scholarly journals Combinatorics on lattice paths in strips

2021 ◽  
Vol 94 ◽  
pp. 103310
Author(s):  
Nancy S.S. Gu ◽  
Helmut Prodinger
Keyword(s):  
Lattice Paths

Enumeration of Lattice paths with infinite types of steps and the Chung-Feller property

2021 ◽  
Vol 344 (8) ◽  
pp. 112452
Author(s):  
Lin Yang ◽  
Sheng-Liang Yang
Keyword(s):  
Lattice Paths ◽  
Feller Property

scholarly journals Random sampling of lattice paths with constraints, via transportation

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Lucas Gerin
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Random Sampling ◽  
Optimal Transport ◽  
Lattice Paths ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Contraction Property ◽  
International Audience

International audience We build and analyze in this paper Markov chains for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. These chains are easy to implement, and sample an "almost" uniform path of length $n$ in $n^{3+\epsilon}$ steps. This bound makes use of a certain $\textit{contraction property}$ of the Markov chain, and is proved with an approach inspired by optimal transport.

scholarly journals Tokuyama's Identity for Factorial Schur $P$ and $Q$ Functions

10.37236/4971 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Angèle M. Hamel ◽  
Ronald C. King
Keyword(s):  
Partition Function ◽  
Lattice Paths ◽  
Schur Function ◽  
Vertex Model ◽  
Shifted Tableaux

A recent paper of Bump, McNamara and Nakasuji introduced a factorial version of Tokuyama's identity, expressing the partition function of  six vertex model as the product of a $t$-deformed Vandermonde and a Schur function. Here we provide an extension of their result by exploiting the language of primed shifted tableaux, with its proof based on the use of non-interesecting lattice paths.

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.

ASYMPTOTICS OF THE WEIGHTED DELANNOY NUMBERS

2012 ◽  
Vol 08 (01) ◽  
pp. 175-188 ◽  
Author(s):  
ROB NOBLE
Keyword(s):  
Number Field ◽  
Asymptotic Expansions ◽  
Lattice Paths ◽  
Delannoy Numbers ◽  
Divisibility Properties ◽  
Special Case

The weighted Delannoy numbers give a weighted count of lattice paths starting at the origin and using only minimal east, north and northeast steps. Full asymptotic expansions exist for various diagonals of the weighted Delannoy numbers. In the particular case of the central weighted Delannoy numbers, certain weights give rise to asymptotic coefficients that lie in a number field. In this paper we apply a generalization of a method of Stoll and Haible to obtain divisibility properties for the asymptotic coefficients in this case. We also provide a similar result for a special case of the diagonal with slope 2.

scholarly journals On the edge set of graphs of lattice paths

2004 ◽  
Vol 2004 (61) ◽  
pp. 3291-3299
Author(s):  
Steven Klee ◽  
Lara Pudwell ◽  
Rick Gillman
Keyword(s):  
Recursive Formula ◽  
Lattice Paths ◽  
New Family ◽  
Fixed Parameter

This note explores a new family of graphs defined on the set of paths of them×nlattice. We let each of the paths of the lattice be represented by a vertex, and connect two vertices by an edge if the corresponding paths share more thanksteps, wherekis a fixed parameter0=k=m+n. Each such graph is denoted byG(m,n,k). Two large complete subgraphs ofG(m,n,k)are described for all values ofm,n, andk. The size of the edge set is determined forn=2, and a complicated recursive formula is given for the size of the edge set whenk=1.

scholarly journals Pairs of Lattice Paths and Positive Trigonometric Sums

2009 ◽  
Vol 32 (1) ◽  
pp. 67-75 ◽  
Author(s):  
Victor J. W. Guo ◽  
Jiang Zeng
Keyword(s):  
Lattice Paths ◽  
Trigonometric Sums
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