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 generating function for the Catalan numbers.
We show bijectively that Dyck paths with all peaks at odd height are counted by the Motzkin numbers and Dyck paths with all peaks at even height are counted by the Riordan numbers.
We show connection between Dyck paths with peaks of bounded height and random walks. The correspondence between a certain class of random walks and such Dyck paths allows us to develop a probabilistic perspective on Chebyshev polynomials.
We provide generating functions for the popularity and the distribution of patterns of length at most three over the set of Dyck paths having a first return decomposition constrained by height.
Raised $k$-Dyck paths are a generalization of $k$-Dyck paths that may both begin and end at a nonzero height. In this paper, we develop closed formulas for the number of raised $k$-Dyck paths from $(0,α)$ to $(\ell,β)$ for all height pairs $α,β\geq 0$, all lengths $\ell \geq 0$, and all $k \geq 2$. We then enumerate raised $k$-Dyck paths with a fixed number of returns to ground, a fixed minimum height, and a fixed maximum height, presenting generating functions (in terms of the generating functions $C_k(t)$ for the $k$-Catalan numbers) when closed formulas aren't tractable. Specializing our re…
A 3-dimensional Catalan word is a word on three letters so that the subword on any two letters is a Dyck path. For a given Dyck path $D$, a recently defined statistic counts the number of Catalan words with the property that any subword on two letters is exactly $D$. In this paper, we enumerate Dyck paths with this statistic equal to certain values, including all primes. The formulas obtained are in terms of Motzkin numbers and Motzkin ballot numbers.
The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from $(0,0)$ to $(n,n)$ which is below the diagonal line $y=x$. One way to generalize the definition of Dyck path is to change the end point of Dyck path, i.e. we define (generalized) Dyck path to be a lattice path from $(0,0)$ to $(m,n) \in \mathbb{N}^2$ which is below the diagonal line $y=\frac{n}{m}x$, and denote by $C(m,n)$ the number of Dyck paths from $(0,0)$ to $(m,n)$. In this paper, we give a formula to calculate $C(m,n)$ for arbitrary $m$ and $n$.
Dyck paths are one of the most important objects in enumerative combinatorics, and there are many papers devoted to counting selected families of Dyck paths. Here we present two approaches for the automatic counting of many such families, using both a "dumb" approach (driven by numeric dynamic programming) that often works in practice, and a "clever" approach, needed for larger problems, driven by "symbolic" dynamic programming. Both approaches are fully automated and implemented in Maple.
We show that the distribution of the number of peaks at height $i$ modulo $k$ in $k$-Dyck paths of a given length is independent of $i\in[0,k-1]$ and is the reversal of the distribution of the total number of peaks. Moreover, these statistics, together with the number of double descents, are jointly equidistributed with any of their permutations. We also generalize this result to generalized Motzkin paths and generalized ballot paths.
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…
For any integer $m\geq 2$ and $r \in \{1,\dots, m\}$, let $f_n^{m,r}$ denote the number of $n$-Dyck paths whose peak's heights are $im+r$ for some integer $i$. We find the generating function of $f_n^{m,r}$ satisfies a simple algebraic functional equation of degree $2$. The $r=m$ case is particularly nice and we give a combinatorial proof. By using the Sulanke and Xin's continued fraction method, we calculate the Hankel determinants for $f_n^{m,r}$. The special case $r=m$ of our result solves a conjecture proposed by Chien, Eu and Fu. We also enriched the class of eventually periodic Hankel de…
We present a new bijection between variants of $m$-Dyck paths (paths with steps in $\{+1,-m\}$ starting and ending at height $0$ and remaining at non-negative height), which generalizes a classical bijection between Dyck prefixes and pointed Łukasiewicz paths. As an application, we present a new random sampling procedure for $m$-Dyck paths with a linear time complexity and using a quasi-optimal number of random bits. This outperforms Devroye's algorithm, which uses $\mathcal O(n\log n)$ random bits.
For any integer $m\ge 2$ and a set $V\subset \{1,\dots,m\}$, let $(m,V)$ denote the union of congruence classes of the elements in $V$ modulo $m$. We study the Hankel determinants for the number of Dyck paths with peaks avoiding the heights in the set $(m,V)$. For any set $V$ of even elements of an even modulo $m$, we give an explicit description of the sequence of Hankel determinants in terms of subsequences of arithmetic progression of integers. There are numerous instances for varied $(m,V)$ with periodic sequences of Hankel determinants. We present a sufficient condition for the set $(m,V)…
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…
We generalize the classical work of de Bruijn, Knuth and Rice (giving the asymptotics of the average height of Dyck paths of length $n$) to the case of $p$--watermelons with a wall (i.e., to a certain family of $p$ nonintersecting Dyck paths; simple Dyck paths being the special case $p=1$.) We work out this asymptotics for the case $p=2$ only, since the computations involved are already quite complicated (but might be of some interest in their own right).
The set of Dyck paths of length $2n$ inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths: \emph{area} (the area under the path) and \emph{rank} (the rank in the lattice). While area for Dyck paths has been studied, pairing it with this rank function seems new, and we get an interesting $(q,t)$-refinement of the Catalan numbers. We present two decompositions of the corresponding generating function: one refines an identity of Carlitz and Riordan; the oth…
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.
Let $\C_n$ be the set of Dyck paths of length $n$. In this paper, by a new automorphism of ordered trees, we prove that the statistic `number of exterior pairs', introduced by A. Denise and R. Simion, on the set $\C_n$ is equidistributed with the statistic `number of up steps at height $h$ with $h\equiv 0$ (mod 3)'. Moreover, for $m\ge 3$, we prove that the two statistics `number of up steps at height $h$ with $h\equiv 0$ (mod $m$)' and `number of up steps at height $h$ with $h\equiv m-1$ (mod $m$)' on the set $\C_n$ are `almost equidistributed'. Both results are proved combinatorially.
We count the number of linear intervals in the Tamari and the Dyck lattices according to their height, using generating series and Lagrange inversion. Surprisingly, these numbers are the same in both lattices. We define a new family of posets on Dyck paths, which we call alt-Tamari posets. Each alt-Tamari poset depends on the choice of an increment function delta in {0,1}^n. We recover the Tamari and the Dyck lattices as extreme cases with delta = 1 and delta = 0, respectively. We prove that all the alt-Tamari posets have the same number of linear intervals of any given height.
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.
We consider the problem of counting the set of $\mathscr{D}_{a,b}$ of Dyck paths inscribed in a rectangle of size $a\times b$. They are a natural generalization of the classical Dyck words enumerated by the Catalan numbers. By using Ferrers diagrams associated to Dyck paths, we derive formulas for the enumeration of $\mathscr{D}_{a,b}$ with $a$ and $b$ non relatively prime, in terms of Catalan numbers.
We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the Dyck pattern poset. Given a Dyck path P, we determine a formula for the number of Dyck paths covered by P, as well as for the number of Dyck paths covering P. We then address some typical pattern-avoidance issues, enumerating some classes of pattern-avoiding Dyck paths. Finally, we offer a conjecture concerning the asymptotic beh…
In \cite{BaDeFePi96} the concept of nondecreasing Dyck paths was introduced. We continue this research by looking at it from the point of view of words, rational languages, planted plane trees, and continued fractions. We construct a bijection with planted plane trees of height $\le 4$ and compute various statistics on trees that are the equivalents of nondecreasing Dyck paths.
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.
In this article we investigate the lattices of Dyck paths of type $A$ and $B$ under dominance order, and explicitly describe their Heyting algebra structure. This means that each Dyck path of either type has a relative pseudocomplement with respect to some other Dyck path of the same type. While the proof that this lattice forms a Heyting algebra is quite straightforward, the explicit computation of the relative pseudocomplements using the lattice-theoretic definition is quite tedious. We give a combinatorial description of the Heyting algebra operations join, meet, and relative pseudocompleme…
In this paper we study a subfamily of a classic lattice path, the \emph{Dyck paths}, called \emph{restricted $d$-Dyck} paths, in short $d$-Dyck. A valley of a Dyck path $P$ is a local minimum of $P$; if the difference between the heights of two consecutive valleys (from left to right) is at least $d$, we say that $P$ is a restricted $d$-Dyck path. The \emph{area} of a Dyck path is the sum of the absolute values of $y$-components of all points in the path. We find the number of peaks and the area of all paths of a given length in the set of $d$-Dyck paths. We give a bivariate generating functio…
We enumerate interlaced pairs of parking functions whose underlying Dyck path has a bounded height. We obtain an explicit formula for this enumeration in the form of a quotient of analogs of Chebicheff polynomials having coefficients in the ring of symmetric functions.
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…
We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength $n$ that is shown to be in bijection with the set of all SYT with $n$ boxes. In addition, we present bijections from certain classes of Motzkin paths to SYT. As a natural framework for some of our bijections, we introduce a class of set partitions which in some sense is dual to the known class of…
School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Department of Mathematics, Institut Teknologi Bandung（万隆理工学院数学系）；Department of IT and Big Data Analytics, Calvin Institute of Technology（卡尔文理工学院信息技术与大数据分析系）
In 2003, Deutsch and Elizalde defined a family of bijective maps between the set of Dyck paths to itself which is induced by some particular permutations. In this paper, we extend the construction of the maps by allowing the permutation to be arbitrary. We characterise the permutations which generate the same map and find all permutations generating a bijection among Dyck paths. Consequently, we give a new combinatorial interpretation of the quantity $(2n-1)!!$ as well as some new statistics of Dyck paths which are equidistributed to some known height statistics via our generalised maps.
We introduce a new concept of permutation avoidance pattern called hatted pattern, which is a natural generalization of the barred pattern. We show the growth rate of the class of permutations avoiding a hatted pattern in comparison to barred pattern. We prove that Dyck paths with no peak at height $p$, Dyck paths with no $ud... du$ and Motzkin paths are counted by hatted pattern avoiding permutations in $\s_n(132)$ by showing explicit bijections. As a result, a new direct bijection between Motzkin paths and permutations in $\s_n(132)$ without two consecutive adjacent numbers is given. These p…
Given a coprime pair $(m,n)$ of positive integers, rational Catalan numbers $\frac{1}{m+n} \binom{m+n}{m,n}$ counts two combinatorial objects:rational $(m,n)$-Dyck paths are lattice paths in the $m\times n$ rectangle that never go below the diagonal; $(m,n)$-cores are partitions with no hook length equal to $m$ or $n$.Anderson established a bijection between $(m,n)$-Dyck paths and $(m,n)$-cores. We define a new transformation, called rank complement, on rational Dyck paths. We show that rank complement corresponds to conjugation of $(m,n)$-cores under Anderson's bijection. This leads to: i) a …
The notion of symmetric and asymmetric peaks in Dyck paths was introduced by Flórez and Rodr\'ıguez, who counted the total number of such peaks over all Dyck paths of a given length. In this paper we generalize their results by giving multivariate generating functions that keep track of the number of symmetric peaks and the number of asymmetric peaks, as well as the widths of these peaks. We recover a formula of Denise and Simion as a special case of our results. We also consider the analogous but more intricate notion of symmetric valleys. We find a continued fraction expression for the gener…
We consider the problem of counting subset of Dyck paths contained in a Ferrers diagram. This enumeration concerns to find the number of the elements in a branch of the Kréwéras tree. Using the Ferrers diagrams associated with Dyck paths, we have developed methods of comparison and decomposition diagrams to obtain enumerative formulas in terms of Catalan numbers. These methods have been developed in the form of algorithms and SAGE coded for verification. ----- Se considerara el problema de contar el subconjunto de caminos de Dyck contenido en un diagrama de Ferrers. Esta enumeración atañe al c…
Grand Dyck paths with air pockets (GDAP) are a generalization of Dyck paths with air pockets by allowing them to go below the $x$-axis. We present enumerative results on GDAP (or their prefixes) subject to various restrictions such as maximal/minimal height, ordinate of the last point and particular first return decomposition. In some special cases we give bijections with other known combinatorial classes.
We present a generating function and a closed counting formula in two variables that enumerate a family of classes of permutations that avoid or contain an increasing pattern of length three and have a prescribed number of occurrences of another pattern of length three. This gives a refinement of some previously studied statistics, most notably one by Noonan. The formula is also shown to enumerate a family of classes of Dyck paths and Standard Young Tableaux, and a bijection is given between the corresponding classes of these two families of objects. Finally, the results obtained are used to s…
Bargraphs are a special class of convex polyominoes. They can be identified with lattice paths with unit steps north, east, and south that start at the origin, end on the $x$-axis, and stay strictly above the $x$-axis everywhere except at the endpoints. Bargraphs, which are used to represent histograms and to model polymers in statistical physics, have been enumerated in the literature by semiperimeter and by several other statistics, using different methods such as the wasp-waist decomposition of Bousquet-Mélou and Rechnitzer, and a bijection with certain Motzkin paths. In this paper we descr…
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.
We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these identities arise from combinatorial interpretations of certain properties of the $β$-Hermite and $β$-Laguerre ensembles of random matrix theory. We conclude by presenting two other identities obtained in the same way, for which finding combinatorial proofs is an open problem.
We consider the system of equations $A_k(x)=p(x)A_{k-1}(x)(q(x)+\sum_{i=0}^k A_i(x))$ for $k\geq r+1$ where $A_i(x)$, $0\leq i \leq r$, are some given functions and show how to obtain a close form for $A(x)=\sum_{k\geq 0}A_k(x)$. We apply this general result to the enumeration of certain subsets of Dyck, Motzkin, skew Dyck, and skew Motzkin paths, defined recursively according to the first return decomposition with a monotonically non-increasing condition relative to the maximal ordinate reached by an occurrence of a given pattern $π$.
We relate the combinatorics of periodic generalized Dyck and Motzkin paths to the cluster coefficients of particles obeying generalized exclusion statistics, and obtain explicit expressions for the counting of paths with a fixed number of steps of each kind at each vertical coordinate. A class of generalized compositions of the integer path length emerges in the analysis.
We investigate nil-Temperley-Lieb algebras of type A. We give a general description of the structure of monomials formed by the generators. We also show that the dimensions of these algebras are the famous Catalan numbers by providing a bijection between the monomials and Dyck paths. We show that the distribution of these monomials by degree is the same as the distribution of Dyck paths by the sum of the heights of the peaks minus the number of peaks.
A circular Pascal array is a periodization of the familiar Pascal's triangle. Using simple operators defined on periodic sequences, we find a direct relationship between the ranges of the circular Pascal arrays and numbers of certain lattice paths within corridors, which are related to Dyck paths. This link provides new, short proofs of some nontrivial formulas found in the lattice-path literature.
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 $2n.$
It is known that the Hilbert space dimensionality for quasiparticles in an SU(2)_k Chern-Simons-Witten theory is given by the number of directed paths in certain Bratteli diagrams. We present an explicit formula for these numbers for arbitrary k. This is on the basis of a relation with Dyck paths and Chebyshev polynomials.
