Products of random matrices and q-Catalan numbers

Symmetric matrices, Catalan paths, and correlations

Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from the cube to the elliptope…

Statistics of Partial Permutations via Catalan matrices

A generalized Catalan matrix $(a_{n,k})_{n,k\ge 0}$ is generated by two seed sequences $\mathbf{s}=(s_0,s_1,\ldots)$ and $\mathbf{t}=(t_1,t_2,\ldots)$ together with a recurrence relation. By taking $s_\ell=2\ell+1$ and $t_\ell=\ell^2$ we can interpret $a_{n,k}$ as the number of partial permutations, which are $n\times n$ $0,1$-matrices of $k$ zero rows with at most one $1$ in each row or column. In this paper we prove that most of fundamental statistics and some set-valued statistics on permutations can also be defined on partial permutations and be encoded in the seed sequences. Results on tw…

A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux

Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We present a unifying perspective on ASMs and other combinatorial objects by studying a certain tetrahedral poset and its subposets. We prove the order ideals of these subposets are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self-complementary plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux, Catalan objects, tournaments, and totally symmetric plane partitions. …

A Catalan Subset of Descending Plane Partitions

Descending plane partitions, alternating sign matrices, and totally symmetric self-complementary plane partitions are equinumerous combinatorial sets for which no explicit bijection is known. In this paper, we isolate a subset of descending plane partitions counted by the Catalan numbers. The proof follows by constructing a generating tree on these descending plane partitions that has the same structure as the generating tree for 231-avoiding permutations. We hope this result will provide insight on the search for a bijection with alternating sign matrices and/or totally symmetric self-complem…

菱形十二面体堆、交替符号矩阵的卡坦 congruences 以及温度利布代数的基础

Heaps of rhombic dodecahedra, catalan congruences on alternating sign matrices, and bases of the Temperley-Lieb algebra

研究了交替符号矩阵格上的卡坦 congruences，证明其与covexillary排列和避免321模式排列的关系，并扩展了温度利布代数的基础。

关于与类 Catalan 数相关序列的 Hankel 行列式的若干结果与猜想

Some results and conjectures about Hankel determinants of sequences which are related to Catalan-like numbers

本文研究类 Catalan 数对应容许矩阵各列序列的 Hankel 行列式，归纳计算机实验揭示的性质，并通过前置零行探讨后向移位情形。

Some observations about determinants which are connected with Catalan numbers and related topics

In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and connections with Hankel determinants. Finally we state some conjectures.

Three-dimensional Catalan numbers and product-coproduct prographs

We present the new combinatorial class of product-coproduct prographs which are planar assemblies of two types of operators: products having two inputs and a single output and coproducts having a single input and two outputs. We show that such graphs are enumerated by the $3$-dimensional Catalan numbers. We present some combinatorial bijections positioning product-coproduct prographs as key objects to probe families of objects enumerated by the $3$-dimensional Catalan numbers.

Catalan Numbers and Jacobi Polynomials

We prove that the inverse of the Hankel matrix of the reciprocals of the Catalan numbers has integer entries. We generalize the result to an infinite family of generalized Catalan numbers. The Hankel matrices that we consider are associated with orthogonal polynomials that are variants of Jacobi polynomials. Our proofs use these polynomials and computer algebra based on Wilf-Zeilberger theory.

The product of trees in the Loday-Ronco algebra through Catalan alternative tableaux

The aim of this note is to show how the introduction of certain tableaux, called Catalan alternative tableaux, provides a very simple and elegant description of the product in the Hopf algebra of binary trees defined by Loday and Ronco. Moreover, we use this description to introduce a new associative product on the space of binary trees.

Combinatoric results for graphical enumeration and the higher Catalan numbers

We summarize some combinatoric problems solved by the higher Catalan numbers. These problems are generalizations of the combinatoric problems solved by the Catalan numbers. The generating function of the higher Catalan numbers appeared recently as an auxiliary function in enumerating maps and explicit computations of the asymptotic expansion of the partition function of random matrices in the unitary ensemble case. We give combinatoric proofs of the formulas for the number of genus 0 and genus 1 maps.

On One-Parameter Catalan Arrays

We present a parametric family of Riordan arrays which are obtained by multiplying any Riordan array with a generalized Pascal array. In particular, we focus on some interesting properties of one-parameter Catalan triangles. We obtain several combinatorial identities that involve two special Catalan matrices, the Chebyshev polynomials of the second kind, some periodic sequences, and the Fibonacci numbers.

广义 Fibonacci 数、Catalan 数与调和数的恒等式及生成函数

Identities and Generating Functions of Products of Generalized Fibonacci numbers, Catalan and Harmonic Numbers

本文研究了广义 Fibonacci 与 Lucas 数类的性质，获得了著名 Fibonacci 恒等式的类比及乘积卷积新恒等式，并写下了其乘积、多截、调和数与 Catalan 数的生成函数。

Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species

The number of flats of a hyperplane arrangement is considered as a generalization of the Bell number and the Stirling number of the second kind. Robert Gill gave the exponential generating function of the number of flats of the extended Catalan arrangements, using species. In this article, we introduce the species of flats of the extended Catalan and Shi arrangements and they are given by iterated substitution of species of sets and lists. Moreover, we enumerate the flats of these arrangements in terms of infinite matrices.

Catalan半环的一种新的布尔矩阵表示

A new Boolean matrix representation for Catalan semirings

本文提出了一种新的布尔矩阵表示方法，用于表示Catalan半环中的顺序保持递减变换。

Enumeration of connected Catalan objects by type

Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects which carry a notion of type. There exists a product formula which enumerates these objects according to type. We define a notion of `connectivity' for these objects and prove an analogous product formula which counts connected objects by type. Our proof of this product formula is combinatorial and bijective. We extend this to a product formula which counts objects with a fixed type and number of connected components. We relate our product formulas to symmetric function…

Expressions for Catalan Kronecker Products

We give some elementary manifestly positive formulae for the Kronecker products s_(d,d) * s_(d+k,d-k) and s_(d,d) * s_(2d-k,1^k). These formulae demonstrate some fundamental properties of the Kronecker coefficients, and we use them to deduce a number of enumerative and combinatorial results.

Chromatic statistics for Catalan and Fuß-Catalan numbers

We refine Catalan numbers and Fuß-Catalan numbers by introducing colour statistics for triangulations of polygons and $d$-dimensional generalisations there-of which we call Fuß-Catalan complexes. Our refinements consist in showing that the number of triangulations, respectively Fuß-Catalan complexes, with a given colour distribution of its vertices is given by closed product formulae. The crucial ingredient in the proof is the Lagrange-Good inversion formula.

Orthogonal polynomials and Smith normal form

Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate matrix that refines one studied by Berlekamp, Carlitz, Roselle, and Scoville. The second argument, which uses orthogonal polynomials, generalizes to a number of other Hankel matrices, Toeplitz matrices, and Gram matrices. It gives new results for q-Catalan numbers, q-Motzkin numbers, q-Schröder numbers, q-Stirling numbers, q-matching numbers, q-factorials, q…

Flow polytopes with Catalan volumes

The Chan-Robbins-Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector $(1, 0, \ldots, 0, -1)$. The normalized volume of the Chan-Robbins-Yuen polytope equals the product of consecutive Catalan numbers, yet there is no combinatorial proof of this fact. We consider a natural generalization of this polytope, namely, the flow polytope of the complete graph with netflow vector $(1,1, 0, \ldots, 0, -2)$. We show that the volume of this polytope is a certain power of $2$ times the product of consecutive Catalan numbers. Our proof uses constant term identities…

Sums of squares of Krawtchouk polynomials, Catalan numbers, and some algebras over the Boolean lattice

Writing the values of Krawtchouk polynomials as matrices, we consider weighted partial sums along columns. For the general case, we find an identity that, in the symmetric case yields a formula for such partial sums. Complete sums of squares along columns involve "Super Catalan" numbers. We look as well for particular values (matrix entries) involving the Catalan numbers. Properties considered and developed in this work are applied to calculations of various dimensions that describe the structure of some *-algebras over the Boolean lattice based on inclusion/superset relations expressed algebr…

Noncommutative Catalan numbers

The goal of this paper is to introduce and study noncommutative Catalan numbers $C_n$ which belong to the free Laurent polynomial algebra in $n$ generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman $(q,t)$-versions, another -- to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices $H_m$ and introduce accompanying noncommutative binomial coefficients.

Catalan Recursion on Externally Ordered Bases of Unit Interval Positroids

The Catalan numbers form a sequence that counts over 200 combinatorial objects. A remarkable property of the Catalan numbers, which extends to these objects, is its recursive definition; that is, we can determine the $n^{th}$ object from previous ones. Matroids are combinatorial objects that generalize the notion of linear independence and have connections with other fields of mathematics. A family of matroids, called unit interval positroids (UIP), are Catalan objects induced by the antiadjacency matrices of unit interval orders. Associated to each UIP is the set of externally ordered bases, …

Catalan-like numbers and succession rules

The ECO method and the theory of Catalan-like numbers introduced by Aigner seems two completely unrelated combinatorial settings. In this work we try to establish a bridge between them, aiming at starting a (hopefully) fruitful study on their interactions. We show that, in a linear algebra context (more precisely, using infinite matrices), a succession rule can be translated into a (generalized) Aigner matrix by means of a suitable change of basis in the vector space of one-variable polynomials. We provide some examples to illustrate this fact and apply it to the study of two particular classe…

A Universal Bijection for Catalan Structures

A Catalan magma is a unique factorisation normed magma with only one irreducible element. The \val partitions the base set into subsets enumerated by Catalan numbers. The primary theorem characterises the conditions which a set a with product map must satisfy in order to be a free magma generated by the irreducible elements. This theorem can be used to prove a set of objects (with a product map) is a Catalan magma. The isomorphism between Catalan magmas gives a "universal" bijection -- essentially one bijection algorithm for all pairs of families. The morphism property ensures the bijection is…

帕斯卡、卡塔兰、莫茨金三角与张量积多重性

Pascal, Catalan, Motzkin triangles and tensor product multiplicities

本文通过表示论探讨了帕斯卡、卡塔兰和莫茨金三角及其一般化，揭示其通过初始条件由帕斯卡法则生成，并讨论了张量积多重性及平方和现象。

Restricted ascent sequences and Catalan numbers

Ascent sequences are those consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it and have been shown to be equinumerous with the (2+2)-free posets of the same size. Furthermore, connections to a variety of other combinatorial structures, including set partitions, permutations, and certain integer matrices, have been made. In this paper, we identify all members of the (4,4)-Wilf equivalence class for ascent sequences corresponding to the Catalan number C_n=\frac{1}{n+1}\binom{2n}{n}. This extends recent work concerning avoidance…

Proofs of two conjectures on Catalan triangle numbers

We prove two conjectures on sums of products of Catalan triangle numbers, which were originally conjectured by Miana, Ohtsuka, and Romero [Discrete Math. 340 (2017), 2388--2397]. The first one is proved by using Zeilberger's algorithm, and the second one is proved by establishing its $q$-analogue.

Generalized Catalan numbers from hypergraphs

The Catalan numbers (C_n)_{n >= 0} = 1,1,2,5,14,42,... form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we define a generalization of the Catalan numbers. In fact we define an infinite collection of generalizations C_n^(m)…

基于半整数二项式系数的调和卡特兰数级数

Harmonic Catalan Number Series via Half-Integer Binomial Coefficients

该研究通过半整数参数的交替二项式恒等式，结合广义二项式系数的显式表示与相关恒等式，推导出含卡特兰数、调和数的闭式求和公式，为转化恒等式提供了系统机制。

Parametric production matrices and weighted succession rules: a Dyck path example

We introduce weighted succession rules and parametric production matrices - simple extensions of the standard ECO method succession rules and production matrices. The purpose is to enumerate combinatorial objects with respect to several variables. We consider one main example, from the theory of Dyck paths. The path statistics primarily considered are peak height, rise height, and certain subwalks. Many classical sequences (such as the Catalan, Motzkin, Narayana and Schroeder numbers) are incorporated in this example.

Catalan matroid decompositions of certain positroids

A positroid is the matroid of a matrix whose maximal minors are all nonnegative. Given a permutation $w$ in $S_n$, the matroid of a generic $n \times n$ matrix whose non-zero entries in row $i$ lie in columns $w(i)$ through $n+i$ is an example of a positroid. We enumerate the bases of such a positroid as a sum of certain products of Catalan numbers, each term indexed by the $123$-avoiding permutations above $w$ in Bruhat order. We also give a similar sum formula for their Tutte polynomials. These are both avatars of a structural result writing such a positroid as a disjoint union of matroids, …

Canonical characters on quasi-symmetric functions and bivariate Catalan numbers

Every character on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character (Aguiar, Bergeron, and Sottile, math.CO/0310016). We obtain explicit formulas for the even and odd parts of the universal character on the Hopf algebra of quasi-symmetric functions. They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers: $$ C(m,n)=\frac{(2m)!(2n)!}{m!(m+n)!n!}. $$ Properties of characters and of quasi-symmetric functions are then used to derive several interesting identities among …

Splitting Kronecker squares, 2-decomposition numbers, Catalan Combinatorics, and the Saxl conjecture

While there has been some progress on the decomposition of Kronecker products of characters of the symmetric groups in recent times, results on the symmetric and alternating part of Kronecker squares are still scarce. Here, new results (and conjectures) are presented on this splitting of the squares that contribute to a refined understanding of the Kronecker squares. Furthermore, connections to 2-modular decomposition numbers, Catalan combinatorics, and to the Saxl conjecture are discussed which further motivate the study of these splittings.

Comparing two matrices of generalized moments defined by continued fraction expansions

We study two matrices $N$ and $M$ defined by the parameters of equivalent $S$- and $J$-continued fraction expansions, and compare them by examining the product $N^{-1}M$. Using examples based on the Catalan numbers, the little Schröder numbers and powers of $q$, we indicate that this matrix product is an object worthy of study. In the case of the little Schröder numbers, we find that the matrix $N$ has an interleaved structure based on two Riordan arrays.

Three interacting families of Fuss-Catalan posets

Three families of posets depending on a nonnegative integer parameter $m$ are introduced. The underlying sets of these posets are enumerated by the $m$-Fuss Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. The three families of posets are related: they fit into a chain for the order extension relation and they share some properties. Two associative algebras are constructed as quotients of generalizations of the Malvenuto-Reutenauer algebra. Their products describe intervals of our analogues of Stanley lattices and…

Random graph states, maximal flow and Fuss-Catalan distributions

For any graph consisting of $k$ vertices and $m$ edges we construct an ensemble of random pure quantum states which describe a system composed of $2m$ subsystems. Each edge of the graph represents a bi-partite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated to a given g…

The Catalan-Qi number of the second kind and a related integral

Li et al. give an integral formula for the Catalan-Qi number of the second kind. They show that this integral can be written as a summation with double factorials. In this paper the integral is reduced to a product of the Catalan number and a hypergeometric function. This hypergeometric function can be written as an associated Legendre polynomial of the first kind. There are also connections with the incomplete Beta function and the Gegenbauer polynomials. In the last section a related integral is calculated.

卡特兰数的树指标和

Tree-indexed sums of Catalan numbers

本文研究由树索引的卡特兰数乘积的无限和，证明其为$1/\pi$的有理系数多项式，并给出有效算法，同时引入参数化提升，将其表示为完全椭圆积分的多项式，次数不超过树顶点数的一半，研究动机源于大型迷宫系统。

集值加泰罗尼亚组合数学

Set-Valued Catalan Combinatorics

本文研究两行集值标准杨表的计数，证明其为Catalan等数的新组合模型，还推广集值余主指标得到新的q-模拟。

利用Fuss-Catalan数求解k-广义斐波那契数

Solutions for $k$-generalized Fibonacci numbers using Fuss-Catalan numbers

该研究提出k-广义斐波那契数Fₖ(n)的新表达式，利用Fuss-Catalan数给出其特征多项式根的显式形式，量化渐近近似精度，提供任意初始条件下的递推基，扩展前人结果并发现多变量Dickson多项式关联及伴随矩阵新恒等式。

Macdonald Polynomials of Type $C_n$ with One-Column Diagrams and Deformed Catalan Numbers

We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $C_n$ degeneration of the Koornwinder polynomials $P_{(1^r)}(x\,|\,a,-a,c,-c\,|\,q,t)$ with one column diagrams, to the type $C_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of three term recursion relations, which can be regarded as a $(a,c,t)$-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type $(C_n,C_n)$ Macdonald polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,b;q,t)= P_{(1^r)}\b…

Volumes of generalized Chan-Robbins-Yuen polytopes

The normalized volume of the Chan-Robbins-Yuen polytope ($CRY_n$) is the product of consecutive Catalan numbers. The polytope $CRY_n$ has captivated combinatorial audiences for over a decade, as there is no combinatorial proof for its volume formula. In their quest to understand $CRY_n$ better, the third author and Morales introduced two natural generalizations of it and conjectured that their volumes are certain powers of $2$ multiplied by a product of consecutive Catalan numbers. Zeilberger proved one of these conjectures. In this paper we present proofs of both conjectures.

Total positivity of recursive matrices

Let $A=[a_{n,k}]_{n,k\ge 0}$ be an infinite lower triangular matrix defined by the recurrence $$a_{0,0}=1,\quad a_{n+1,k}=r_{k}a_{n,k-1}+s_{k}a_{n,k}+t_{k+1}a_{n,k+1},$$ where $a_{n,k}=0$ unless $n\ge k\ge 0$ and $r_k,s_k,t_k$ are all nonnegative. Many well-known combinatorial triangles are such matrices, including the Pascal triangle, the Stirling triangle (of the second kind), the Bell triangle, the Catalan triangles of Aigner and Shapiro. We present some sufficient conditions such that the recursive matrix $A$ is totally positive. As applications we give the total positivity of the above me…

Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials: Polynomiality, Unimodality and Log-concavity

We will look at the Catalan numbers from the {\it Rigged Configurations} point of view originated \cite{Kir} from an combinatorial analysis of the Bethe Ansatz Equations associated with the higher spin anisotropic Heisenberg models . Our strategy is to take a combinatorial interpretation of Catalan numbers $C_n$ as the number of standard Young tableaux of rectangular shape $(n^2)$, or equivalently, as the Kostka number $K_{(n^2),1^{2n}}$, as the starting point of research. We observe that the rectangular (or multidimensional) Catalan numbers $ C(m,n)$ introduced and studied by P. MacMahon \cit…

From Entanglement Witness to Generalized Catalan Numbers

The problem of entanglement detection for arbitrary spin systems is analyzed. We demonstrate how a single measurement of the squared total spin can probabilistically discern separable from entangled many-particle states. For achieving this goal, we construct a tripartite analogy between the degeneracy of entanglement witness eigenstates, tensor products of $SO(3)$ representations and classical lattice walks with special constraints. Within this framework, degeneracies are naturally given by generalized Catalan numbers and determine the fraction of states that are decidedly entangled and also k…

Combinatorial interpretations of Lucas analogues of binomial coefficients and Catalan numbers

The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the q-integers [n]_q. Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue by replacing each factor of n in the expression with {n}. It is then natural to ask if the resulting rational function is actually a polynomial in s and t with nonnegative integer coefficients and, if so, what it counts. The…

与杨图相关的随机矩阵

Random matrices associated to Young diagrams

该研究分析杨图形状分块随机矩阵的奇异值分布，证明其平方经验分布收敛于矩为推广卡特兰数的极限分布，给出超几何表示，并在特例中复现经典测度，还得到矩的复变量因式分解推广。

Set-Valued Young Tableaux and Product-Coproduct Prographs

Standard set-valued Young tableaux are a generalization of standard Young tableaux where cells can contain unordered sets of integers, with the added condition that every integer at position $(i,j)$ must be smaller that every integer at both $(i+1,j)$ and $(i,j+1)$. In this paper, we explore properties of standard set-valued Young tableaux with three rows and a fixed number of integers in every cell of each row (referred to as set-valued tableaux with row-constant density). Our primary focus is on standard set-valued Young tableaux with $1$ integer in each first-row cell, $k-1$ integers in eac…

The polytope of Tesler matrices

We introduce the Tesler polytope Tes_n(a_1,a_2,...,a_n), whose integer points are the Tesler matrices of size n with nonnegative integer hook sums a_1,a_2,...,a_n. We show that Tes_n(a) is a flow polytope and therefore the number of Tesler matrices is counted by the type A_n Kostant partition function evaluated at (a_1,a_2,...,a_n,-a_1-...-a_n). We describe the faces of this polytope in terms of "Tesler tableaux" and characterize when the polytope is simple. We prove that the h-vector of Tes_n(a) when all a_i>0 is given by the Mahonian numbers and calculate the volume of Tes_n(1,1,...,1) to be…

精确计算Catalan数C(2050572903)

Exact Computation of the Catalan Number $C(2,050,572,903)$

本文提出了一种两阶段算法，用于精确计算大规模Catalan数C(2050572903)，通过素数指数域运算和高效数据结构实现高精度计算。

Some determinants of path generating functions

We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have numerous corollaries. In particular, they cover numerous determinant evaluations of combinatorial numbers - most notably of Catalan, ballot, and of Motzkin numbers - that appeared previously in the literature.

Hessenberg-Toeplitz Matrix Determinants with Schroder and Fine Number Entries

In this paper, we find determinant formulas of several Hessenberg-Toeplitz matrices whose nonzero entries are derived from the small and large Schroder and Fine number sequences. Algebraic proofs of these results can be given which make use of Trudi's formula and the generating function of the associated sequence of determinants. We also provide direct arguments of our results that utilize various counting techniques, among them sign-changing involutions, on combinatorial structures related to classes of lattice paths enumerated by the Schroder and Fine numbers. As a consequence of our results…

Representations of the Temperley-Lieb Algebra via a New Inner Product on Half-Diagrams

We describe an inner product on the diagrams on which the Temperley-Lieb algebra can be represented. We exhibit several constructions which are in natural combinatorial bijection with these diagrams, which are generalizations of various constructions counted by the Catalan numbers. We use a method similar to the existing ones for orthogonalizing the Temperley-Lieb algebra to construct an orthogonal basis for the vector space over these diagrams.