Lattice Path Combinatorics Research Articles

Bounded Dyck paths, bounded alternating sequences, orthogonal polynomials, and reciprocity

The theme of this article is a “reciprocity” between bounded up-down paths and bounded alternating sequences. Roughly speaking, this “reciprocity” manifests itself by the fact that the extension of the sequence of numbers of paths of length n, consisting of diagonal up- and down-steps and being confined to a strip of bounded width, to negative n produces numbers of alternating sequences of integers that are bounded from below and from above. We show that this reciprocity extends to families of non-intersecting bounded up-down paths and certain arrays of alternating sequences which we call alternating tableaux. We provide as well weighted versions of these results. Our proofs are based on Viennot’s theory of heaps of pieces and on the combinatorics of non-intersecting lattice paths. An unexpected application leads to a refinement of a result of Bousquet-Mélou and Viennot on the width-height-area generating function of parallelogram polyominoes. Finally, we exhibit the relation of the arising alternating tableaux to plane partitions of strip shapes.

Cryptographic keys exchange model for smart city applications

This study presents a secret key sharing protocol that establishes cryptographically secured communication between two entities. A new symmetric key exchange scenario for smart city applications is presented in this research. The protocol is based on the specific properties of the Fuss-Catalan numbers and the Lattice Path combinatorics. The proposed scenario consists of three phases: generating a Fuss-Catalan object based on the grid dimension, defining the movement in the Lattice Path Grid and defining the key equalisation rules. In the experimental part, the authors present the security analysis of the protocol as well as its test. Also, they examine the equivalence of the proposed with Maurer's satellite scenario and suggest a new scenario that implements an information-theoretical protocol for the public key distribution. Additionally, a comparison with related studies and methods is provided, as well as a comparison with satellite scenario, which proves the advantages of solution presented by the authors. Finally, they propose further research directions regarding key management in smart city applications.

Analytic Combinatorics of Lattice Paths with Forbidden Patterns, the Vectorial Kernel Method, and Generating Functions for Pushdown Automata

In this article we develop a vectorial kernel method—a powerful method which solves in a unified framework all the problems related to the enumeration of words generated by a pushdown automaton. We apply it for the enumeration of lattice paths that avoid a fixed word (a pattern), or for counting the occurrences of a given pattern. We unify results from numerous articles concerning patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. This refines the study by Banderier and Flajolet from 2002 on enumeration and asymptotics of lattice paths: we extend here their results to pattern-avoiding walks/bridges/meanders/excursions. We show that the autocorrelation polynomial of this forbidden pattern, as introduced by Guibas and Odlyzko in 1981 in the context of rational languages, still plays a crucial role for our algebraic languages. En passant, our results give the enumeration of some classes of self-avoiding walks, and prove several conjectures from the On-Line Encyclopedia of Integer Sequences. Finally, we also give the trivariate generating function (length, final altitude, number of occurrences of the pattern p), and we prove that the number of occurrences is normally distributed and linear with respect to the length of the walk: this is what Flajolet and Sedgewick call an instance of Borges’s theorem.

The genesis of involutions (Polarizations and lattice paths)

The number of Borel orbits in the symmetric space SLn∕S(GLp×GLq) is analyzed, various (bivariate) generating functions are found. Relations to lattice path combinatorics are explored.

Total Idle Time Density Function of M/C2/1 Systems under (0,k) Policy

The aim of this paper is to derive the probability density function (pdf) of the total idle time of busy period of M/C2/1 queues operating under control policies through lattice path combinatorics (LPC) approach. The service distribution is approximated by Coxian two-phase distribution. We focus on deriving the pdf of total idle time of M/C2/1 queues under (0,k) control policy, wherein the server goes on the vacation when the system becomes empty and re-opens for service immediately at the arrival of the kthcostumer. We present an important result which is the theorem of the pdf of total idle time when system is in busy period that ends with a departure.

Affine geometric crystal of [formula omitted] and limit of Kirillov–Reshetikhin perfect crystals

Let g be an affine Lie algebra with index set I={0,1,2,⋯,n} and gL be its Langlands dual. It is conjectured in [16] that for each k∈I∖{0} the affine Lie algebra g has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for gL. Motivated by this conjecture we construct a positive geometric crystal for the affine Lie algebra g=An(1) for each Dynkin index k∈I∖{0} and show that its ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for An(1) given in [24]. In the process we develop and use some lattice-path combinatorics.

Entanglement and correlation functions of the quantum Motzkin spin-chain

We present exact results on the exactly solvable spin chain of Bravyi et al. [Phys. Rev. Lett. 109, 207202 (2012)]. This model is a spin one chain and has a Hamiltonian that is local and translationally invariant in the bulk. It has a unique (frustration free) ground state with an energy gap that is polynomially small in the system’s size (2n). The half-chain entanglement entropy of the ground state is 12logn+const.[Bravyi et al., Phys. Rev. Lett. 109, 207202 (2012)]. Here we first write the Hamiltonian in the standard spin-basis representation. We prove that at zero temperature, the magnetization is along the z-direction, i.e., ⟨sx⟩=⟨sy⟩=0 (everywhere on the chain). We then analytically calculate ⟨sz⟩ and the two-point correlation functions of sz. By analytically diagonalizing the reduced density matrices, we calculate the Schmidt rank, von Neumann, and Rényi entanglement entropies for the following: 1. Any partition of the chain into two pieces (not necessarily in the middle) and 2. L consecutive spins centered in the middle. Further, we identify entanglement Hamiltonians (Eqs. (49) and (59)). We prove a small lemma (Lemma (1)) on the combinatorics of lattice paths using the reflection principle to relate and calculate the Motzkin walk “height” to spin expected values. We also calculate the, closely related (scaled), correlation functions of Brownian excursions. The known features of this model are summarized in a table in Sec. I.

A Lattice Path Combinatorial Approach to Rothe Numbers and Related Convolution Results

Rothe numbers (aka generalized Catalan numbers) have a natural interpretation in terms of lattice paths. Making use of this interpretation this paper aims at showing how some classical convolution identities involving Rothe numbers can also be proved conveniently using lattice path combinatorics.

Half-lattice Paths and Virasoro Characters

We first briefly review the role of lattice paths in the derivation of fermionic expressions for the M(p,p') minimal model characters of the Virasoro Lie algebra. We then focus on the recently introduced half-lattice paths for the M(p,2p+/-1) characters, reformulating them in such a way that the two cases may be treated uniformly. That the generating functions of these half-lattice paths are indeed M(p,2p+/-1) characters is proved by describing weight preserving bijections between them and the corresponding RSOS lattice paths. Here, the M(p,2p-1) case is derived for the first time. We then apply the methods of Bressoud and Warnaar to these half-lattice paths to derive fermionic expressions for the Virasoro characters X^{p,2p+/-1}_{1,2} that differ from those obtained from the RSOS paths. This work is an extension of that presented by the third author at the "7th International Conference on Lattice Path Combinatorics and Applications", Siena, Italy, July 2010.

Lattices of paths: Representation theory and valuations

We study some distributive lattices arising in the combinatorics of lattice paths. In particular, for the Dyck, Motzkin and Schroder lattices we describe the spectrum and we determine explicitly the Euler characteristic in terms of natural parameters of lattice paths.

On batch queueing systems: A combinatorial approach

We consider batch queueing systems M / M H / 1 and M H / M / 1 with catastrophes. The transient probability functions of these queueing systems are obtained by a Lattice Path Combinatorics approach that utilizes randomization and dual processes. Steady state distributions are also determined. Generalization to systems having batches of different sizes are discussed.

Lattice path counting and the theory of queues

In this paper we will show how recent advances in the combinatorics of lattice paths can be applied to solve interesting and nontrivial problems in the theory of queues. The problems we discuss range from classical ones like M a / M b / 1 systems to open tandem systems with and without global blocking and to queueing models that are related to random walks in a quarter plane like the Flatto–Hahn model or systems with preemptive priorities.

Lattice path combinatorics for multiple product identities

Lattice paths are enumerated as walks on a lattice under the Delannoy criterion of vertical, horizontal and upward diagonal steps. Delannoy recursions are generalized for arbitrary weights in each of these directions, where the row elements of the Delannoy triangles correspond to lattice paths for all walks at a given length. This paper surveys known connections between generating functions for Delannoy paths, orthogonal polynomials, and multiple product identities. Generating functions for these sets of weighted lattice paths are derived for the individual triangles as well as for multiple triangle overlays. Recursive algorithms and series expansions are used to describe the basic structure of generalized Delannoy triangles. Ordinary generating functions are expanded to yield lacunary sequences in the Delannoy row summations. The lattice paths are enumerated by generalized Chebyshev polynomials of the first and second kind, as well as being evaluated by Jacobi polynomials, Legendre polynomials, and Zernike polynomials. The Jacobi triple product and Watson quintuple product identities are extended to balanced product identities. Elements of Delannoy number triangles are derived from these multiple product identities as coefficients in their power series expansions. Results for lacunary Fibonacci and Lucas number sequences are highlighted as special cases for many of the multiple product identities. For the most generalized variable selection, Jacobi polynomials and hypergeometric functions are used to derive expansion terms in the triple and quintuple products. Interpretation of the lattice path weights in exponential terms is shown to generate trigonometric theta functions.

Busy period analysis of [formula omitted] queue: Lattice path approach

In this paper, we find the busy period density of C 2 / C 2 b / 1 queues in explicit computational form, through lattice path (LP) approach. Both the arrival and service time distributions are approximated by 2-phase Cox distribution C 2 , which has a Markovian property enabling us to use LP combinatorics. Since any distribution with rational Laplace–Stieltjes transform (LST) and square coefficient of variation ( C V 2 ) lying in [ 1 / 2 , ∞ ) can be approximated by a C 2 ([M. Agarwal, K. Sen, B. Borkakaty, Busy period density of queueing system C 3 / M / 1 , Journal of Combinatorics, Information and Systems Sciences 31 (1–4) (2006) 127–161]), the results obtained would be applicable to a very wide class of distributions occurring in real life.

Narayana numbers and Schur–Szegő composition

In the present paper we find a new interpretation of Narayana polynomials N n ( x ) which are the generating polynomials for the Narayana numbers N n , k = 1 n C n k − 1 C n k where C j i stands for the usual binomial coefficient, i.e. C j i = j ! i ! ( j − i ) ! . They count Dyck paths of length n and with exactly k peaks, see e.g. [R.A. Sulanke, The Narayana distribution, in: Lattice Path Combinatorics and Applications (Vienna, 1998), J. Statist. Plann. Inference 101 (1–2) (2002) 311–326 (special issue)] and they appeared recently in a number of different combinatorial situations, see for e.g. [T. Doslic, D. Syrtan, D. Veljan, Enumerative aspects of secondary structures, Discrete Math. 285 (2004) 67–82; A. Sapounakis, I. Tasoulas, P. Tsikouras, Counting strings in Dyck paths, Discrete Math. 307 (2007) 2909–2924; F. Yano, H. Yoshida, Some set partitions statistics in non-crossing partitions and generating functions, Discrete Math. 307 (2007) 3147–3160]. Strangely enough Narayana polynomials also occur as limits as n → ∞ of the sequences of eigenpolynomials of the Schur–Szegő composition map sending ( n − 1 ) -tuples of polynomials of the form ( x + 1 ) n − 1 ( x + a ) to their Schur–Szegő product, see below. We present below a relation between Narayana polynomials and the classical Gegenbauer polynomials which implies, in particular, an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence { N n ( x ) } .

Gambler's Ruin with Catastrophes and Windfalls

Abstract We compute ruin probabilities, in both infinite-time and finite-time, for a Gambler's Ruin problem with both catastrophes and windfalls in addition to the customary win/loss probabilities. For constant transition probabilities, the infinite-time ruin probabilities are derived using difference equations. Finite-time ruin probabilities of a system having constant win/loss probabilities and variable catastrophe/windfall probabilities are determined using lattice path combinatorics. Formulae for expected time till ruin and the expected duration of gambling are also developed. The ruin probabilities (in infinite-time) for a system having variable win/loss/catastrophe probabilities but no windfall probability are found. Finally, the infinite-time ruin probabilities of a system with variable win/loss/catastrophe/windfall probabilities are determined.

Characters of graded parafermion conformal field theory

The graded parafermion conformal field theory at level k is a close cousin of the much-studied Z(k) parafermion model. Three character formulas for the graded parafermion theory are presented, one bosonic, one fermionic (both previously known), and one of spinon type (which is new). The main result of this paper is a proof of the equivalence of these three forms using q-series methods combined with the combinatorics of lattice paths. The pivotal step in our approach is the observation that the graded parafermion theory - which is equivalent to the coset (osp) over cap (1, 2)(k)/(u) over cap (1) - can be factored as ((osp) over cap (1, 2)(k)/(su) over cap (2)(k)) x ((su) over cap (2)(k)/(u) over cap (1)), with the two cosets on the right equivalent to the minimal model M(k + 2, 2k + 3) and the Z(k) parafermion model, respectively. This factorization allows for a new combinatorial description of the graded parafermion characters in terms of the one-dimensional con. guration sums of the (k + 1)-state Andrews-Baxter-Forrester model.

Analytic Combinatorics of Lattice Paths: Enumeration and Asymptotics for the Area

This paper tackles the enumeration and asymptotics of the area below directed lattice paths (walks on $\mathbb{N}$ with a finite set of jumps). It is a nice surprise (obtained via the "kernel method'') that the generating functions of the moments of the area are algebraic functions, expressible as symmetric functions in terms of the roots of the kernel. For a large class of walks, we give full asymptotics for the average area of excursions ("discrete'' reflected Brownian bridge) and meanders ("discrete'' reflected Brownian motion). We show that drift is not playing any role in the first case. We also generalise previous works related to the number of points below a path and to the area between a path and a line of rational slope.

Dual processes to solve single server systems

A new approach is used to determine the transient probability functions of the classical queueing systems: M/M/1, M/M/1/H, and M/M/1/H with catastrophes. This new solution method uses dual processes, randomization and lattice path combinatorics. The method reveals that the transient probability functions for M/M/1/H and M/M/1/H with catastrophes have the same mathematical form.

István Vincze (1912–1999) and his contribution to lattice path combinatorics and statistics

A brief account of the life and work of István Vincze, a prominent Hungarian statistician, is given. His contributions in various topics are discussed. They include empirical distribution, Kolmogorov–Smirnov statistics, information theory, Cramér–Fréchet–Rao inequality, estimation of density, and a characterization problem.