Abstract. This article is not a research paper, but a little note on the history of combinatorics: We present here a tentative short biography of Henri Delannoy, and a survey of his most notable works. This answers to the question raised in the title, as these works are related to lattice paths enumeration, to the so-called Delannoy numbers, and were the first general way to solve Ballot-like problems.

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A Delannoy path is a minimal path with diagonal steps in Z 2 between two arbitrary points. We extend this notion to the n dimensions space Z n and identify such paths with words on a special kind of alphabet: an S-alphabet. We show that the set of all the words corresponding to Delannoy paths going from one point to another is exactly one class in the congruence generated by a Thue system that we exhibit. This Thue system induces a partial order on this set that is isomorphic to the set of ordered partitions of a fixed multiset where the blocks are sets with a natural order relation. Our main result is that this poset is a lattice. Introduction. A Delannoy path[11] is given as a path that can be drawn on a rectangular grid, starting from the southwest corner, going to the northeast corner, using only three kinds of elementary steps: north, east, and northeast. Hence they are minimal paths with diagonal steps. We generalize the notion of a Delannoy path to the hyperspace Z n , considering a hyperparallelipedic grid as a set of elementary steps: a step in each direction and the combinations of several of them, the diagonal steps.We prove that, in a very natural way, an S-alphabet can be associated with the possible elementary steps in a Delannoy path in Z n , and consequently S-words with Delannoy paths themselves. These notions were introduced by Schwer [8], in a completely different context, for treating simultaneity problems.We then define a Thue system on the set of S-words that turns out to be noetherian and confluent. This Thue system induces both an ordering on S-words and a congruence. Our main goal is to prove that each equivalence class for this congruence is with this order relation a lattice (Theorem 5.5). (This lattice is a nondistributive lattice as soon as n > 2.) An equivalence class can be viewed as the set of all ordered partitions of a fixed multiset where the blocks are sets (not multisets). There is a transparent bijection between an equivalence class and an element of this set, and the order relation over partitions derived is a very natural one. In [9] are given some links between S-words and others mathematical objects.Moreover, we exhibit a characterization of the S-words of a class (and so of generalized Delannoy paths going from a point to another) with a family of matrices having its coefficients in {−1, 0, 1} (Theorem 4.2), and we prove that the order on S-words can be exactly transposed as the componentwise order on matrices induced by −1 < 0 < 1 (Theorem 4.6).

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