Franz Lehner
A formula expressing free cumulants in terms of the Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and Lagrange inversion. For the converse we discuss Gessel-Viennot theory to express Hankel determinants in terms of various cumulants.
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Non-intersecting path explanation for block Pfaffians and applications into skew-orthogonal polynomials
本文将 Stembridge 的结果推广到块 Pfaffian，用非相交路径给出组合解释，并建立其生成函数与斜正交多项式及多重部分斜正交多项式之间的联系。
Decomposition of free cumulants
引入适应Motzkin路径的非交叉分拆格，定义算子值Motzkin累积量，通过Möbius反演公式细化自由累积量与布尔累积量的关系，并应用于自由概率得到自由累积量的加性分解。
Generalized Schröder paths arising from a combinatorial interpretation of generalized Laurent bi-orthogonal polynomials
本文定义并研究ℓ-Schröder路径，推广了多种已知路径，并揭示其与广义Laurent双正交多项式矩的组合联系，且给出非相交路径生成函数的闭式分解。
The aim of the dissertation is threefold: the first two parts are devoted to explore the Fourth Moment Theorem and universality properties for homogeneous sums, while the last part approaches the classical theory of orthogonal polynomials via the invariant theory of binary forms. Specifically, Part I deals with the Lindberg method of influence functions in free probability spaces. By providing a general multidimensional invariance principle for homogeneous sums in freely independent variables, a class of universal laws for semicircular and free Poisson approximations will be derived that satis…
The subject of this paper are polynomials in multiple non-commuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favard-type recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in non-commuting variables with a resolvent-type generating function. Among such families, we describe the ones that are orthogonal. Their recursion relations have a more special form; the best way to describe them is in terms of the free cumulant generating function of the state of orthogonality, which turns out to satisfy a type of second-or…
$d$-orthogonal polynomials, Fuss-Catalan matrices and lattice paths
本文利用$d$-正交多项式定义Fuss-Catalan-Riordan阵列，并建立其与Fuss Catalan数和格路的联系，强调生产矩阵的作用。
A Discrete Logarithm Construction for Orthogonal Double Covers of the Complete Graph by Hamiltonian Paths
本文通过离散对数方法，利用哈密顿路径构造完全图的正交双重覆盖，拓展了已知的ODC应用范围。
Combinatorics of orthogonal polynomials on the unit circle
本文从组合视角研究单位圆上正交多项式的矩，给出三种路径解释，并据此推导圆Jacobi与Rogers–Szegő多项式的广义矩显式公式，还引入并解释了广义线性化系数。
The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolvent-type form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, …
We study the higher-order Euler polynomials and give the corresponding monic orthogonal polynomials, which are Meixner-Pollaczek polynomials with certain arguments and constant factors. Moreover, through a general connection between moments of random variables and the generalized Motzkin numbers, we can obtain a new recurrence formula and a matrix representation for the higher-order Euler polynomials, interpreting them as weighted lattice paths.
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…
We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalising known results for Catalan numbers.
If a sequence indexed by nonnegative integers satisfies a linear recurrence without constant terms, one can extend the indices of the sequence to negative integers using the recurrence. Recently, Cigler and Krattenthaler showed that the negative version of the number of bounded Dyck paths is the number of bounded alternating sequences. In this paper we provide two methods to compute the negative versions of sequences related to moments of orthogonal polynomials. We give a combinatorial model for the negative version of the number of bounded Motzkin paths. We also prove two conjectures of Cigle…
Bounded Dyck paths, bounded alternating sequences, orthogonal polynomials, and reciprocity
本文研究有界上下路径与有界交错序列间的“互反性”，将其推广至不相交路径与交错表，基于碎片堆理论证明，并应用于细化多连块生成函数及平面分拆研究。
We derive explicit expressions for $q$-orthogonal polynomials arising in the enumeration of area-weighted Dyck paths with restricted height.
Bidiagonal matrix factorisations associated with symmetric multiple orthogonal polynomials and lattice paths
本文利用双对角矩阵分解、格路径与分支连分式联系，刻画 (r+1) 重对称 r-正交多项式及其与 r-Dyck 路径生产矩阵的关系，并给出 Appell 序列和测度密度公式。
Free Meixner states are a class of functionals on non-commutative polynomials introduced in math.CO/0410482. They are characterized by a resolvent-type form for the generating function of their orthogonal polynomials, by a recursion relation for those polynomials, or by a second-order non-commutative differential equation satisfied by their free cumulant functional. In this paper, we construct an operator model for free Meixner states. By combinatorial methods, we also derive an operator model for their free cumulant functionals. This, in turn, allows us to construct a number of examples. Many…
Skew symplectic and orthogonal characters through lattice paths
针对斜辛与正交Schur函数行列式表达式研究较少的问题，研究者基于Koike和Terada表模型的格点路径描述，用组合方法建立其对偶Jacobi-Trudi与Lascoux-Pragacz公式的类似形式，并代数推导得普通Jacobi-Trudi公式。
Orthogonality between acyclic subdigraphs and paths in digraphs
本文研究了有向图中无环子图与路径的正交性，证明了在替换稳定集为诱导无环子图后，可以放松两个开放猜想。
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 compute the bulk limit of the correlation functions for the uniform measure on lozenge tilings of a hexagon. The limiting determinantal process is a translation invariant extension of the discrete sine process, which also describes the ergodic Gibbs measure of an appropriate slope.
Non-commutative probability insights into the double-scaling limit SYK Model with constant perturbations: moments cumulants and $q$-independence
本文研究含固定数目常数对角扰动的 DSSYK 模型，通过矩—累积量关系揭示其矩公式中由 $\ ilde{q}$ 控制的经典与 Boolean 独立性混合，并联系组合结构及引力路径积分。
A classic result of Asplund and Grünbaum states that intersection graphs of axis-aligned rectangles in the plane are $χ$-bounded. This theorem can be equivalently stated in terms of path-decompositions as follows: There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every graph that has two path-decompositions such that each bag of the first decomposition intersects each bag of the second in at most $k$ vertices has chromatic number at most $f(k)$. Recently, Dujmović, Joret, Morin, Norin, and Wood asked whether this remains true more generally for two tree-decompositions. In this note…
A combinatorial theory for type $R_I$ orthogonal polynomials is given. The ingredients include weighted generalized Motzkin paths, moments, continued fractions, determinants, and histories. Several explicit examples in the Askey scheme are given.
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. …
Polynomial sequences $p_n(x)$ of binomial type are a principal tool in the umbral calculus of enumerative combinatorics. We express $p_n(x)$ as a \emph{path integral} in the ``phase space'' $\Space{N}{} \times {[-π,π]}$. The Hamiltonian is $h(ϕ)=\sum_{n=0}^\infty p_n'(0)/n! e^{inϕ}$ and it produces a Schrödinger type equation for $p_n(x)$. This establishes a bridge between enumerative combinatorics and quantum field theory. It also provides an algorithm for parallel quantum computations. Keywords: Feynman path integral, umbral calculus, polynomial sequence of binomial type, token, Schrödinger …
The main objects of the investigation presented in this paper are branched-continued-fraction representations of ratios of contiguous hypergeometric series and type II multiple orthogonal polynomials on the step-line with respect to linear functionals or measures whose moments are ratios of products of Pochhammer symbols. This is an interesting case study of the recently found connection between multiple orthogonal polynomials and branched continued fractions that gives a clear example of how this connection leads to considerable advances on both topics. We obtain new results about generating …
This thesis deals with some $(1+1)$-dimensional lattice path models from the KPZ universality class: the directed random polymer with inverse-gamma weights (known as log-gamma polymer) and its zero temperature degeneration, i.e. the last passage percolation model, with geometric or exponential waiting times. We consider three path geometries: point-to-line, point-to-half-line, and point-to-line with paths restricted to stay in a half-plane. Through exact formulas, we establish new connections between integrable probabilistic models and the ubiquitous Whittaker and Schur functions. More in deta…
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