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.

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 Counting Segmented Permutations Using Bicoloured Dyck Paths

10.37236/1936 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Anders Claesson
Keyword(s):  
Generating Function ◽  
Chebyshev Polynomials ◽  
Dyck Path ◽  
Dyck Paths ◽  
One To One ◽  
Bivariate Generating Function

A bicoloured Dyck path is a Dyck path in which each up-step is assigned one of two colours, say, red and green. We say that a permutation $\pi$ is $\sigma$-segmented if every occurrence $o$ of $\sigma$ in $\pi$ is a segment-occurrence (i.e., $o$ is a contiguous subword in $\pi$). We show combinatorially the following two results: The $132$-segmented permutations of length $n$ with $k$ occurrences of $132$ are in one-to-one correspondence with bicoloured Dyck paths of length $2n-4k$ with $k$ red up-steps. Similarly, the $123$-segmented permutations of length $n$ with $k$ occurrences of $123$ are in one-to-one correspondence with bicoloured Dyck paths of length $2n-4k$ with $k$ red up-steps, each of height less than $2$. We enumerate the permutations above by enumerating the corresponding bicoloured Dyck paths. More generally, we present a bivariate generating function for the number of bicoloured Dyck paths of length $2n$ with $k$ red up-steps, each of height less than $h$. This generating function is expressed in terms of Chebyshev polynomials of the second kind.

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 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 Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers

10.37236/6376 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Paul Drube
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Young Tableau ◽  
Young Tableaux ◽  
Dyck Paths ◽  
Standard Tableau ◽  
Ballot Numbers ◽  
Standard Young Tableaux

An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard. Such a tableau with precisely $k$ inversion pairs is said to be a $k$-inverted semistandard Young tableau. Building upon earlier work by Fresse and the author, this paper develops generating functions for the numbers of $k$-inverted semistandard Young tableaux of various shapes $\lambda$ and contents $\mu$. An easily-calculable generating function is given for the number of $k$-inverted semistandard Young tableaux that "standardize" to a fixed semistandard Young tableau. For $m$-row shapes $\lambda$ and standard content $\mu$, the total number of $k$-inverted standard Young tableaux of shape $\lambda$ is then enumerated by relating such tableaux to $m$-dimensional generalizations of Dyck paths and counting the numbers of "returns to ground" in those paths. In the rectangular specialization of $\lambda = n^m$ this yields a generating function that involves $m$-dimensional analogues of the famed Ballot numbers. Our various results are then used to directly enumerate all $k$-inverted semistandard Young tableaux with arbitrary content and two-row shape $\lambda = a^1 b^1$, as well as all $k$-inverted standard Young tableaux with two-column shape $\lambda=2^n$.

scholarly journals The Double Riordan Group

10.37236/2034 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
Dennis E. Davenport ◽  
Louis W. Shapiro ◽  
Leon C. Woodson
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Triangular Matrices ◽  
Ordered Trees ◽  
Dyck Paths ◽  
Motzkin Numbers ◽  
The Matrix ◽  
Riordan Group ◽  
Lower Triangular ◽  
Lower Triangular Matrices

The Riordan group is a group of infinite lower triangular matrices that are defined by two generating functions, $g$ and $f$. The kth column of the matrix has the generating function $gf^k$. In the Double Riordan group there are two generating function $f_1$ and $f_2$ such that the columns, starting at the left, have generating functions using $f_1$ and $f_2$ alternately. Examples include Dyck paths with level steps of length 2  allowed at even height and also ordered trees with differing degree possibilities at even and odd height(perhaps representing summer and winter). The Double Riordan group is a generalization not of the Riordan group itself but of the checkerboard subgroup. In this context both familiar and far less familiar sequences occur such as the Motzkin numbers and the number of spoiled child trees. The latter is a slightly enhanced cousin of ordered trees which are counted by the Catalan numbers.

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 Generalized Small Schröder Numbers

10.37236/4827 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
JiSun Huh ◽  
SeungKyung Park
Keyword(s):  
Lattice Path ◽  
Dyck Paths ◽  
Schröder Numbers ◽  
Schröder Paths ◽  
Positive Integers

We study generalized small Schröder paths in the sense of arbitrary sizes of steps. A generalized small Schröder path is a generalized lattice path from $(0,0)$ to $(2n,0)$ with the step set of  $\{(k,k), (l,-l), (2r,0)\, |\, k,l,r \in {\bf P}\}$, where ${\bf P}$ is the set of positive integers, which never goes below the $x$-axis, and with no horizontal steps at level 0.  We find a bijection between 5-colored Dyck paths and generalized small Schröder paths, proving that the number of generalized small Schröder paths is equal to $\sum_{k=1}^{n} N(n,k)5^{n-k}$ for $n\geq 1$.

scholarly journals Difference Equations and Generating Functions for some Lattice Path Problems

2019 ◽  
pp. 551-559
Author(s):  
Sreelatha Chandragiri
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Generating Functions ◽  
Lattice Paths ◽  
Lattice Path ◽  
Dyck Paths ◽  
Constant Coefficients ◽  
Novel Method

An identity for generating functions is proved in this paper. A novel method to compute the number of restricted lattice paths is developed on the basis of this identity. The method employs a difference equation with non-constant coefficients. Dyck paths, Schr¨oder paths, Motzkins path and other paths are computed to illustrate this method

scholarly journals The location of the first maximum in the first sojourn of a Dyck path

2008 ◽  
Vol Vol. 10 no. 3 (Analysis of Algorithms) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Random Walks ◽  
Generating Functions ◽  
Analysis Of Algorithms ◽  
Dyck Path ◽  
Dyck Paths ◽  
Asymptotic Expressions ◽  
International Audience

Analysis of Algorithms International audience For Dyck paths (nonnegative symmetric) random walks, the location of the first maximum within the first sojourn is studied. Generating functions and explicit resp. asymptotic expressions for the average are derived. Related parameters are also discussed.

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