scholarly journals An Area-to-Inv Bijection Between Dyck Paths and 312-avoiding Permutations

10.37236/1584 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Jason Bandlow ◽  
Kendra Killpatrick
Keyword(s):  
Catalan Number ◽  
Dyck Path ◽  
Combinatorial Objects ◽  
Dyck Paths

The symmetric $q,t$-Catalan polynomial $C_n(q,t)$, which specializes to the Catalan polynomial $C_n(q)$ when $t=1$, was defined by Garsia and Haiman in 1994. In 2000, Garsia and Haglund described statistics $a(\pi)$ and $b(\pi)$ on Dyck paths such that $C_n(q,t) = \sum_{\pi} q^{a(\pi)}t^{b(\pi)}$ where the sum is over all $n \times n$ Dyck paths. Specializing $t=1$ gives the Catalan polynomial $C_n(q)$ defined by Carlitz and Riordan and further studied by Carlitz. Specializing both $t=1$ and $q=1$ gives the usual Catalan number $C_n$. The Catalan number $C_n$ is known to count the number of $n \times n$ Dyck paths and the number of $312$-avoiding permutations in $S_n$, as well as at least 64 other combinatorial objects. In this paper, we define a bijection between Dyck paths and $312$-avoiding permutations which takes the area statistic $a(\pi)$ on Dyck paths to the inversion statistic on $312$-avoiding permutations. The inversion statistic can be thought of as the number of $(21)$ patterns in a permutation $\sigma$. We give a characterization for the number of $(321)$, $(4321)$, $\dots$, $(k\cdots21)$ patterns that occur in $\sigma$ in terms of the corresponding Dyck path.

scholarly journals Bicoloured Dyck Paths and the Contact Polynomial for $n$ Non-Intersecting Paths in a Half-Plane Lattice

10.37236/1728 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
R. Brak ◽  
J. W. Essam
Keyword(s):  
Recurrence Relation ◽  
Half Plane ◽  
Generating Function ◽  
Catalan Numbers ◽  
Catalan Number ◽  
Lattice Paths ◽  
Product Form ◽  
Dyck Path ◽  
Combinatorial Interpretation ◽  
Dyck Paths

In this paper configurations of $n$ non-intersecting lattice paths which begin and end on the line $y=0$ and are excluded from the region below this line are considered. Such configurations are called Hankel $n-$paths and their contact polynomial is defined by $\hat{Z}^{\cal{H}}_{2r}(n;\kappa)\equiv \sum_{c= 1}^{r+1} |{\cal H}_{2r}^{(n)}(c)|\kappa^c$ where ${\cal H}_{2r}^{(n)}(c)$ is the set of Hankel $n$-paths which make $c$ intersections with the line $y=0$ the lowest of which has length $2r$. These configurations may also be described as parallel Dyck paths. It is found that replacing $\kappa$ by the length generating function for Dyck paths, $\kappa(\omega) \equiv \sum_{r=0}^\infty C_r \omega^r$, where $C_r$ is the $r^{th}$ Catalan number, results in a remarkable simplification of the coefficients of the contact polynomial. In particular it is shown that the polynomial for configurations of a single Dyck path has the expansion $\hat{Z}^{\cal{H}}_{2r}(1;\kappa(\omega)) = \sum_{b=0}^\infty C_{r+b}\omega^b$. This result is derived using a bijection between bi-coloured Dyck paths and plain Dyck paths. A bi-coloured Dyck path is a Dyck path in which each edge is coloured either red or blue with the constraint that the colour can only change at a contact with the line $y=0$. For $n>1$, the coefficient of $\omega^b$ in $\hat{Z}^{\cal{W}}_{2r}(n;\kappa(\omega))$ is expressed as a determinant of Catalan numbers which has a combinatorial interpretation in terms of a modified class of $n$ non-intersecting Dyck paths. The determinant satisfies a recurrence relation which leads to the proof of a product form for the coefficients in the $\omega$ expansion of the contact polynomial.

scholarly journals A General Theory of Wilf-Equivalence for Catalan Structures

10.37236/5629 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Michael Albert ◽  
Mathilde Bouvel
Keyword(s):  
General Theory ◽  
Equivalence Relation ◽  
Generating Functions ◽  
Asymptotic Estimate ◽  
Equivalence Classes ◽  
Catalan Number ◽  
Combinatorial Structures ◽  
Dyck Paths ◽  
Significant Interest ◽  
Wilf Equivalence

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences or generating functions of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal subclasses defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal subclasses have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of size $n$ and show that it is exponentially smaller than the $n^{th}$ Catalan number. In other words these "coincidental" equalities are in fact very common among principal subclasses. Our results also allow us to prove in a unified and bijective manner several known Wilf-equivalences from the literature.

scholarly journals A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

2012 ◽  
Vol 64 (4) ◽  
pp. 822-844 ◽  
Author(s):  
J. Haglund ◽  
J. Morse ◽  
M. Zabrocki
Keyword(s):  
Building Blocks ◽  
Main Diagonal ◽  
Combinatorial Theory ◽  
Dyck Path ◽  
Dyck Paths ◽  
Littlewood Polynomials ◽  
Diagonal Harmonics ◽  
Littlewood Polynomial ◽  
Theory Operator ◽  
Shuffle Conjecture

Abstract We introduce a q, t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇ en[X]. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of q, t-Catalan sequences, and we prove a number of identities involving these functions.

scholarly journals Constructing combinatorial operads from monoids

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Samuele Giraudo
Keyword(s):  
Rooted Trees ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
International Audience ◽  
Motzkin Paths ◽  
Planar Rooted Trees

International audience We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.

scholarly journals Short note on the number of 1-ascents in dispersed Dyck paths

2017 ◽  
Vol 09 (06) ◽  
pp. 1750077
Author(s):  
Kairi Kangro ◽  
Mozhgan Pourmoradnasseri ◽  
Dirk Oliver Theis
Keyword(s):  
Generating Function ◽  
Short Note ◽  
Lattice Path ◽  
Dyck Path ◽  
Closed Formula ◽  
Dyck Paths ◽  
Maximal Sequence

A dispersed Dyck path (DDP) of length [Formula: see text] is a lattice path on [Formula: see text] from [Formula: see text] to [Formula: see text] in which the following steps are allowed: “up” [Formula: see text]; “down” [Formula: see text]; and “right” [Formula: see text]. An ascent in a DDP is an inclusion-wise maximal sequence of consecutive up steps. A 1-ascent is an ascent consisting of exactly 1 up step. We give a closed formula for the total number of 1-ascents in all dispersed Dyck paths of length [Formula: see text], #A191386 in Sloane’s OEIS. Previously, only implicit generating function relations and asymptotics were known.

scholarly journals Ascent Sequences Avoiding Pairs of Patterns

10.37236/4479 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Andrew M. Baxter ◽  
Lara K. Pudwell
Keyword(s):  
Pattern Avoidance ◽  
Exact Enumeration ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
Generating Trees ◽  
Precise Bound ◽  
Ascent Sequences ◽  
Pattern Avoiding Permutations

Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and Steingrímsson. In this paper, we consider ascent sequences of length $n$ avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the Erdős-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound.

scholarly journals Rational Catalan Combinatorics: The Associahedron

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Drew Armstrong ◽  
Brendon Rhoades ◽  
Nathan Williams
Keyword(s):  
Simplicial Complex ◽  
Rational Number ◽  
Catalan Number ◽  
Positive Rational Number ◽  
Dyck Paths ◽  
Narayana Numbers ◽  
International Audience ◽  
Positive Integers ◽  
Coprime Positive Integers ◽  
Product Formulas

International audience Each positive rational number $x>0$ can be written $\textbf{uniquely}$ as $x=a/(b-a)$ for coprime positive integers 0<$a$<$b$. We will identify $x$ with the pair $(a,b)$. In this extended abstract we use $\textit{rational Dyck paths}$ to define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass} (x)=\mathsf{Ass} (a,b)$ called the $\textit{rational associahedron}$. It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the $\textit{rational Catalan number}$ $\mathsf{Cat} (x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)! }{ a! b!}.$ 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 Fomin-Reading. We prove that $\mathsf{Ass} (a,b)$ is shellable and give nice product formulas for its $h$-vector (the $\textit{rational Narayana numbers}$) and $f$-vector (the $\textit{rational Kirkman numbers}$). We define $\mathsf{Ass} (a,b)$ .

scholarly journals On Parking Functions and the Zeta Map in Types B, C and D

10.37236/6714 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Robin Sulzgruber ◽  
Marko Thiel
Keyword(s):  
Hilbert Series ◽  
Root Systems ◽  
The Other ◽  
Lattice Paths ◽  
Square Lattice ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Coxeter Number ◽  
Dyck Paths ◽  
Diagonal Harmonics

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.

scholarly journals Dyck path triangulations and extendability (extended abstract)

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Cesar Ceballos ◽  
Arnau Padrol ◽  
Camilo Sarmiento
Keyword(s):  
Cartesian Product ◽  
Explicit Construction ◽  
Oriented Matroids ◽  
Oriented Matroid ◽  
Dyck Path ◽  
Matroid Theory ◽  
Dyck Paths ◽  
Nous Montrons ◽  
International Audience

International audience We introduce the Dyck path triangulation of the cartesian product of two simplices $\Delta_{n-1}\times\Delta_{n-1}$. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of $\Delta_{r\ n-1}\times\Delta_{n-1}$ using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever$m\geq k>n$, any triangulations of $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ extends to a unique triangulation of $\Delta_{m-1}\times\Delta_{n-1}$. Moreover, with an explicit construction, we prove that the bound $k>n$ is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory. Nous introduisons la triangulation par chemins de Dyck du produit cartésien de deux simplexes $\Delta_{n-1}\times\Delta_{n-1}$. Les simplexes maximaux de cette triangulation sont donnés par des chemins de Dyck, et cette construction se généralise de façon naturelle pour produire des triangulations $\Delta_{r\ n-1}\times\Delta_{n-1}$ qui utilisent des chemins de Dyck rationnels. Notre étude de la triangulation par chemins de Dyck est motivée par des problèmes de prolongement de triangulations partielles de produits de deux simplexes. On montre que $m\geq k>n$ alors toute triangulation de $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ se prolonge en une unique triangulation de $\Delta_{m-1}\times\Delta_{n-1}$. De plus, avec une construction explicite, nous montrons que la borne $k>n$ est optimale. Nous présentons aussi des interprétations de nos résultats dans le langage des matroïdes orientés tropicaux, qui sont analogues aux résultats classiques de la théorie des matroïdes orientés.

scholarly journals General Results on the Enumeration of Strings in Dyck Paths

10.37236/561 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
K. Manes ◽  
A. Sapounakis ◽  
I. Tasoulas ◽  
P. Tsikouras
Keyword(s):  
Generating Function ◽  
Lattice Path ◽  
Dyck Path ◽  
Dyck Paths

Let $\tau$ be a fixed lattice path (called in this context string) on the integer plane, consisting of two kinds of steps. The Dyck path statistic "number of occurrences of $\tau$" has been studied by many authors, for particular strings only. In this paper, arbitrary strings are considered. The associated generating function is evaluated when $\tau$ is a Dyck prefix (or a Dyck suffix). Furthermore, the case when $\tau$ is neither a Dyck prefix nor a Dyck suffix is considered, giving some partial results. Finally, the statistic "number of occurrences of $\tau$ at height at least $j$" is considered, evaluating the corresponding generating function when $\tau$ is either a Dyck prefix or a Dyck suffix.

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