We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary `decorated' weights as well as an arbitrary `background' weight. Our CT theorem, like Viennot's lattice path theorem from which it is derived primarily by a change of variable lemma, is expressed in terms of orthogonal polynomials which in our applications of interest often turn out to be non-classical. Hence we also present an efficient method for finding explicit closed form polynomial expressions for these non-classical orthogonal polynomials. Our method for finding the closed form polynomial expressions relies on simple combinatorial manipulations of Viennot's diagrammatic representation for orthogonal polynomials. In the course of the paper we also provide a new proof of Viennot's original orthogonal polynomial lattice path theorem. The new proof is of interest because it uses diagonalization of the transfer matrix, but gets around difficulties that have arisen in past attempts to use this approach. In particular we show how to sum over a set of implicitly defined zeros of a given orthogonal polynomial, either by using properties of residues or by using partial fractions. We conclude by applying the method to two lattice path problems important in the study of polymer physics as models of steric stabilization and sensitized flocculation.
We say that a permutation $π$ is a Motzkin permutation if it avoids 132 and there do not exist $a<b$ such that $π_a<π_b<π_{b+1}$. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing and decreasing subsequences and the number of rises and descents. We also enumerate Motzkin permutations with additional restrictions, and study the distribution of occurrences of fairly general patterns in this class of permutations.
Chebyshev quotients, Demazure multiplicities, and Dyck-path models
School of Mathematical Sciences National Institute of Science Education and Research (NISER)（国立科学教育与研究学院数学院）；Homi Bhabha National Institute Training School Complex（霍米·巴巴国立学院培训学院）；Axiom Math（Axiom数学）
Recently, a new class of words, denoted by L_n, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality given by the (n-1)-st Catalan number. Here, we consider statistics on L_n recording the number of occurrences of a letter i. In the cases i = 0 and i = 1, we are able to determine explicit expressions for the number of members of L_n containing a given number of zeros or ones, which generalizes the prior result. To do so, we make use of recurrences to derive a functional equation satisfied by the generating function, which we solve by a new method employi…
We study the level $m$--Demazure flag of a level $\ell$--Demazure module for $\frak{sl}_2[t]$. We define the generating series $A_n^{\ell \rightarrow m}(x,q)$ which encodes the $q$--multiplicity of the level $m$ Demazure module of weight $n$. We establish two recursive formulae for these functions. We show that the specialization to $q=1$ is a rational function involving the Chebyshev polynomials. We give a closed form for $A_n^{\ell \rightarrow \ell+1}(x,q)$ and prove that it is given by a rational function. In the case when $m=\ell+1$ and $\ell=1,2$, we relate the generating series to partia…
In chemistry, Cyvin-Gutman enumerates Kekulé numbers for certain benzenoids and record it as $A050446$ on OEIS. This number is exactly the two variable array $T(n,m)$ defined by the recursion $T(n, m) = T(n, m-1) + \sum^{\lfloor\frac{n-1}{2}\rfloor}_{k=0} T(2k, m-1)T(n-1-2k, m)$, where $T(n,0)=T(0,m)=1$ for all nonnegative integers $m,n$. Interestingly, this number also appeared in the context of weighted graphs, graph polytopes, magic labellings, and unit primitive matrices, studied by different authors. Several interesting conjectures were made on the OEIS. These conjectures are related to b…
We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials -- the Chebyshev, Hermite, and Laguerre polynomials -- can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogona…
Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the expected height are analyzed asymptotically. Additionally, we use a bijection between admissible random walks and special binary sequences to prove a recent conjecture by Zhao on ballot sequences.
A Dyck path is a lattice path in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of steps (1,1) and (1,-1), which never passes below the x-axis. A peak at height k on a Dyck path is a point on the path with coordinate y=k that is immediately preceded by a (1,1) step and immediately followed by a (1,-1) step. In this paper we find an explicit expression to the generating function for the number of Dyck paths starting at (0,0) and ending at (2n,0) with exactly r peaks at height k. This allows us to express this function via Chebyshev polynomials of the second kind and generatin…
A weighted graph is a graph in which every edge is assigned a non-negative real number. In a weighted graph, the weight of a path is the sum of the weights of its edges, and the weighed degree of a vertex is the sum of the weights of the edges incident with it. In this paper we give three weighted degree conditions for the existence of heavy or Hamilton paths with one or two given end-vertices in 2-connected weighted graphs.
Kim and Drake used generating functions to prove that the number of 2-distant noncrossing matchings, which are in bijection with little Schroeder paths, is the same as the weight of Dyck paths in which downsteps from even height have weight 2. This work presents bijections from those Dyck paths to little Schroeder paths, and from a similar set of Dyck paths to big Schroeder paths. We show the effect of these bijections on the corresponding matchings, find generating functions for two new classes of lattice paths, and demonstrate a relationship with 231-avoiding permutations.
This paper concentrates on the set $\mathcal{V}_n$ of weighted Dyck paths of length $2n$ with special restrictions on the level of valleys. We first give its explicit formula of the counting generating function in terms of certain weight functions. When the weight functions are specialized, some connections are builded between $\mathcal{V}_n$ and other classical combinatorial structures such as $(a,b)$-Motzkin paths, $q$-Schröder paths, Delannoy paths and complete $k$-ary trees. Some bijections are also established between these settings and $\mathcal{V}_n$ subject to certain special weight fu…
In this article, we evaluate determinants of block hook matrices, which are block matrices consist of hook matrices. In particular, we deduce that the determinant of a block hook matrix factorizes nicely. In addition we give a combinatorial interpretation of the aforesaid factorization property by counting weighted paths in a suitable weighted digraph.
Consider a weighted graph G with n vertices, numbered by the set {1,...,n}. For any path p in G, we call w_G(p) the sum of the weights of the edges of the path and we define the multiset {\cal D}_{i,j} (G) = {w_G(p) | p simple path between i and j} We establish a criterion to say when, given a multisubset of the set of the real numbers there exists a weighted complete graph G such that the multisubset is equal to {\cal D}_{i,j} (G) for some i,j vertices of G. Besides we establish a criterion to say when, given for any i, j in {1,...,n} a multisubset of the set of the real numbers,{\cal D}_{i,j…
We prove a conjecture of Drake and Kim: the number of $2$-distant noncrossing partitions of $\{1,2,...,n\}$ is equal to the sum of weights of Motzkin paths of length $n$, where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is combinatorial.
A weighted coloured-edge graph is a graph for which each edge is assigned both a positive weight and a discrete colour, and can be used to model transportation and computer networks in which there are multiple transportation modes. In such a graph paths are compared by their total weight in each colour, resulting in a Pareto set of minimal paths from one vertex to another. This paper will give a tight upper bound on the cardinality of a minimal set of paths for any weighted coloured-edge graph. Additionally, a bound is presented on the expected number of minimal paths in weighted bicoloured-ed…
A partial Motzkin path is a path from $(0, 0)$ to $(n, k)$ in the $XOY$-plane that does not go below the $X$-axis and consists of up steps $U=(1, 1)$, down steps $D=(1, -1)$ and horizontal steps $H=(1, 0)$. A weighted partial Motzkin path is a partial Motzkin path with the weight assignment that all up steps and down steps are weighted by 1, the horizontal steps are endowed with a weight $x$ if they are lying on $X$-axis, and endowed with a weight $y$ if they are not lying on $X$-axis. Denote by $M_{n,k}(x, y)$ to be the weight function of all weighted partial Motzkin paths from $(0, 0)$ to $(…
We introduce and study the weighted $r$-path ideal of a weighted graph $G_ω$, which is a common generalization of Conca and De Negri's $r$-path ideal for unweighted graphs and Paulsen and Sather-Wagstaff's edge ideal of the weighted graph. Over a field, we explicitly describe primary decompositions of these ideals, and we characterize Cohen-Macaulayness of these ideals for trees (with arbitrary $r$) and complete graphs (for $r=2$).
We revisit $R$-polynomials with introducing the new idea ``shifted $R$-polynomials" (or Bruhat weight) for all Bruhat intervals in finite Coxeter groups. Then, we apply these polynomials to weighted counting of Bruhat paths. Further, we prove a new criterion of irregularity of lower intervals as analogy of Carrell-Peterson's and Dyer's results. Also, we present the upper bound of shifted $R$-polynomials for Bruhat intervals of fixed length by Jacobsthal numbers.
We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about α^* \log n$ edges where α^* ~ 3.5911 is the unique solution of the equation $alpha log(alpha) - α=1. This answers a question posed by Janson (1999).
The Gaussian polynomial in variable $q$ is defined as the $q$-analog of the binomial coefficient. In addition to remarkable implications of these polynomials to abstract algebra, matrix theory and quantum computing, there is also a combinatorial interpretation through weighted lattice paths. This interpretation is equivalent to weighted board tilings, which can be used to establish Gaussian polynomial identities. In particular, we prove duals of such identities and evaluate related sums.
In the first part of this paper, we survey results that are associated with three types of Laplacian matrices:difference, normalized, and signless. We derive eigenvalue and eigenvector formulaes for paths and cycles using circulant matrices and present an alternative proof for finding eigenvalues of the adjacency matrix of paths and cycles using Chebyshev polynomials. Even though each results is separately well known, we unite them, and provide uniform proofs in a simple manner. The main objective of this study is to solve the problem of finding graphs, on which spectral clustering methods and…
Consider a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$. The weight of the shortest (minimum-weight) path $P_1$ between two given vertices is known to be $\ln n / n$, asymptotically. Define a second-shortest path $P_2$ to be the shortest path edge-disjoint from $P_1$, and consider more generally the shortest path $P_k$ edge-disjoint from all earlier paths. We show that the cost $X_k$ of $P_k$ converges in probability to $2k/n+\ln n/n$ uniformly for all $k \leq n-1$. We show analogous results when the edge weights are drawn from an exponential …
A simple method for transmitting quantum states within a quantum computer is via a quantum spin chain---that is, a path on $n$ vertices. Unweighted paths are of limited use, and so a natural generalization is to consider weighted paths; this has been further generalized to allow for loops (\emph{potentials} in the physics literature). We study the particularly important situation of perfect state transfer with respect to the corresponding adjacency matrix or Laplacian through the use of orthogonal polynomials. Low-dimensional examples are given in detail. Our main result is that PST with respe…
We say that a digraph is essentially cyclic if its Laplacian spectrum is not completely real. The essential cyclicity implies the presence of directed cycles, but not vice versa. The problem of characterizing essential cyclicity in terms of graph topology is difficult and yet unsolved. Its solution is important for some applications of graph theory, including that in decentralized control. In the present paper, this problem is solved with respect to the class of digraphs with ring structure, which models some typical communication networks. It is shown that the digraphs in this class are essen…
