Symmetric Dyck paths and q-Narayana numbers
Johann Cigler
本文证明当q=-1时，q-Narayana数按谷的数量计数对称Dyck路径。
我们证明，当q=-1时，q-Narayana数按照路径谷的数量来计数对称Dyck路径。
We show that the q-Narayana numbers for q=-1 count symmetric Dyck paths according to the number of their valleys.
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We show that an identity of Kreweras for the Narayana numbers counts Dyck paths with a given number of peaks by number of peak plateaus, where a peak plateau is a run of consecutive peaks that is immediately preceded by an upstep and followed by a downstep.
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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。
We give a simple proof that the the number of Dyck paths of semilength $n$ with $i$ returns to ground level and $j$ peaks is the generalized Narayana number $\frac{i}{n} \binom{n}{j} \binom{n - i - 1}{j - i}$.
It is known that both the number of Dyck paths with $2n$ steps and $k$ peaks, and the number of Dyck paths with $2n$ steps and $k$ steps at odd height follow the Narayana distribution. In this paper we present a bijection which explicitly illustrates this equinumeracy. Moreover, we extend this bijection to bilateral Dyck paths. The restriction to Dyck paths preserves the number of contacts.
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A combinatorial proof of a symmetry for a refinement of the Narayana numbers
本文利用循环引理给出了加细Narayana数关于Dyck路径因子的对称性的组合证明，推广了该对称性并研究了相关多项式的实根性、γ-正性等性质。
Dyck paths on colored lattices
研究彩色格上戴克路径，推广弗里德的计数结论，证明对于列模\(c\geq2\)的着色，给出半长度为\(n\)且\(c\)个余数类权重相等的戴克路径数量为拉尼数\(\Raney_{c + 1,r}(m)\)（\(n = cm + r - 1\)）。
In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schröder paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order $n$ is one half of the number of super Dyck paths of order $n$. He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order $n$ with $k$ peaks is the Narayana number. By double counting super Schröder paths, we also get an identity involving produc…
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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 …
Fibonacci Polyominoes: Refined Enumeration and Dyck-Path Bijections
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本文定义盒路径及更一般的 $k$-盒路径，建立其与多元 $(k+2)$ 叉树、$k_t$-Dyck 路径和阈值序列子族的双射，并给出按返回与长上升的精细化枚举。
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本文研究了有理Dyck路径上的run和ratio-run统计量，证明了其与return统计量的对称联合分布，并推广了Li和Lin的结果。
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本文研究了一类新的回文下降多项式，通过Dyck路径和排列生成因果排列，并探讨了其多项式性质及组合解释。
The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian-Wachs $q$-analogue have important connections to Hessenberg varieties, diagonal harmonics, and LLT polynomials. In the case of, so-called, abelian Dyck paths they are also curiously related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet (2013). For the $q$-analogue, these results have been generalized by Abreu-Nigro (2020) and Guay-Paquet (private communication), using $q$-hit numbers, which are a variant of the ones introduced by Garsia and Remmel. Among our main re…
A new generalization of the Narayana numbers inspired by linear operators on associative $d$-ary algebras
本文提出了一种受关联$d$-元代数启发的Narayana数泛化，并给出了其九种组合解释及与经典组合对象的双射关系。
We define a map $ν$ between the symmetric group $S_n$ and the set of pairs of Dyck paths of semilength $n$. We show that the map $ν$ is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map.
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The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian-Wachs $q$-analogue have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are also curiously related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet (2013). For the $q$-analogue, these results have been generalized by Abreu-Nigro (2020) and Guay-Paquet (private communication), using $q$-hit numbers. Among our main results is a new proof of Guay-Paquet's elegant identity expressing the $q$-CSFs i…
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In this paper, we study symmetric lattice paths. Let $d_{n}$, $m_{n}$, and $s_{n}$ denote the number of symmetric Dyck paths, symmetric Motzkin paths, and symmetric Schröder paths of length $2n$, respectively. By using Riordan group methods we obtain six identities relating $d_{n}$, $m_{n}$, and $s_{n}$ and also give two of them combinatorial proofs. Finally, we investigate some relations satisfied by the generic element of some special Riordan arrays and get the average mid-height and the average number of points on the x-axis of symmetric Dyck paths of length $2n.$
Major Index on Catalan Combinatorics
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Heaps of pieces for lattice paths
本文引入格路径的I型与II型片段堆，建立对偶、双射及高维对应，并给出生成函数的行列式与显式公式。
Each positive rational number x>0 can be written uniquely as x=a/(b-a) for coprime positive integers 0<a<b. We will identify x with the pair (a,b). In this paper we define for each positive rational x>0 a simplicial complex \Ass(x)=\Ass(a,b) called the {\sf rational associahedron}. It is a pure simplicial complex of dimension a-2, and its maximal faces are counted by the {\sf rational Catalan number} \Cat(x)=\Cat(a,b):=\frac{(a+b-1)!}{a!\,b!}. The cases (a,b)=(n,n+1) and (a,b)=(n,kn+1) recover the classical associahedron and its "Fuss-Catalan" generalization studied by Athanasiadis-Tzanaki and…
Generalized Dyck paths (or discrete excursions) are one-dimensional paths that take their steps in a given finite set S, start and end at height 0, and remain at a non-negative height. Bousquet-Mélou showed that the generating function E_k of excursions of height at most k is of the form F_k/F_{k+1}, where the F_k are polynomials satisfying a linear recurrence relation. We give a combinatorial interpretation of the polynomials F_k and of their recurrence relation using a transfer matrix method. We then extend our method to enumerate discrete meanders (or paths that start at 0 and remain at a n…
Narayana numbers appear in many places in combinatorics and probability, and it is known that they are asymptotically normal. Using Stein's method of exchangeable pairs, we provide an error of approximation in total variation to a symmetric binomial distribution of order~$n^{-1}$, which also implies a Kolmogorov bound of order~$n^{-1/2}$ for the normal approximation. Our exchangeable pair is based on a birth-death chain and has remarkable properties, which allow us to perform some otherwise tricky moment computations. Although our main interest is in Narayana numbers, we show that our main abs…
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Symmetry of the refined $q,t$-Catalan polynomials for $\vec{k}$-Dyck paths
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The aim of this work is to study the quotient ring R_n of the ring Q[x_1,...,x_n] over the ideal J_n generated by non-constant homogeneous quasi-symmetric functions. We prove here that the dimension of R_n is given by C_n, the n-th Catalan number. This is also the dimension of the space SH_n of super-covariant polynomials, that is defined as the orthogonal complement of J_n with respect to a given scalar product. We construct a basis for R_n whose elements are naturally indexed by Dyck paths. This allows us to understand the Hilbert series of SH_n in terms of number of Dyck paths with a given …
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Simplicial Complexes of Antichains in Root Posets and Related Combinatorics of Dyck Paths
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Simultaneous core partitions have been widely studied since Anderson's work on the enumeration of $(s,t)$-core partitions. Amdeberhan and Leven showed that the number of $(s,s+1, \ldots, s+k)$-core partitions is equal to the number of $(s, k)$-Dyck paths. In this paper, we prove that self-conjugate $(s,s+1, \ldots, s+k)$-core partitions are equinumerous with symmetric $(s, k)$-Dyck paths, confirming a conjecture posed by Cho, Huh and Sohn.
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The degree of symmetry of a combinatorial object, such as a lattice path, is a measure of how symmetric the object is. It typically ranges from zero, if the object is completely asymmetric, to its size, if it is completely symmetric. We study the behavior of this statistic on Dyck paths and grand Dyck paths, with symmetry described by reflection along a vertical line through their midpoint; partitions, with symmetry given by conjugation; and certain compositions interpreted as bargraphs. We find expressions for the generating functions for these objects with respect to their degree of symmetry…
The nonsymmetric shuffle theorem
本文引入非对称 $\boldsymbol{\nabla}$ 算子与非对称组合 Hall–Littlewood 多项式，结合双 Dyck 路径代数理论，证明组合 shuffle 定理的非对称版本，并给出 atom 正性猜想。
We introduce natural binary set-theoretical products on the set of all $m$-Dyck paths, which led us to define a non-symmetric algebraic operad $\Dy^m$, described on the vector space spanned by $m$-Dyck paths. Our construction is closely related to the $m$-Tamari lattice, so the products defining $\Dy^m$ are given by intervals in this lattice. For $m=1$, we recover the notion of dendriform algebra introduced by J.-L. Loday in \cite{Lod}, and there exists a natural operad morphism from the operad ${\mbox {\it Ass}}$ of associative algebras into the operad $\Dy^m$, consequently $\Dy ^m$ is a Hopf…
We study the compatibility of the action of the DAHA of type GL with two inverse systems of polynomial rings obtained from the standard Laurent polynomial representations. In both cases, the crucial analysis is that of the compatibility of the action of the Cherednik operators. Each case leads to a representation of a limit structure (the +/- stable limit DAHA) on a space of almost symmetric polynomials in infinitely many variables (the standard representation). As an application, we show that the defining representation of the double Dyck path algebra arises from the standard representation o…
Peaks and peak-nestings on unit interval graphs
本文在单位区间图上引入峰嵌套统计量，通过Dyck路径与双色Motzkin路径的双射细化Touchard恒等式，研究各类子图的多项式性质，并联系格路径拟阵。
Fuss--Catalan algebras on generalized Dyck paths via non-crossing partitions
本研究通过非交叉划分研究广义Dyck路径上的Fuss–Catalan代数，建立了相关双射与代数表示的相容性，引入带边界的推广形式，并在r=2时得到反射方程的新解。
Enumerating runs, valleys, and peaks in Catalan words
该研究针对 Catalan 词中的多类游程、谷与峰统计量，给出对应生成函数、计数公式及渐近结果，并通过与受限 Dyck 路径和有序树的双射迁移统计量。
The \emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Haglund and Haiman discovered combinatorial formulas for $C_n(q,t)$ as weighted sums of Dyck paths (or equivalently, integer partitions contained in a staircase shape). This paper undertakes a combinatorial investigation of the joint symmetry property $C_n(q,t)=C_n(t,q)$. We conjecture some structural decompositions of Dyck objects into "mutually op…
Area and water-capacity statistics for upper hulls of Dyck paths
研究Dyck路径在面积和蓄水容量双重约束下的计数，通过上壳分解得到耦合的面积-容量替换，给出精确的四变量高度展开，并证明长度半径由分母分支最小值决定。
It is well-known that the length generating function E(t) of Dyck paths (excursions with steps +1 and -1) satisfies 1-E+t^2E^2=0. The generating function E^(k)(t) of Dyck paths of height at most k is E^(k)=F_k/F_{k+1}, where the F_k are polynomials in t given by F_0=F_1=1 and F_{k+1}= F_k-t^2F_{k-1}. This means that the generating function of these polynomials is \sum_{k\ge 0} F_k z^k= 1/(1-z+t^2z^2). We note that the denominator of this fraction is the minimal polynomial of the algebraic series E(t). This pattern persists for walks with more general steps. For any finite set of steps S, the g…
Exchange identities and symmetric slices of the valley Delta conjecture
本文证明了山谷Delta猜想中生成函数在t^0和t^1项的对称性，通过相邻交换恒等式并引入q-变形双变量代数中的转移算子演算，将交换简化为两个标量对称级数恒等式。
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