scholarly journals Viewing Determinants as Nonintersecting Lattice Paths yields Classical Determinantal Identities Bijectively

10.37236/2530 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Markus Fulmek
Keyword(s):  
Generating Functions ◽  
Point Of View ◽  
Lattice Paths ◽  
Binet Formula ◽  
Plücker Relations ◽  
Nonintersecting Lattice Paths ◽  
Determinantal Identities ◽  
Determinantal Identity

In this paper, we show how general determinants may be viewed as generating functions of nonintersecting lattice paths, using the Lindström-Gessel-Viennot-method and the Jacobi Trudi identity together with elementary observations.After some preparations, this point of view provides "graphical proofs'' for classical determinantal identities like the Cauchy-Binet formula, Dodgson's condensation formula, the Plücker relations, Laplace's expansion and Turnbull's identity. Also, a determinantal identity generalizing Dodgson's condensation formula is presented, which might be new.

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.

On fluctuation problems in the theory of queues

1976 ◽  
Vol 8 (03) ◽  
pp. 548-583 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Markov Chains ◽  
Markov Processes ◽  
Generating Functions ◽  
Finite Differences ◽  
Historical Development ◽  
Banach Algebras ◽  
Point Of View ◽  
Mathematical Methods ◽  
Operator Calculus ◽  
Complex Functions

This paper gives a survey of the historical development of the solutions of various fluctuation problems in the theory of queues from the point of view of the mathematical methods used. These methods include Markov chains, Markov processes, integral equations, complex functions, generating functions, Laplace transforms, factorization of functions, operator calculus, Banach algebras and some particular methods, such as calculus of finite differences and combinatorics. In addition, the paper contains several recent results of the author for semi-Markov queuing processes.

scholarly journals The Hermite polynomials and the Bessel functions from a general point of view

2003 ◽  
Vol 2003 (57) ◽  
pp. 3633-3642 ◽  
Author(s):  
G. Dattoli ◽  
H. M. Srivastava ◽  
D. Sacchetti
Keyword(s):  
Difference Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Bessel Functions ◽  
Hermite Polynomials ◽  
Point Of View ◽  
General Point ◽  
Ordinary Case

We introduce new families of Hermite polynomials and of Bessel functions from a point of view involving the use of nonexponential generating functions. We study their relevant recurrence relations and show that they satisfy differential-difference equations which are isospectral to those of the ordinary case. We also indicate the usefulness of some of these new families.

scholarly journals Counting pairs of nonintersecting lattice paths with respect to weighted turns

1996 ◽  
Vol 153 (1-3) ◽  
pp. 189-198 ◽  
Author(s):  
Christian Krattenthaler ◽  
Robert A. Sulanke
Keyword(s):  
Lattice Paths ◽  
Nonintersecting Lattice Paths

scholarly journals An extension to overpartitions of Rogers-Ramanujan identities for even moduli

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Sylvie Corteel ◽  
Jeremy Lovejoy ◽  
Olivier Mallet
Keyword(s):  
Generating Functions ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Lattice Paths ◽  
Infinite Products ◽  
International Audience

International audience We investigate class of well-poised basic hypergeometric series $\tilde{J}_{k,i}(a;x;q)$, interpreting these series as generating functions for overpartitions defined by multiplicity conditions. We also show how to interpret the $\tilde{J}_{k,i}(a;1;q)$ as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases $(a,q) \to (1/q,q)$, $(1/q,q^2)$, and $(0,q)$, where some of the functions $\tilde{J}_{k,i}(a;x;q)$ become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.

scholarly journals Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences

10.37236/7375 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Nicholas R. Beaton ◽  
Mathilde Bouvel ◽  
Veronica Guerrini ◽  
Simone Rinaldi
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Kernel Method ◽  
Lattice Paths ◽  
Schröder Numbers ◽  
Special Cases ◽  
Generating Tree

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.

scholarly journals Vertically Constrained Motzkin-Like Paths Inspired by Bobbin Lace

10.37236/7799 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Veronika Irvine ◽  
Stephen Melczer ◽  
Frank Ruskey
Keyword(s):  
Mathematical Model ◽  
Generating Functions ◽  
Finite Lattice ◽  
Lattice Paths ◽  
Change Direction ◽  
Quarter Plane ◽  
Directed Paths ◽  
Schröder Paths ◽  
Motzkin Paths

Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\mathfrak{A}=$$\{\rightarrow,$ $\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$ with the restriction that vertical steps $(\uparrow, \downarrow)$ cannot be consecutive. The set $\mathfrak{A}$ is the union of the well known Motzkin step vectors $\mathfrak{M}=$$\{\rightarrow,$ $\nearrow,$ $\searrow\}$ with the vertical steps $\{\uparrow, \downarrow\}$. An explicit bijection $\phi$ between the exhaustive set of vertically constrained paths formed from $\mathfrak{A}$ and a bisection of the paths generated by $\mathfrak{M}S$ is presented. In a similar manner, paths with the step vectors $\mathfrak{B}=$$\{\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$, the union of Dyck step vectors and constrained vertical steps, are examined.  We show, using the same $\phi$ mapping, that there is a bijection between vertically constrained $\mathfrak{B}$ paths and the subset of Motzkin paths avoiding horizontal steps at even indices.  Generating functions are derived to enumerate these vertically constrained, partially directed paths when restricted to the half and quarter-plane.  Finally, we extend Schröder and Delannoy step sets in a similar manner and find a bijection between these paths and a subset of Schröder paths that are smooth (do not change direction) at a regular horizontal interval.

scholarly journals Summing a Family of Generalized Pell Numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Linear Combination ◽  
Computer Algebra ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Partial Sums ◽  
Pell Numbers ◽  
New Family ◽  
Binet Formula

AbstractA new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P lnis expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.

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