Written: March 29, 2011
This is Adieu to my cher ami, maître, and guru, Philippe Flajolet. I have only met you a few times, but all these times were great fun, and I learned so much from you!
In some ways you were typically French, for example, you were un fumeur invétéré, and you loved to dine and wine. You also had a great sense of humor, but not quite French, it was of the gentle-irony kind rather than the biting-sarcasm, moqueur, kind.
I probably met you for the first time in the Oberwolfach Spring 1982 conference, on combinatorics, organized by Dominique Foata, where you talked about your beautiful, seminal, work on the combinatorial interpretation of continued fractions. This is mentioned as one of the most influential articles that appeared in Peter Hammer's journal Discrete Mathematics. This was still combinatorial-clean-fun, without analytic contamination. This work I purely loved. I also loved your innovative approach to proving the purely combinatorial facts that certain formal languages are inherently ambiguous, by looking at the asymptotics of the coefficients of their counting (formal!) power-series, and citing the Chomsky-Schützenberger theorem that the counting series of a formal language that is generated by a non-ambiguous context-free grammar must be an algebraic formal power series. Then (to my horror!, as John Riordan used to say) you looked at these formal power series as analytic series representing complex-analytic functions, and noting that these have very tame asymptotic behavior, while the asymptotics of the counting sequences for the examined languages had much wilder behavior.
But then you really annoyed me by turning into a full-fledged analyst. But once I read your beautiful, seminal work on singular combinatorics, and browsed in your magnum opus (with Robert Sedgewick) I almost started liking analysis. In your hands it became almost as pretty as combinatorics, and thanks to your beautiful and powerful methods we now know how to count, approximately, so many combinatorial objects that we would never know how to count exactly.
Thanks also for calling me, in this lovely page, the inspired poet of modern combinatorial analysis. What you say is what you are! C'est toi who is the vrai poète. In particular, you allowed yourself some poetic license, and you deviated from your legendary precision. So let me take this opportunity to sharpen (and update) your estimates. First, good news! Your Wiles number (and more important Wiles's Flajolet number) is 3, not 4, via Odlyzko and Skinner. Your Zeilberger number, unfortunately (for me!) is indeed 3, (but this can be reduced, all I need is to collaborate with one of your seventy coauthors, (recorded in 1998, I am sure that there are many more today). But your Einstein number is ≤ 4, and that path does not go via myself, but via Ernst Strauss, Carl Pomerance, and John Robson.
But the most blatant error was the statement that your "Flajolet number" is 0. Of course it is tautologically so, but, granting some poetic license, you probably have a finite grand-père Philippe Flajolet number. One of the coauthors of that astronomy article that you found in the Library of Congress catalog was Monsieur Jean Merlin, of crible de Merlin fame, that was later immortalized by Viggo Brun. Merlin died believing that he proved the Goldbach conjecture, and was killed, like so many bright (and of course not so bright) young French (and German, and British, and American, ...) men in that madhouse called World War I. He must have sent his manuscript to Jacques Hadamard, who published it, along with a commentary, and pointing out the error, at the bulletin de science mathematique. This definitely could be counted as a collaboration. One of Hadamard's students was Maurice Fréchet, so assuming that this counts as an edge, Flajolet grandfather has Maurice Fréchet number ≤ 3 (F->Merlin->Hadamard->Fréchet). According to MathSciNet, Fréchet has Flajolet number ≤ 4 (via Nobelist Ilya Prigogine, M. Courbage, and Jean-Paul Allouche). So your Flajolet number is ≤ 7 (in addition to being 0).
I last heard from you more than two years ago when you sent me the following Email message, dated Jan. 3, 2009:
MathIsFun MathIsFun MathIsFun MathIsFun MathIsFun MathIsFun Dear Doron, A happy new year full of identities and combinatorial discoveries. (The card attached has an intriguing one, discovered empirically just recently.) BEST 2009 from your fan, Philippe
The members of the Académie Française are called Les immortels, and by analogy, also the members of the Académie des sciences. Of course, in most cases, this is not quite true, but in your case, I truly believe it, and your death, a week ago, was a removable singularity. Your vision and approach, taming analysis to count interesting things, will make you truly immortal!