Delannoy Numbers Research Articles

Off-diagonally symmetric domino tilings of the Aztec diamond of odd order

We study the enumeration of off-diagonally symmetric domino tilings of odd-order Aztec diamonds in two directions: (1) with one boundary defect, and (2) with maximally-many zeroes on the diagonal. In the first direction, we prove a symmetry property which states that the numbers of off-diagonally symmetric domino tilings of the Aztec diamond of order 2n−1 are equal when the boundary defect is at the kth position and the (2n−k)th position on the boundary, respectively. This symmetry property proves a special case of a recent conjecture by Behrend, Fischer, and Koutschan.In the second direction, a Pfaffian formula is obtained for the number of “nearly” off-diagonally symmetric domino tilings of odd-order Aztec diamonds, where the entries of the Pfaffian satisfy a simple recurrence relation. The numbers of domino tilings mentioned in the above two directions do not seem to have a simple product formula, but we show that these numbers satisfy simple matrix equations in which the entries of the matrix are given by Delannoy numbers. The proof of these results involves the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths. Finally, we propose conjectures concerning the log-concavity and asymptotic behavior of the number of off-diagonally symmetric domino tilings of odd-order Aztec diamonds.

Polynomial reduction for holonomic sequences and applications in π-series and congruences

Recently, Hou, Mu and Zeilberger introduced a new process of polynomial reduction for hypergeometric terms, which can be used to prove and generate hypergeometric identities automatically. In this paper, we extend this polynomial reduction to holonomic sequences. As applications, we describe an algorithmic way to prove and generate new multi-summation identities. Especially we present new families of π-series involving Domb numbers and Franel numbers, and new families of congruences for Franel numbers and Delannoy numbers.

A new orthogonal polynomial associated with a generalization of Delannoy numbers

In this paper, we define a general class of orthogonal polynomials Dn(r)(x,y) and derive a relationship between Dn(r)(x,y) and generalized Delannoy polynomials dn(r)(x,y). Also, we deduce a generating function for Dn(r)(x,y) from which we obtain several identities for it. Moreover, some further relations for generalized Delannoy polynomials dn(r)(x,y) are achieved.

Combinatorial aspects of sandpile models on wheel and fan graphs

We study combinatorial aspects of the sandpile model on wheel and fan graphs, seeking bijective characterisations of the model’s recurrent configurations on these families. For wheel graphs, we exhibit a bijection between these recurrent configurations and the set of subgraphs of the cycle graph which maps the level of the configuration to the number of edges of the subgraph. This bijection relies on two key ingredients. The first consists in considering a stochastic variant of the standard Abelian sandpile model (ASM), rather than the ASM itself. The second ingredient is a mapping from a given recurrent state to a canonical minimal recurrent state, exploiting similar ideas to previous studies of the ASM on complete bipartite graphs and Ferrers graphs. We also show that on the wheel graph with 2n vertices, the number of recurrent states with level n is given by the first differences of the central Delannoy numbers. Finally, using similar tools, we exhibit a bijection between the set of recurrent configurations of the ASM on fan graphs and the set of subgraphs of the path graph containing the right-most vertex of the path. We show that these sets are also equinumerous with certain lattice paths, which we name Kimberling paths after the author of the corresponding entry in the Online Encyclopedia of Integer Sequences.

Some analytical properties of the matrix related to q-coloured Delannoy numbers

AbstractThe $q$-coloured Delannoy numbers $D_{n,k}(q)$ count the number of lattice paths from $(0,\,0)$ to $(n,\,k)$ using steps $(0,\,1)$, $(1,\,0)$ and $(1,\,1)$, among which the $(1,\,1)$ steps are coloured with $q$ colours. The focus of this paper is to study some analytical properties of the polynomial matrix $D(q)=[d_{n,k}(q)]_{n,k\geq 0}=[D_{n-k,k}(q)]_{n,k\geq 0}$, such as the strong $q$-log-concavity of polynomial sequences located in a ray or a transversal line of $D(q)$ and the $q$-total positivity of $D(q)$. We show that the zeros of all row sums $R_n(q)=\sum \nolimits _{k=0}^{n}d_{n,k}(q)$ are in $(-\infty,\, -1)$ and are dense in the corresponding semi-closed interval. We also prove that the zeros of all antidiagonal sums $A_n(q)=\sum \nolimits _{k=0}^{\lfloor n/2 \rfloor }d_{n-k,k}(q)$ are in the interval $(-\infty,\, -1]$ and are dense there.

A class of weighted Delannoy numbers

The weighted Delannoy numbers are defined by the recurrence relation fm,n = ? fm?1,n +? fm,n?1 + ? fm?1,n?1 if mn > 0, with fm,n = ?m?n if nm = 0. In this work, we study a generalization of these numbers considering the same recurrence relation but with fm,n = AmBn if nm = 0. More particularly, we focus on the diagonal sequence fn,n. With some ingenuity, we are able to make use of well-established methods by Pemantle and Wilson, and by Melczer in order to determine its asymptotic behavior in the case A, B, ?, ?, ? ? 0. In addition, we also study its P-recursivity with the help of symbolic computation tools.

On Some Identities Involving Cauchy Products of Central Delannoy Numbers

In this paper, we have examined Cauchy products of central Delannoy numbers. Moreover, using their recurrence relation we have derived some important identities such as the Cassini and Catalan's identities which contain these products.

A determinantal expression and a recursive relation of the Delannoy numbers

Abstract In the paper, by a general and fundamental, but non-extensively circulated, formula for derivatives of a ratio of two differentiable functions and by a recursive relation of the Hessenberg determinant, the author finds a new determinantal expression and a new recursive relation of the Delannoy numbers. Consequently, the author derives a recursive relation for computing central Delannoy numbers in terms of related Delannoy numbers.

Hankel determinants of linear combinations of moments of orthogonal polynomials, II

We present a formula that expresses the Hankel determinants of a linear combination of length d+1 of moments of orthogonal polynomials in terms of a dtimes d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

Superbinomial coefficients

We investigate several families of polynomials that are related to certain Euler type summation operators. Being integer valued at integral points, they satisfy combinatorial properties and nearby symmetries, due to triangle recursion relations involving squares of polynomials. Some of these interpolate the Delannoy numbers. The results are motivated by and strongly related to our study of irreducible Lie supermodules, where dimension polynomials in many cases show similar features.

Relationships between spherical and bispherical harmonics, and an electrostatic T-matrix for dimers

Bispherical harmonics are the solutions to Laplace’s equation in bispherical coordinates. We investigate the relationships between spherical harmonics and bispherical harmonics in terms of radial inversion and derive new series expansions between the harmonics. The series coefficients are the Delannoy numbers encountered in combinatorics, and we exploit the series to derive for them a new second order homogeneous recurrence relation. We use the T-matrix/null field method to solve for the potential of two different spheres in an arbitrary static external electric field, as a series of bispherical harmonics where the coefficients are related via matrix transformations. The matrix elements are expressed as surface integrals of bispherical harmonics, for which analytic expressions are derived in terms of Legendre functions. The resonant values of the dielectric function are computed via an eigenvalue problem which is found to be more stable than the solution using difference equations. The rate of convergence of the matrix formulation is compared to the re-expansion method using spherical harmonics for a uniform field an a dipole in the gap; each series converges faster in a different region of space, which we discuss in terms of the boundaries of convergence of each solution and the image singularities of the scattered field.

A new recursive formula arising from a determinantal expression for weighted Delannoy numbers

A new recursive formula arising from a determinantal expression for weighted Delannoy numbers

Some aspects of Neyman triangles and Delannoy arrays

This note considers some number theoretic properties of the orthonormal Neyman polynomials which are related to Delannoy numbers and certain complex Delannoy numbers.

A New Generalization of Delannoy Numbers

In this paper, a new extension for Delannoy numbers is presented. Also, several identities such as generating function and recurrence relations for these numbers are obtained.

Determinantal forms and recursive relations of the Delannoy two-functional sequence

In the paper, the authors establish closed forms for the Delannoy two-functional sequence and its difference in terms of the Hessenberg determinants, derive recursive relations for the Delannoy two-functional sequence and its difference, and deduce closed forms, in terms of the Hessenberg determinants, and recursive relations for the Delannoy one-functional sequence, the Delannoy numbers, and central Delannoy numbers. In the paper, the authors establish closed forms for the Delannoy two-functional sequence and its difference in terms of the Hessenberg determinants, derive recursive relations for the Delannoy two-functional sequence and its difference, and deduce closed forms, in terms of the Hessenberg determinants, and recursive relations for the Delannoy one-functional sequence, the Delannoy numbers, and central Delannoy numbers.

Unimodality and linear recurrences associated with rays in the Delannoy triangle

In this paper, we study the unimodality of sequences located in the infinite transversals of the Delannoy triangle. We establish recurrence relations associated with the sum of elements laying along the finite transversals of the cited triangle and we give the generating function of the established sum. Moreover, new identities for the odd and even terms of the Tribonacci sequence are given. Finally, we define a $q$-analogue for the Delannoy numbers and we propose a $q$-deformation of the Tribonacci sequence.

Catalan-like number sequences and Hausdorff moment sequences

Many well-known Catalan-like sequences turn out to be Stieltjes moment sequences (Liang et al. (2016)). However, a Stieltjes moment sequence is in general not determinate; Liang et al. suggested a further analysis about whether these moment sequences are determinate and how to obtain the associated measures. In this paper we find necessary conditions for a Catalan-like sequence to be a Hausdorff moment sequence. As a consequence, we will see that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers, are Hausdorff moment sequences. We can also identify the smallest interval including the support of the unique representing measure. Since Hausdorff moment sequences are determinate and a representing measure for above mentioned sequences are already known, we could almost complete the analysis raised by Liang et al. In addition, subsequences of Catalan-like number sequences are also considered; we will see a necessary and sufficient condition for subsequences of Stieltjes Catalan-like number sequences to be Stieltjes Catalan-like number sequences. We will also study a representing measure for a linear combination of consecutive terms in Catalan-like number sequences.

Analytic aspects of Delannoy numbers

The Delannoy numbers d(n,k) count the number of lattice paths from (0,0) to (n−k,k) using steps (1,0),(0,1) and (1,1). We show that the zeros of all Delannoy polynomials dn(x)=∑k=0nd(n,k)xk are in the open interval (−3−22,−3+22) and are dense in the corresponding closed interval. We also show that the Delannoy numbers d(n,k) are asymptotically normal (by central and local limit theorems).

Delannoy Numbers and Preferential Arrangements

A preferential arrangement on [ [ n ] ] = { 1 , 2 , … , n } is a ranking of the elements of [ [ n ] ] where ties are allowed. The number of preferential arrangements on [ [ n ] ] is denoted by r n . The Delannoy number D ( m , n ) is the number of lattice paths from ( 0 , 0 ) to ( m , n ) in which only east ( 1 , 0 ) , north ( 0 , 1 ) , and northeast ( 1 , 1 ) steps are allowed. We establish a symmetric identity among the numbers r n and D ( p , q ) by means of algebraic and combinatorial methods.

Alternative proofs of some formulas for two tridiagonal determinants

Abstract In the paper, the authors provide five alternative proofs of two formulas for a tridiagonal determinant, supply a detailed proof of the inverse of the corresponding tridiagonal matrix, and provide a proof for a formula of another tridiagonal determinant. This is a companion of the paper [F. Qi, V. Čerňanová,and Y. S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), in press.