q-Catalan numbers and q-Narayana polynomials

$q$-超级Catalan数：组合恒等式、生成函数与Narayana细化

$q$-Super Catalan Numbers: Combinatorial identities, Generating Functions, and Narayana Refinements

本文推导$q$-超级Catalan数的组合恒等式，引入Narayana型细化并证明其$\gamma$-正性及$q$-模拟的$q$-$\gamma$-正性，推广已知恒等式。

Some elementary observations on Narayana polynomials and related topics II: q-Narayana polynomials

We show that q-Catalan numbers, q- central binomial coefficients and q- Narayana polynomials are moments of q-analogues of Fibonacci and Lucas polynomials and related polynomials.

Boolean-Narayana 数

Boolean-Narayana numbers

引入 Boolean-Narayana 数作为 Boolean-Catalan 数的细化，给出显式公式，证明序列的单峰性、对数凹性和实根性，并推导生成多项式的三项递推关系。

与Narayana多项式及q=-1时的q-Narayana多项式相关的一些序列和数三角

Some sequences and number triangles which are related to Narayana polynomials and to q-Narayana polynomials for q=-1

本研究围绕与Narayana多项式及q=-1时q-Narayana多项式相关的数序列、数三角展开，探究有界Dyck路径的对应权重，获得相关结果与猜想。

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。

Narayana数的对称性与根偏序的行空化

Symmetry of Narayana numbers and rowvacuation of root posets

本文证明对于经典类型的根偏序，行空化算子实现了Panyushev提出的对合，从而展示了W-Narayana数的对称性。

Catalan 数卷积幂的 Hankel 行列式再探

Hankel determinants of convolution powers of Catalan numbers revisited

本文利用推广的 Andrews-Wimp 定理，简洁证明了 Catalan 数卷积幂平移的 Hankel 行列式，并推广至 Narayana 多项式。

Immanant Positivity for Catalan-Stieltjes Matrices

In this paper we give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices. As applications, we provide a unified way to produce inequalities for many combinatorial polynomials, such as the Eulerian polynomials, Schröder polynomials and Narayana polynomials.

Kirillov's unimodality conjecture for the rectangular Narayana polynomials

In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillov's unimodality conjecture with the help of Newton's inequality. By using an equidistribution property between descent numbers and ascent numbers on ballot paths due to Sulanke and a bijection between lattice words and standard Young tableaux, we show that the rectangular Narayana polynomial is equal to the descent generating function on standard Young tableaux of…

关于加权和有界多维卡特兰数

On Weighted and Bounded Multidimensional Catalan Numbers

本文定义了多维卡特兰数的加权版本，推导了矩阵递推和周期性条件，并引入两个新三角数组分别枚举精确高度的平衡选票路径和推广的 Narayana 数。

Schur Positivity and the $q$-Log-convexity of the Narayana Polynomials

Using Schur positivity and the principal specialization of Schur functions, we provide a proof of a recent conjecture of Liu and Wang on the $q$-log-convexity of the Narayana polynomials, and a proof of the second conjecture that the Narayana transformation preserves the log-convexity. Based on a formula of Bränd$\mathrm{\acute{e}}$n which expresses the $q$-Narayana numbers as the specializations of Schur functions, we derive several symmetric function identities using the Littlewood-Richardson rule for the product of Schur functions, and obtain the strong $q$-log-convexity of the Narayana pol…

Laurent biorthogonal polynomials, q-Narayana polynomials and domino tilings of the Aztec diamonds

A Töplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schröder paths. As an application, a new proof is given to the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp concerning domino tilings of the Aztec diamonds. The proof is based on the correspondence with non-intersecting Schröder paths developed by Eu and Fu.

Positivity of Narayana polynomials and Eulerian polynomials

Gamma-positivity appears frequently in finite geometries, combinatorics and number theory. Motivated by the recent work of Sagan and Tirrell (Adv. Math., 374 (2020), 107387), we study the relationships between gamma-positivity and alternating gamma-positivity. As applications, we derive several alternatingly gamma-positive polynomials related to Narayana polynomials and Eulerian polynomials. In particular, we show the alternating gamma-positivity and Hurwitz stability of a combination of the modified Narayana polynomials of types A and B. By using colored $2\times n$ Young diagrams, we present…

Narayana polynomials and some generalizations

In this note, by counting some colored plane trees we obtain several binomial identities. These identities can be viewed as specific evaluations of certain generalizations of the Narayana polynomials. As consequences, it provides combinatorial proofs for a bijective problem in Stanley's collection "Bijective Proof Problems", a new formula for the Narayana polynomials as well as a new expression for the Harer-Zagier formula enumerating unicellular maps, in a unified way. Furthermore, we identify a class of plane trees, whose enumeration is closely connected to the Schröder numbers. Many other b…

Catalan numbers, Hankel determinants and Fibonacci polynomials

This (partly expository) paper originated from the study of Hankel determinants of convolution powers of Catalan numbers and of Narayana polynomials. This led to some Hankel determinants of signed Catalan numbers whose values are multiples of Fibonacci numbers and to some Hankel determinants of signed central binomial coefficients whose values are multiples of Lucas numbers. Most proofs are computational but we also include a combinatorial one due to Christian Krattenthaler. Finally we formulate some conjectures.

The Real-rootedness of Generalized Narayana Polynomials

In this paper, we prove the real-rootedness of two classes of generalized Narayana polynomials: one arising as the $h$-polynomials of the generalized associahedron associated to the finite Weyl groups, the other arising in the study of the infinite log-concavity of the Boros-Moll polynomials. For the former, Brändén has already proved that these $h$-polynomials have only real zeros. We establish certain recurrence relations for the two classes of Narayana polynomials, from which we derive the real-rootedness. To prove the real-rootedness, we use a sufficient condition, due to Liu and Wang, to …

On the positive zeros of generalized Narayana polynomials related to the Boros-Moll polynomials

The generalized Narayana polynomials $N_{n,m}(x)$ arose from the study of infinite log-concavity of the Boros-Moll polynomials. The real-rootedness of $N_{n,m}(x)$ had been proved by Chen, Yang and Zhang. They also showed that when $n\geq m+2$, each of the generalized Narayana polynomials has one and only one positive zero and $m$ negative zeros, where the negative zeros of $N_{n,m}(x)$ and $N_{n+1,m+1}(x)$ have interlacing relations. In this paper, we study the properties of the positive zeros of $N_{n,m}(x)$ for $n\geq m+2$. We first obtain a new recurrence relation for the generalized Naray…

A planar network proof for Hankel total positivity of type $B$ Narayana polynomials

The Hankel matrix of type B Narayana polynomials was proved to be totally positive by Wang and Zhu, and independently by Sokal. Pan and Zeng raised the problem of giving a planar network proof of this result. In this paper, we present such a proof by constructing a planar network allowing negative weights, applying the Lindström-Gessel-Viennot lemma and establishing an involution on the set of nonintersecting families of directed paths.

多重集欧拉-那拉亚纳多项式的零点与交错性

Zeros and interlacing for multiset Eulerian-Narayana polynomials

本文证明了多重集欧拉-那拉亚纳多项式的实根性猜想，通过森林分解和拉格朗日反演得到微分算子分解，进而证明零点非递减性及转移矩阵的完全非负性，并建立相邻列生成多项式零点的严格交错性。

一个涉及偏序集关联多面体的h-多项式与B型Narayana多项式的恒等式

An identity involving $h$-polynomials of poset associahedra and type B Narayana polynomials

本文针对偏序集关联多面体的h-多项式，通过将链状自治子偏序集替换为反链构造新偏序集，建立了其与B型Narayana多项式的恒等式，并由此推导出涉及Narayana多项式、Eulerian多项式和栈排序原像的多个恒等式。

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…

Narayana, Mandelbrot, and A New Kind of Companion Matrix

We demonstrate a new kind of companion matrix, for polynomials of the form $c(λ) = λa(λ)b(λ) + c_0$ where upper Hessenberg companions are known for the polynomials $a(λ)$ and $b(λ)$. This construction can generate companion matrices with smaller entries than the Fiedler or Frobenius forms. This generalizes Piers Lawrence's Mandelbrot companion matrix. We motivate the construction by use of Narayana-Mandelbrot polynomials, which are also new to this paper.

Narayana多项式卷积幂的Hankel行列式

Hankel determinants for convolution powers of Narayana polynomials

本文通过符号计算程序验证并推广了Narayana多项式卷积幂的Hankel行列式猜想，采用猜测与证明策略，利用Hankel连分数技术，最终给出简洁闭合表达式。

稳定的多元 Narayana 多项式与标号平面树

Stable multivariate Narayana polynomials and labeled plane trees

本文研究 A 型与 B 型 Narayana 多项式的多元稳定推广，借助标号平面树插入算法、非恰当边及上下文无关文法证明实稳定性，并给出二阶 Eulerian 多项式的新型多元稳定加细。

Some elementary observations on Narayana polynomials and related topics

We give an elementary account of generalized Fibonacci and Lucas polynomials whose moments are Narayana polynomials of type A and type B.

Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers

We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [arXiv:1208.0024]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.

Some identities for the Narayana polynomial and its derivatives

In this note, by employing a nice property of semicircular distributions, we derive some identities for the Narayana polynomial and its derivatives.

关于Pell-Narayana序列在某些群中的周期

On the period of Pell-Narayana sequence in some groups

本文研究模m的Pell-Narayana序列，定义2-生成群的Pell-Narayana轨道，考察其周期长度，并给出多面体群及若干群（如Q8及其扩张）的周期结果。

Factors of alternating sums of powers of $q$-Narayana numbers

The $q$-Narayana numbers $N_q(n,k)$ and $q$-Catalan numbers $C_n(q)$ are respectively defined by $$ N_q(n,k)=\frac{1-q}{1-q^n}{n\brack k}{n\brack k-1}\quad\text{and}\quad C_n(q)=\frac{1-q}{1-q^{n+1}}{2n\brack n}, $$ where ${n\brack k}=\prod_{i=1}^{k}\frac{1-q^{n-i+1}}{1-q^i}$. We prove that, for any positive integers $n$ and $r$, there holds \begin{align*} \sum_{k=-n}^{n}(-1)^{k}q^{jk^2+{k\choose 2}}N_q(2n+1,n+k+1)^r \equiv 0 \pmod{C_n(q)}, \end{align*} where $0\leqslant j\leqslant 2r-1$. We also propose several related conjectures.

关于q=-1时的q-纳拉亚多项式的若干评论

Some remarks about q-Narayana polynomials for q=-1

本文研究了q=-1时的q-纳拉亚多项式性质，并与q=1时的性质进行比较。

关于半对称高度和加权卡特兰数的多维推广

On Semisymmetric Height and a Multidimensional Generalization of Weighted Catalan Numbers

本文引入了多维半对称加权卡特兰数及其变体，通过半对称高度定义，并证明了其模数的周期性和推导了相关公式，还推导了多维设置下的新类比。

由避免模式的斯托伊梅诺夫匹配和其他菲什伯恩对象产生的卡特兰结构

Catalan structures arising from pattern-avoiding Stoimenow matchings and other Fishburn objects

解决贝万等人2025年提出的关于识别由卡特兰数计数的斯托伊梅诺夫匹配子集的问题，给出五种基于避免模式匹配的解决方案，还考虑相关模式族及统计量分布等结果。

A Rational Catalan Formula for $(m,3)$-Hikita Polynomials

Building upon a recent formula for $(3,m)$-Catalan polynomials, we describe a formula for $(3,m)$-Hikita polynomials in terms related to Catalan polynomials. This formula shows a surprising relation among coefficients of Hikita polynomials and implies deeper recursive relations and proves the $q,t$-symmetry of $(3,m)$-Hikita polynomials.

关于多项式方程的超加泰罗尼亚级数解与测地线的习题

Exercises for A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode

本文为“多项式方程的超加泰罗尼亚级数解与测地线”一文配套编写习题并给出解答，以帮助读者理解和练习相关内容。

Tuenter polynomials and a Catalan triangle

We consider Tuenter polynomials as linear combinations of descending factorials and show that coefficients of these linear combinations are expressed via a Catalan triangle of numbers. We also describe a triangle of coefficients in terms of some polynomials.

A probabilistic interpretation of a sequence related to Narayana polynomials

A sequence of coefficients appearing in a recurrence for the Narayana polynomials is generalized. The coefficients are given a probabilistic interpretation in terms of beta distributed random variables. The recurrence established by M. Lasalle is then obtained from a classical convolution identity. Some arithmetical properties of the generalized coefficients are also established.

Bi-banded Paths, a Bijection and the Narayana Numbers

We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating bi-banded Dyck paths, which class of paths has arisen naturally in previous literature in a solution of the stationary state of the `TASEP' stochastic process.

$s$-Catalan numbers and Littlewood-Richardson polynomials

In this note, we study two generalizations of the Catalan numbers, namely the $s$-Catalan numbers and the spin $s$-Catalan numbers. These numbers first appeared in relation to quantum physics problems about spin multiplicities. We give a combinatorial description for these numbers in terms of Littlewood-Richardson coefficients, and explain some of the properties they exhibit in terms of Littlewood-Richardson polynomials.

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.

On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths

We use the inversion of coefficient arrays to define dual polynomials to the Fibonacci and Catalan-Fibonacci polynomials, and we explore the properties of these new polynomials sequences. Many of the arrays involved are Riordan arrays. Direct links to the counting of Motzkin paths by different statistics emerge.

On super Catalan polynomials

We present a $q$-analog of the super Catalan number $(2m)!(2n)!/2m!n!(m+n)!$, which also generalizes the $q$-Catalan numbers $c_n(λ)$, due to Fürlinger and Hofbauer, for $λ=0$ and $λ=1$. We give a combinatorial interpretation for this analog when $m=2$.

Narayana多项式与Motzkin多项式的新加细

New refinements of Narayana polynomials and Motzkin polynomials

该研究引入新的加细多项式进一步加细Narayana多项式与Motzkin多项式，推导二者生成函数，得到Coker公式的新加细及相关推论，发展了统一语法方法研究平面树统计量相关多项式。

Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial

We classify recurrent configurations of the sandpile model on the complete bipartite graph K_{m,n} in which one designated vertex is a sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a m*n rectangle. Several special types of recurrent configurations and their properties via this bijection are examined. For example, recurrent configurations whose sum of heights is minimal are shown to correspond to polyominoes of least area. Two other classes of recurrent configurations are shown to be related to bicomposition matrice…

A Recursion for the FiboNarayana and the Generalized Narayana Numbers

The Lucas polynomials, $\{n\}$, are polynomials in $s$ and $t$ given by $\{ n \} = s \{ n-1 \} + t \{ n-2 \}$ for $n \geq 2$ with $ \{ 0 \} = 0$ and $\{ 1 \} = 1$. The lucanomial coefficients, an analogue of the binomial coefficients, are given by \[ \Bigl\{ \begin{array}{c} n\\k \end{array} \Bigr \} = \frac{ \{n\}! }{ \{k\}! \{n-k\}!}. \] When $s = t = 1$ then $\{ n \} = F_n$ and the lucanomial coefficient becomes the fibonomial coefficient \[ \binom{n}{k}_F = \frac{F_n!}{F_k! F_{n-k}!}. \] The well-known Narayana numbers, $N_{n,k}$ satisfy the equation \[ N_{n,k} = \frac{1}{n} \binom{n}{k} \…

解析与组合方法用于加权Catalan求和

Analytic and combinatorial approaches to a weighted Catalan sum

本文研究加权Catalan数的卷积，通过解析、组合和概率方法推导出闭式表达式，并探讨其与随机游走返回概率的关系。

$d$-正交多项式、Fuss-Catalan矩阵与格路

$d$-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。

Multivariate Fuss-Narayana polynomials and their application to random matrices

It has been shown recently that the limit moments of $W(n)=B(n)B^{*}(n)$, where B(n) is a product of $p$ independent rectangular random matrices, are certain homogenous polynomials in the asymptotic dimensions of these matrices. Using the combinatorics of noncrossing partitions, we explicitly determine these polynomials and show that they are closely related to polynomials which can be viewed as multivariate Fuss-Narayana polynomials. Using this result, we compute the moments of the n-fold free multiplicative convolution of Marchenko-Pastur distributions with arbitrary shape parameters.

Hughes平面的PTR多项式及涉及Catalan数的新一类置换多项式

A PTR polynomial for the Hughes planes and a new class of permutation polynomials involving Catalan numbers

本文确定了Hughes平面的PTR多项式，并发现Catalan数与广义Catalan数在系数中出现，从而得到新的无限类置换多项式。

Narayana numbers and Schur-Szego 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} counting Dyck paths of length n and with exactly k peaks. Strangely enough Narayana polynomials also occur as limits as n->oo of the sequences of eigenpolynomials of the Schur-Szego composition map sending (n-1)-tuples of polynomials of the form (x+1)^{n-1}(x+a) to their Schur-Szego product, see below. As a corollary we obtain that every N_n(x) has all roots real and non-positive. Additionally, we present an explicit formula for the density and …

(k,m)-Catalan Numbers and Hook Length Polynomials for Plane Trees

Motivated by a formula of A. Postnikov relating binary trees, we define the hook length polynomials for m-ary trees and plane forests, and show that these polynomials have a simple binomial expression. An integer value of this expression is C_{k,m}(n)=\frac{1}{mn+1}{(mn+1)k \choose n}, which we call the (k,m)-Catalan number. For proving the hook length formulas, we also introduce a combinatorial family, (k,m)-ary trees, which are counted by the (k,m)-Catalan numbers.

Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials

We show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and indicate that Somos $4$ sequences often arise. We exhibit relations between recurrences, Riordan arrays, elliptic curves and Somos $4$ sequences. We furthermore indicate how one can associate a family of orthogonal polynomials to a point on an elliptic curve, whose moments are related to recurrence solutions.

Narayana变换

The Narayana transformation

本文定义Narayana多项式并证明其诱导的基变换保持实根性，通过多项式矩形加法卷积证明，并应用于下三角矩阵乘积的行生成函数，恢复经典三角矩阵的实根性。

New identities for binary Krawtchouk polynomials, binomial coefficients and Catalan numbers

We obtain new combinatorial identities for integral values of binary Krawtchouk polynomials $K^{2m}_p(x)$, $0\le p\le 2m$, by computing the characters of the $p$-exterior representations on certain elements of order 2 of $\mathrm{SO}(2m)$. From this identities, we deduce several new relations for binomial coefficients and Catalan numbers.

A proof of the $4$-variable Catalan polynomial of the Delta conjecture

In The Delta Conjecture (arxiv:1509.07058), Haglund, Remmel and Wilson introduced a four variable $q,t,z,w$ Catalan polynomial, so named because the specialization of this polynomial at the values $(q,t,z,w) = (1,1,0,0)$ is equal to the Catalan number $\frac{1}{n+1}\binom{2n}{n}$. We prove the compositional version of this conjecture (which implies the non-compositional version) that states that the coefficient of $s_{r,1^{n-r}}$ in the expression $Δ_{h_\ell} \nabla C_α$ is equal to a weighted sum over decorated Dyck paths.