  What I love about this paper is that it was a collaborative effort between math.SE users Listing, Peter Taylor, J.M., and myself on Listing’s question “nth derivative of $e^{1/x}$.”  The paper itself gives five proofs that the Lah numbers are the coefficients in the nth derivative of $e^{1/x}$, using different properties of the Lah numbers and drawing from several different areas of mathematics.  This is the first paper I’ve published that resulted from an online conversation, and I find myself wondering how much new mathematics is going to be created in the future by online collaborations mediated by sites like math.SE, Math Overflow, and others.