On the lattice path method in convolution-type combinatorial identities (II)—The weighted counting function method on lattice paths

1989 ◽  
Vol 10 (12) ◽  
pp. 1131-1135 ◽  
Author(s):  
Chu Wen-chang
Keyword(s):  
Function Method ◽  
Counting Function ◽  
Combinatorial Identities ◽  
Lattice Paths ◽  
Convolution Type ◽  
Lattice Path

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 Some Convolution Identities and an Inverse Relation Involving Partial Bell Polynomials

10.37236/2476 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Daniel Birmajer ◽  
Juan B. Gil ◽  
Michael D. Weiner
Keyword(s):  
Combinatorial Identities ◽  
Convolution Type ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Inverse Relation

We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting multinomial formula for the binomial coefficients. The inverse relation is deduced from a parametrization of suitable identities that facilitate dealing with nested compositions of partial Bell polynomials.

scholarly journals $n$-color overpartitions, lattice paths, and multiple basic hypergeometric series

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Olivier Mallet
Keyword(s):  
Generating Functions ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Combinatorial Identities ◽  
Lattice Paths ◽  
Multiple Series ◽  
Infinite Products ◽  
Nous Montrons ◽  
International Audience ◽  
Multiple Basic Hypergeometric Series

International audience We define two classes of multiple basic hypergeometric series $V_{k,t}(a,q)$ and $W_{k,t}(a,q)$ which generalize multiple series studied by Agarwal, Andrews, and Bressoud. We show how to interpret these series as generating functions for special restricted lattice paths and for $n$-color overpartitions with weighted difference conditions. We also point out that some specializations of our series can be written as infinite products, which leads to combinatorial identities linking $n$-color overpartitions with ordinary partitions or overpartitions. Nous définissons deux classes de séries hypergéométriques basiques multiples $V_{k,t}(a,q)$ et $W_{k,t}(a,q)$ qui généralisent des séries multiples étudiées par Agarwal, Andrews et Bressoud. Nous montrons comment interpréter ces séries comme les fonctions génératrices de chemins avec certaines restrictions et de surpartitions $n$-colorées vérifiant des conditions de différences pondérées. Nous remarquons aussi que certaines spécialisations de nos séries peuvent s'écrire comme des produits infinis, ce qui conduit à des identités combinatoires reliant les surpartitions $n$-colorées aux partitions ou surpartitions ordinaires.

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.

scholarly journals On the Number of Turns in Reduced Random Lattice Paths

2013 ◽  
Vol 50 (2) ◽  
pp. 499-515
Author(s):  
Yunjiang Jiang ◽  
Weijun Xu
Keyword(s):  
Asymptotic Expansion ◽  
Limit Theorems ◽  
Lattice Paths ◽  
Lattice Path ◽  
Large Deviations Principle ◽  
Random Lattice ◽  
Symmetric Random Walk ◽  
Mean And Variance ◽  
The Mean ◽  
Lower Order Terms

We consider the tree-reduced path of a symmetric random walk on ℤd. It is interesting to ask about the number of turns Tn in the reduced path after n steps. This question arises from inverting the signatures of lattice paths: Tn gives an upper bound of the number of terms in the signature needed to reconstruct a ‘random’ lattice path with n steps. We show that, when n is large, the mean and variance of Tn in the asymptotic expansion have the same order as n, while the lower-order terms are O(1). We also obtain limit theorems for Tn, including the large deviations principle, central limit theorem, and invariance principle. Similar techniques apply to other finite patterns in a lattice path.

On Lattice Paths with Several Diagonal Steps

1968 ◽  
Vol 11 (4) ◽  
pp. 537-545 ◽  
Author(s):  
S.G. Mohanty ◽  
B.R. Handa
Keyword(s):  
Positive Integer ◽  
Line Segment ◽  
Lattice Point ◽  
Lattice Points ◽  
Lattice Paths ◽  
Lattice Path ◽  
Vertical Step ◽  
Horizontal Step ◽  
Minimal Lattice

In this note we consider the enumeration of unrestricted and restricted minimal lattice paths from (0, 0) to (m, n), with the following (μ + 2) moves, μ being a positive integer. Let the line segment between two lattice points on which no other lattice point lies be called a step. A lattice path at any stage can have either (1) a vertical step denoted by S0, or (2) a diagonal step parallel to the line x = ty (t = 1,…, μ), denoted by St, or (3) a horizontal step, denoted by Sμ+1.

scholarly journals Difference Equations and Generating Functions for some Lattice Path Problems

2019 ◽  
pp. 551-559
Author(s):  
Sreelatha Chandragiri
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Generating Functions ◽  
Lattice Paths ◽  
Lattice Path ◽  
Dyck Paths ◽  
Constant Coefficients ◽  
Novel Method

An identity for generating functions is proved in this paper. A novel method to compute the number of restricted lattice paths is developed on the basis of this identity. The method employs a difference equation with non-constant coefficients. Dyck paths, Schr¨oder paths, Motzkins path and other paths are computed to illustrate this method

scholarly journals On the Number of Turns in Reduced Random Lattice Paths

2013 ◽  
Vol 50 (02) ◽  
pp. 499-515
Author(s):  
Yunjiang Jiang ◽  
Weijun Xu
Keyword(s):  
Asymptotic Expansion ◽  
Limit Theorems ◽  
Lattice Paths ◽  
Lattice Path ◽  
Large Deviations Principle ◽  
Random Lattice ◽  
Symmetric Random Walk ◽  
Mean And Variance ◽  
The Mean ◽  
Lower Order Terms

We consider the tree-reduced path of a symmetric random walk on ℤ d . It is interesting to ask about the number of turns T n in the reduced path after n steps. This question arises from inverting the signatures of lattice paths: T n gives an upper bound of the number of terms in the signature needed to reconstruct a ‘random’ lattice path with n steps. We show that, when n is large, the mean and variance of T n in the asymptotic expansion have the same order as n, while the lower-order terms are O(1). We also obtain limit theorems for T n, including the large deviations principle, central limit theorem, and invariance principle. Similar techniques apply to other finite patterns in a lattice path.

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