scholarly journals A Generalization of the $k$-Bonacci Sequence from Riordan Arrays

10.37236/4618 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
José L. Ramírez ◽  
Víctor F. Sirvent
Keyword(s):  
Lattice Paths ◽  
Lattice Path ◽  
Combinatorial Interpretation ◽  
Polynomial Sequence ◽  
Lucas Polynomials ◽  
Riordan Arrays ◽  
Weighted Lattice Paths

In this article, we introduce  a family of weighted lattice paths, whose step set is $\{H=(1,0), V=(0,1), D_1=(1,1), \dots, D_{m-1}=(1,m-1)\}$. Using these lattice paths, we define a family of Riordan arrays whose sum on the rising diagonal is the $k$-bonacci sequence. This construction generalizes the Pascal and Delannoy Riordan arrays, whose sum on the rising diagonal is the Fibonacci and tribonacci sequence, respectively.  From this family of Riordan arrays we introduce a generalized $k$-bonacci polynomial sequence, and we give a lattice path combinatorial interpretation of these polynomials. In particular, we find a combinatorial interpretation of tribonacci and tribonacci-Lucas polynomials.

scholarly journals Some Properties of the(p,q)-Fibonacci and(p,q)-Lucas Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
GwangYeon Lee ◽  
Mustafa Asci
Keyword(s):  
Generating Functions ◽  
Combinatorial Identities ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
Pascal Matrix ◽  
Riordan Arrays ◽  
Combinatorial Sums

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called(p,q)-Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving(p,q)-Fibonacci polynomials.

scholarly journals A Continuous Analogue of Lattice Path Enumeration

10.37236/8788 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Quang-Nhat Le ◽  
Sinai Robins ◽  
Christophe Vignat ◽  
Tanay Wakhare
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lattice Paths ◽  
Lattice Path ◽  
Continuous Version ◽  
Continuous Analog ◽  
Continuous Analogue ◽  
Lattice Path Enumeration ◽  
Multinomial Coefficients ◽  
General Method

Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.  

scholarly journals Lattice Paths, Sampling Without Replacement, and Limiting Distributions

10.37236/156 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
M. Kuba ◽  
A. Panholzer ◽  
H. Prodinger
Keyword(s):  
Generating Functions ◽  
Probability Distributions ◽  
Random Variable ◽  
Phase Changes ◽  
Lattice Paths ◽  
Sampling Without Replacement ◽  
Sample Paths ◽  
Quarter Plane ◽  
Explicit Formulae ◽  
Weighted Lattice Paths

In this work we consider weighted lattice paths in the quarter plane ${\Bbb N}_0\times{\Bbb N}_0$. The steps are given by $(m,n)\to(m-1,n)$, $(m,n)\to(m,n-1)$ and are weighted as follows: $(m,n)\to(m-1,n)$ by $m/(m+n)$ and step $(m,n)\to(m,n-1)$ by $n/(m+n)$. The considered lattice paths are absorbed at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$. We provide explicit formulæ for the sum of the weights of paths, starting at $(m,n)$, which are absorbed at a certain height $k$ at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$, using a generating functions approach. Furthermore these weighted lattice paths can be interpreted as probability distributions arising in the context of Pólya-Eggenberger urn models, more precisely, the lattice paths are sample paths of the well known sampling without replacement urn. We provide limiting distribution results for the underlying random variable, obtaining a total of five phase changes.

scholarly journals On weighted lattice paths

1973 ◽  
Vol 14 (1) ◽  
pp. 21-29 ◽  
Author(s):  
R.D Fray ◽  
D.P Roselle
Keyword(s):  
Lattice Paths ◽  
Weighted Lattice Paths

scholarly journals Sects and Lattice Paths over the Lagrangian Grassmannian

10.37236/8664 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Aram Bingham ◽  
Özlem Uğurlu
Keyword(s):  
Symmetric Space ◽  
Borel Subgroup ◽  
Cell Decomposition ◽  
Bruhat Order ◽  
Lattice Paths ◽  
Lagrangian Grassmannian ◽  
A Cell ◽  
Weighted Lattice Paths

We examine Borel subgroup orbits in the classical symmetric space of type $CI$, which are parametrized by skew symmetric $(n, n)$-clans. We describe bijections between such clans, certain weighted lattice paths, and pattern-avoiding signed involutions, and we give a cell decomposition of the symmetric space in terms of collections of clans called sects. The largest sect with a conjectural closure order is isomorphic (as a poset) to the Bruhat order on partial involutions.

scholarly journals A Combinatorial Interpretation of the Area of Schröder Paths

10.37236/1472 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
E. Pergola ◽  
R. Pinzani
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Lattice Path ◽  
Combinatorial Interpretation ◽  
Schröder Paths

An elevated Schröder path is a lattice path that uses the steps $(1,1)$, $(1,-1)$, and $(2,0)$, that begins and ends on the $x$-axis, and that remains strictly above the $x$-axis otherwise. The total area of elevated Schröder paths of length $2n+2$ satisfies the recurrence $f_{n+1}=6f_n-f_{n-1}$, $n \geq 2$, with the initial conditions $f_0=1$, $f_1=7$. A combinatorial interpretation of this recurrence is given, by first introducing sets of unrestricted paths whose cardinality also satisfies the recurrence relation and then establishing a bijection between the set of these paths and the set of triangles constituting the total area of elevated Schröder paths.

scholarly journals A Determinant Expression for the Generalized Bessel Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Sheng-liang Yang ◽  
Sai-nan Zheng
Keyword(s):  
Catalan Numbers ◽  
Polynomial Sequence ◽  
Bessel Polynomials ◽  
Determinant Formula ◽  
Riordan Arrays ◽  
Bessel Polynomial

Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.

Lattice Paths and a Sergeev-Pragacz Formula for Skew Supersymmetric Functions

1995 ◽  
Vol 47 (2) ◽  
pp. 364-382 ◽  
Author(s):  
A. M. Hamel ◽  
I. P. Goulden
Keyword(s):  
Lattice Paths ◽  
Lattice Path

AbstractWe obtain a new version of the Sergeev–Pragacz formula for supersymmetric functions of standard shape–one applicable to arbitrary skew shape. The result involves an antisymmetrized sum of determinants that are themselves flagged supersymmetric functions. The proof is combinatorial, and follows by means of lattice path transformations.

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