Why Delannoy numbers?

Banderier, Cyril; Schwer, Sylviane

doi:10.1016/j.jspi.2005.02.004

Cited by 98 publications

(114 citation statements)

References 43 publications

Supporting

Mentioning

114

Contrasting

Order By: Relevance

“…The significance of these numbers is explained within a historic context in the paper "Why Delannoy numbers?" [2] by Banderier and Schwer. The diagonal elements (d n,n : n ≥ 0) in this array are the (central) Delannoy numbers (A001850 of Sloane [17]).…”

Section: Delannoy Numbersmentioning

confidence: 99%

“…Banderier and Schwer [2] note that there is no "natural" correspondence between Legendre polynomials and the original lattice path enumeration problem associated to the Delannoy array, while Sulanke [20] states that "the definition of Legendre polynomials does not appear to foster any combinatorial interpretation leading to enumeration". The present author has found a combinatorial interpretation in [11]; this interpretation involves, however, a variant of the Delannoy numbers whose table coincides on the main diagonal only.…”

Section: Legendre Polynomials and Their Connection To The Delannoy Numentioning

confidence: 99%

See 1 more Smart Citation

Delannoy Orthants of Legendre Polytopes

Hetyei

2008

Discrete Comput Geom

View full text Add to dashboard Cite

We construct an n-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of (x − 1)/2 in the nth Legendre polynomial. We show that the noncentral Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open generalized orthant. We thus provide a geometric interpretation of a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago (Good in Proc. Camb. Philos. Soc. 54:39-42, 1958; Lawden in Math. Gaz. 36:193-196, 1952; Moser and Zayachkowski in Scr. Math. 26:223-229, 1963). The polytopes we construct are closely related to the root polytopes introduced by Gelfand et al. (Arnold-Gelfand mathematical seminars: geometry and singularity theory, pp. 205-221. Birkhauser, Boston, 1996).

show abstract

Section: Delannoy Numbersmentioning

confidence: 99%

Section: Legendre Polynomials and Their Connection To The Delannoy Numentioning

confidence: 99%

Delannoy Orthants of Legendre Polytopes

Hetyei

2008

Discrete Comput Geom

View full text Add to dashboard Cite

show abstract

“…Many properties and applications of Delannoy numbers have been discussed [1,10,12,13,20,21]. In combinatorics, we regard weight as the number of colors and normalize by setting w = 1.…”

Section: Generalized Delannoy Matrixmentioning

confidence: 99%

“…The generic term f n,k = ( k n−k ) of the Fibonacci matrix F = (f n,k ) n,k≥0 counts the number of lattice paths from (0, 0) to (n − k, k) with steps (0, 1) and (1,1), and the entries of the Fibonacci matrix F satisfy the recurrence relation f n+1,k+1 = f n,k + f n−1,k .…”

Section: Introductionmentioning

confidence: 99%

Schröder matrix as inverse of Delannoy matrix

Yang

Zheng

2013

Linear Algebra and its Applications

View full text Add to dashboard Cite

Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turn out that Delannoy matrix, Pascal matrix, and Fibonacci matrix are all special cases of the generalized Delannoy matrices, meanwhile Schröder matrix and Catalan matrix also arise in involving inverses of the generalized Delannoy matrices. These connections are the focus of our paper.The half of generalized Delannoy matrix is also considered. In addition, we obtain a combinatorial interpretation for the generalized Fibonacci numbers.

show abstract

“…1 [16, p. 185]. We refer the reader to Banderier and Schwer [1] and to the references given there. In this case we have…”

Section: Introductionmentioning

confidence: 99%

Counting Lattice Paths With Four Types of Steps

Dziemiańczuk

2013

Graphs and Combinatorics

View full text Add to dashboard Cite

The main purpose of this paper is to derive generating functions for the numbers of lattice paths running from (0, 0) to any (n, k) in Z × N consisting of four types of steps: horizontal, and sloping L = (−1, 1). These paths generalize the well-known Delannoy paths which consist of steps H, V , and D. Several restrictions are considered. However, we mainly treat with those which will be needed to get the generating function for the numbers R(n, k) of these lattice paths whose points lie in the integer rectangle {(x, y) ∈ N 2 : 0 ≤ x ≤ n, 0 ≤ y ≤ k}. Recurrence relation, generating functions and explicit formulas are given. We show that most of considered numbers define Riordan arrays.

show abstract

Why Delannoy numbers?

Cited by 98 publications

References 43 publications

Delannoy Orthants of Legendre Polytopes

Delannoy Orthants of Legendre Polytopes

Schröder matrix as inverse of Delannoy matrix

Counting Lattice Paths With Four Types of Steps

Contact Info

Product

Resources

About