Dyck Paths with Peak‐ and Valley‐Avoiding Sets

In this paper, we focus on Dyck paths with peaks and valleys, avoiding an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We also discuss a special case that requires all peak and valley heights to avoid congruence classes modulo k. We study the shift equivalence on sequences, which in turn induces an equivalence relation on avoiding sets.     

Dyck paths with a first return decomposition constrained by height

We study the enumeration of Dyck paths having a first return decomposition with special properties based on a height constraint. We exhibit new restricted sets of Dyck paths counted by the Motzkin numbers, and we give a constructive bijection between these objects and Motzkin paths. As a byproduct, we provide a generating function for the number of Motzkin paths of height $k$ with a flat (resp. with no flats) at the maximal height.     

Counting Dyck Paths by Area and Rank

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: area (the area under the path) and 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 other refines the notion of γ-nonnegativity, and is based on a decomposition of the lattice of noncrossing partitions due to Simion and Ullman. Further, Biane’s correspondence and a result of Stump allow us to conclude that the joint distribution of area and rank for Dyck paths equals the joint distribution of length and reflection length for the permutations lying below the n-cycle (12· · ·n) in the absolute order on the symmetric group.     

A combinatorial classification of 2-regular simple modules for Nakayama algebras

Enomoto showed for finite dimensional algebras that the classification of exact structures on the category of finitely generated projective modules can be reduced to the classification of 2-regular simple modules. In this article, we give a combinatorial classification of 2-regular simple modules for Nakayama algebras and we use this classification to answer several natural questions such as when there is a unique exact structure on the category of finitely generated projective modules for Nakayama algebras. We also classify 1-regular simple modules, quasi-hereditary Nakayama algebras and Nakayama algebras of global dimension at most two. It turns out that most classes are enumerated by well-known combinatorial sequences, such as Fibonacci, Riordan and Narayana numbers. We first obtain interpretations in terms of the Auslander-Reiten quiver of the algebra using homological algebra, and then apply suitable bijections to relate these to combinatorial statistics on Dyck paths.     

A Simple and Unusual Bijection for Dyck Paths and its Consequences

In this paper we introduce a new bijection from the set of Dyck paths to itself. This bijection has the property that it maps statistics that appeared recently in the study of patternavoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. We also present a generalization of the bijection, as well as several applications of it to enumeration problems of statistics in restricted permutations.AMS Subject Classification: 05A15, 05A05.     

Tiling bijections between paths and Brauer diagrams

There is a natural bijection between Dyck paths and basis diagrams of the Temperley–Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the two-dimensional integer lattice. We show that there is a natural bijection, extending the above tiling construction, between overhang paths and basis diagrams of the Brauer algebra.     

Two bijections for the area of Dyck paths

The sum of the areas of (2n+2)-length Dyck paths, or total area, is equal to the number of points with ordinate 1 in Grand-Dyck paths of length 2n+2, n⩾0. A bijective proof of this correspondence is shown by passing through an auxiliary class of marked paths. The sequence of numbers 1,6,29,130,562,… counts the total area of (2n+2)-length Dyck paths as well as the number of points having ordinate 0 and which satisfy an additional condition, on 2n-length paths made up of rise and fall steps. First, a bijection between these points and the triangles constituting the total area of (2n+2)-length Dyck paths is established, and then the correspondence between the above-mentioned points and the points with ordinate 1 on (2n+2)-length Grand-Dyck paths is shown.     

Some set partition statistics in non-crossing partitions and generating functions

In this paper we shall give the generating functions for the enumeration of non-crossing partitions according to some set partition statistics explicitly, which are based on whether a block is singleton or not and is inner or outer. Using weighted Motzkin paths, we find the continued fraction form of the generating functions. There are bijections between non-crossing partitions, Dyck paths and non-nesting partitions, hence we can find applications in the enumeration of Dyck paths and non-nesting partitions. We shall also study the integral representation of the enumerating polynomials for our statistics. As an application of integral representation, we shall give some remarks on the enumeration of inner singletons in non-crossing partitions, which is equivalent to one of udu's at high level in Dyck paths investigated in [Y. Sun, The statistic “number of udu's” in Dyck paths, Discrete Math. 284 (2004) 177-186].     

A new statistic on Dyck paths for counting 3-dimensional Catalan words

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.     

Double Catalan monoids

In this paper, we define and study what we call the double Catalan monoid. This monoid is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations. We show that the double Catalan monoid provides an algebraization of the (combinatorial) set of 4321-avoiding permutations and relate its combinatorics to various off-shoots of both the combinatorics of Catalan numbers and the combinatorics of permutations. In particular, we give an algebraic interpretation of the first derivative of the Kreweras involution on Dyck paths, of 4321-avoiding involutions and of recent results of Barnabei et al. on admissible pairs of Dyck paths. We compute a presentation and determine the minimal dimension of an effective representation for the double Catalan monoid. We also determine the minimal dimension of an effective representation for the 0-Hecke monoid.     

Restricted Dumont permutations, Dyck paths, and noncrossing partitions

We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding certain patterns of length 3 and 4 and give a natural bijection between 3142-avoiding Dumont permutations of the second kind and noncrossing partitions that uses cycle decomposition, as well as bijections between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths. Finally, we enumerate Dumont permutations of the first kind simultaneously avoiding certain pairs of 4-letter patterns and another pattern of arbitrary length.     

Knight's paths towards Catalan numbers

We provide enumerating results for partial knight's paths of a given size. We prove algebraically that zigzag knight's paths of a given size ending on the x-axis are enumerated by the generalized Catalan numbers, and we give a constructive bijection with peakless Motzkin paths of a given length. After enumerating partial knight's paths of a given length, we prove that zigzag knight's paths of a given length ending on the x-axis are counted by the Catalan numbers. Finally, we give a constructive bijection with Dyck paths of a given length.     

On the combinatorics of the Pfaff identity

Recently, there has been a revival of interest in the Pfaff identity on hypergeometric series because of the specialization of Simons and a generalization of Munarini. We present combinatorial settings and interpretations of the specialization and the generalization; one is based on free Dyck paths and free Schröder paths, and the other relies on a correspondence of Foata and Labelle between the Meixner endofunctions and bicolored permutations, and an extension of the technique developed by Labelle and Yeh for the Pfaff identity. Applying the involution on weighted Schröder paths, we derive a formula for the Narayana numbers as an alternating sum of the Catalan numbers.     

Multiresolution Equivalence and Path Connectedness

An equivalence relation between multiresolution analyses was first introduced in 1996; an analogous definition for generalized multiresolution analyses was given in 2010. This article describes the relationship between the two notions and shows that both types of equivalence classes are path connected in an operator-theoretic sense. The GMRA paths are restricted to canonical GMRAs, and it is shown that whenever two MRAs in L ²(?) are equivalent, the GMRA path construction between their corresponding canonical GMRAs yields the natural analog of the MRA path. Examples are provided of GMRA paths that are distinct from MRA paths.     

The butterfly decomposition of plane trees

We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition, which turns out to have many applications. From the butterfly decomposition we obtain a one-to-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the Catalan numbers and the central binomial coefficients. We also establish a one-to-one correspondence between leaf-colored doubly rooted plane trees and free Schröder paths. The classical Chung-Feller theorem as well as some generalizations and variations follow quickly from the butterfly decomposition. We next obtain two involutions on free Dyck paths and free Schröder paths, leading to parity results and combinatorial identities. We also use the butterfly decomposition to give a combinatorial treatment of Klazar's generating function for the number of chains in plane trees. Finally we study the total size of chains in plane trees with n edges and show that the average size of such chains tends asymptotically to (n+9)/6.     

Ordering sequences by permutation transducers

To extend a natural concept of equivalence of sequences to two-sided infinite sequences, the notion of permutation transducer is introduced. Requiring the underlying automaton to be deterministic in two directions, it provides the means to rewrite bi-infinite sequences. The first steps in studying the ensuing hierarchy of equivalence classes of bi-infinite sequences are taken, by describing the classes of ultimately periodic two-sided infinite sequences. It is important to make a distinction between unpointed and pointed sequences, that is, whether or not sequences are considered equivalent up to shifts. While one-sided ultimately periodic sequences form a single equivalence class under ordinary transductions, which is shown to split into two under permutation transductions, in the two-sided case there are three unpointed and seven pointed equivalence classes under permutation transduction.     

Enumerating several aspects of non-decreasing Dyck paths

A Dyck path is non-decreasing if the $y$-coordinates of its valleys form a non-decreasing sequence. In this paper we give enumerative results and some statistics of several aspects of non-decreasing Dyck paths. We give the number of pyramids at a fixed level that the paths of a given length have, count the number of primitive paths, count how many of the non-primitive paths can be expressed as a product of primitive paths, and count the number of paths of a given height and a given length. We present and prove our results using combinatorial arguments, generating functions (using the symbolic method) and parameterize the results studied here using the Riordan arrays. We use known bijections to connect direct column-convex polyominoes, Elena trees, and non-decreasing Dyck paths.     

Identities from weighted Motzkin paths

Based on a weighted version of the bijection between Dyck paths and 2-Motzkin paths, we find combinatorial interpretations of two identities related to the Narayana polynomials and the Catalan numbers. These interpretations answer two questions posed recently by Coker.     

峰严格递增的Dyck路的计数

本文考虑了由最高峰的高度为m,并且峰的高度沿着Dyck路严格递增的所有Dyck路组成的集合,即集合Dm的子集的计数问题．利用双射、生成树以及Riordan阵的方法来对集合Dm的一些子集进行计数,得到了一些以经典的序列如Catalan数、Narayana数、Motzkin数、Fibonacci数、Schroder数以及第一类无符号Stirling数来计数的组合结构．特别地,我们给出了两个新的Catalan结构,它们并没有明显地出现在Stanley关于Catalan结构的列表中．