Why Delannoy numbers?

doi:10.1016/j.jspi.2005.02.004

Journal of Statistical Planning and Inference

Volume 135, Issue 1, 1 November 2005, Pages 40-54

https://doi.org/10.1016/j.jspi.2005.02.004 Get rights and content

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 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.

Section snippets

Classical lattice paths

Before tackling the question of Delannoy numbers and Delannoy lattice paths, note that the classical number sequences or lattice paths have the name of a mathematician: the Italian Leonardo Fibonacci ( $\sim 1170$ – $\sim 1250$ ), the French Blaise Pascal (1623–1662), the Swiss Jacob Bernoulli (1654–1705), the Scottish James Stirling (1692–1770), the Swiss Leonhard Euler (1707–1783), the Belgian Eugène Catalan (1814–1894), the German Ernst Schröder (1841–1902), the German Walther von Dyck (1856–1934), the

Delannoy numbers

Delannoy is another “famous” name which is associated to an integer sequence related to lattice paths enumeration. Delannoy's numbers indeed correspond to the sequence $(D_{n, k})_{n, k \in N}$ , the number of walks from $(0, 0)$ to $(n, k)$ , with jumps $(0, 1)$ , $(1, 1)$ , or $(1, 0)$ . $\begin{matrix} 1 & 19 & 181 & 1159 & 5641 & 22363 & 75517 & 224143 & 598417 & 1462563 \\ 1 & 17 & 145 & 833 & 3649 & 13073 & 40081 & 108545 & 265729 & 598417 \\ 1 & 15 & 113 & 575 & 2241 & 7183 & 19825 & 48639 & 108545 & 224143 \\ 1 & 13 & 85 & 377 & 1289 & 3653 & 8989 & 19825 & 40081 & 75517 \\ 1 & 11 & 61 & 231 & 681 & 1683 & 3653 & 7183 & 13073 & 22363 \\ 1 & 9 & 41 & 129 & 321 & 681 & 1289 & 2241 & 3649 & 5641 \\ 1 & 7 & 25 & 63 & 129 & 231 & 377 & 575 & 833 \end{matrix}$

Henri Auguste Delannoy (1833–1915)

Some people suggested that “Delannoy” was related either to the French mathematician Charles Delaunay (like in Delaunay triangulations) or to the Russian mathematician Boris Nikolaevich Delone, but this is not the case, as we shall see.

It is true that “Delannoy” sounds like (and actually is) a French family name [d $\land$ lanoa] in approximative phonetic alphabet ( $\land$ like in “duck” and a in “have”). There are in fact thousands of Delannoy, mostly in the North of France and in Belgium. This toponym

Delannoy's mathematical work

Delannoy began his mathematical life reading the mathematical recreations that Lucas began to publish in 1879 in La Revue Scientifique. He was in contact with him in 1880 and began immediately to work with him, answering to letters of mathematicians transmitted by Lucas.

The first mention, in a mathematical work, to Delannoy is in an article by Lucas “Figurative arithmetics and permutations (1883)” (Lucas, 1883), which deals with enumeration of configurations of 8 queen-like problems (the

Other Delannoy's works

Besides mathematics, Delannoy painted watercolors and, perhaps more importantly, studied history. Indeed, from 1897 to 1914, he published 29 accurate archaeological/historical articles in the Mémoires de la Société des Sciences Naturelles et archéologiques de la Creuse.

Let us give a taste of Delannoy's writer talent: here are some titles of his articles: “On the signification of word ieuru”, “One more word about ieuru”, “A riot in Guéret in 1705”, “Aubusson's tapestries”, “A bigamist in

Acknowledgements

The first author's interest to Delannoy numbers comes from a talk that Marko Petkovšek gave in the Algorithms Seminar at INRIA in 1999 (a summary of this talk can be found in Banderier (2000)). As an example, he was dealing with chess king moves (his general result about the nature of different multidimensional recurrences can be found in the article (Bousquet-Mélou and Petkovšek, 2000)). M. Petkovšek asked the first author what he knew about Delannoy, and C. Banderier then started to conduct

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^☆: This article is an extended version of the abstract submitted (February 2002) by the first author to the 5th lattice path combinatorics and discrete distributions conference (Athens, June 5–7, 2002).

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Why Delannoy numbers?☆

Abstract

Section snippets

Classical lattice paths

Delannoy numbers

Henri Auguste Delannoy (1833–1915)

Delannoy's mathematical work

Other Delannoy's works

Acknowledgements

Discrete Math.

Theor. Comput. Sci.

J. Statist. Plann. Inference

Discrete Math.

J. Combin. Theory Ser. A

J. Combin. Theory Ser. A.

Discrete Math.

Adv. Math.

Discrete Math.

J. Mol. Struct. (Theochem)

Discrete Appl. Math.

Discrete Math.

Discrete Math.

Discrete Appl. Math.

C.R. Acad. Sci. Paris

A propos de l’interprétation géométrique du problème du scrutin

L’Enseignement Math.

Calcul des probabilités. Solution directe du problème résolu par M. Bertrand

C.R. Acad. Sci.

On generalized Delannoy paths

SIAM J. Discrete Math.

Henri-Auguste Delannoy et la publication des œuvres posthumes d’édouard Lucas

Gazette des Math.

Path transformations connecting Brownian bridge, excursion and meander

Bull. Sci. Math.

The average height of planted plane trees

Note sur une équation aux différences finies

J. M. Pures Appl.

On the analytical forms called trees, with applications to the theory of chemical combinations

Rep. Brit. Ass.

On the analytical forms called trees

Sylv., Amer. J. IV.

Analyse Combinatoire. Tomes I, II

The art of finite and infinite expansions. Advanced Combinatorics

L’arithméticien Édouard Lucas (1842–1891): théorie et instrumentation

Revue d’Histoire des Mathématiques

Emploi de l’échiquier pour la résolution de problèmes arithmétiques

Assoc. Franç. Nancy

Sur la durée du jeu

S. M. F. Bull. XVI

Emploi de l’échiquier pour la résolution de divers problèmes de probabilité

Assoc. Franç. Paris XVIII

Formules relatives aux coefficients du binôme

Assoc. Franç. Limoges XIX

Problèmes divers concernant le jeu

Assoc. Franç. Limoges XIX

Sur les arbres géométriques et leur emploi dans la théorie des combinaisons chimiques

Assoc. Franç. Caen XXIII

Emploi de l’échiquier pour la résolution de certains problèmes de probabilités

Assoc. Franç. Bordeaux XXIV

Sur une question de probabilités traitée par d’Alembert

S. M. F. Bull. XXIII