As a form of reply, let me tell you some stories related to those discoveries. First, the Cartier-Foata episode. I had been very privileged to have seen my thesis refereed by Pierre Cartier, when I applied for a junior professorship in Strasbourg. My tutor was Paul-Andre' Meyer, the late great probabilist, whom I first met in Spring 1959 in Paris, when I attended a seminar in Probability led by Michel Loeve, who was on leave for a semester from his home Berkeley University . At that time Andre', as we called him, had worked with Brelot and Deny, on, say, Potential Theory and hard analysis. Thanks to the presence of Loeve in Paris, he got very much interested in Probability. Loeve seeing his great talent, sent him to Urbana (Illinois University) to work with Doob. There and on his way back to Paris, he completed his State Doctorate very rapidly.
When I was back from the States in Summer 1961, I was drafted in the French Navy, to be appointed in the Operations Research Bureau. There, whom I met: Paul-Andre' Meyer. We have remained close friends since, until he died back in 2003 of a heart attack. In 1965 I could complete my State Doctorate painfully, under the illuminating guidance of Schutzenberger. A year earlier, I could get a provisional position at Lille, and was immediately invited by Paul-Andre' to come over to Strasbourg, where he had been hired the previous year. I was lucky to get the job in october 1965. The first thing I did for my research work was to re-write my Doctoral dissertation, to put it in a more contemporary way, with some diagrams, and universal problems. Jean-Louis Verdier, who had been hired at the same time (and already belonged to the Bourbaki group) was kind enough to make some interesting remarks, but more essentially gave a copy to Pierre Cartier.
Pierre Cartier had just been writing a first draft on graphs, functors and categories for Bourbaki, a draft that had been rejected by the Bourbaki establishment. He was a bit disappointed, but when he read my draft, he was pleased to see that he could use the techniques developed in his own draft to present my work in a more algebraic way. By the way, the great Knuth told me that he had preferred my own thesis style, compared to the Bourbaki flavor of the Cartier-Foata ! Completing the joint monograph took two years, but it was well-received by several groups. Having Cartier as a co-author is a great privilege.
As for the exponential formula, I must say that the Mehler formula proof is a very rewarding subject. Not too complicated, and still spectacular. Proved by means of two transparencies, as Xavier Viennot would say. Himself gave a talk in Montreal some years ago, and surprisingly reproduced the combinatorial proof of the Jacobi polynomial generating function, a work I had done with Pierre Leroux,
Polynomes de Jacobi, interpretation combinatoire et fonction generatrice, Proc. Amer. Math. Soc., 87, 1983, p. 47-53.
Dick Askey had just written a paper giving several proofs of such a generating function, saying that it was an important result, as he sometimes says emphatically. My response was that it was not such a great accomplishment, as it was a way of writing two combinatorial objects in two different manners.