scholarly journals A Bijection on Dyck Paths and its Cycle Structure

10.37236/946 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
David Callan
Keyword(s):  
Catalan Numbers ◽  
Complete Analysis ◽  
Cycle Structure ◽  
Pascal Matrix ◽  
Dyck Paths ◽  
Motzkin Numbers

The known bijections on Dyck paths are either involutions or have notoriously intractable cycle structure. Here we present a size-preserving bijection on Dyck paths whose cycle structure is amenable to complete analysis. In particular, each cycle has length a power of 2. A new manifestation of the Catalan numbers as labeled forests crops up en route as does the Pascal matrix mod 2. We use the bijection to show the equivalence of two known manifestations of the Motzkin numbers. Finally, we consider some statistics on the new Catalan manifestation and the identities they interpret.

scholarly journals Two Statistics Linking Dyck Paths and Non-crossing Partitions

10.37236/570 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Haijian Zhao ◽  
Zheyuan Zhong
Keyword(s):  
Catalan Numbers ◽  
Dyck Paths ◽  
Narayana Numbers

We introduce a pair of statistics, maj and sh, on Dyck paths and show that they are equidistributed. Then we prove that this maj is equivalent to the statistics $ls$ and $rb$ on non-crossing partitions. Based on non-crossing partitions, we give the most obvious $q$-analogue of the Narayana numbers and the Catalan numbers.

scholarly journals A Short Approach to Catalan Numbers Modulo $2^r$

10.37236/664 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Guoce Xin ◽  
Jing-Feng Xu
Keyword(s):  
Catalan Numbers ◽  
Binary Expansion ◽  
Dyck Paths ◽  
Combinatorial Interpretations ◽  
Recent Classification ◽  
Motzkin Paths

We notice that two combinatorial interpretations of the well-known Catalan numbers $C_n=(2n)!/n!(n+1)!$ naturally give rise to a recursion for $C_n$. This recursion is ideal for the study of the congruences of $C_n$ modulo $2^r$, which attracted a lot of interest recently. We present short proofs of some known results, and improve Liu and Yeh's recent classification of $C_n$ modulo $2^r$. The equivalence $C_{n}\equiv_{2^r} C_{\bar n}$ is further reduced to $C_{n}\equiv_{2^r} C_{\tilde{n}}$ for simpler $\tilde{n}$. Moreover, by using connections between weighted Dyck paths and Motzkin paths, we find new classes of combinatorial sequences whose $2$-adic order is equal to that of $C_n$, which is one less than the sum of the digits of the binary expansion of $n+1$.

scholarly journals A Classification of Motzkin Numbers Modulo 8

10.37236/7092 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Ying Wang ◽  
Guoce Xin
Keyword(s):  
Short Proof ◽  
Recursive Formula ◽  
Catalan Numbers ◽  
Motzkin Numbers ◽  
Full Classification ◽  
Factorial Representation

The well-known Motzkin numbers were conjectured by Deutsch and Sagan to be nonzero when modulo $8$. The conjecture was first proved by Sen-Peng Eu, Shu-chung Liu and Yeong-Nan Yeh by using the factorial representation of the Catalan numbers. We present a short proof by finding a recursive formula for Motzkin numbers modulo $8$. Moreover, such a recursion leads to a full classification of Motzkin numbers modulo $8$. An addendum was added on April 3 2018.

scholarly journals Several Explicit and Recursive Formulas for the Generalized Motzkin Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Recursive Formula ◽  
Catalan Numbers ◽  
Explicit Formulas ◽  
Motzkin Numbers ◽  
Recursive Formulas

In the paper, the authors find two explicit formulas and recover a recursive formula for the generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.

scholarly journals Another bijection between $2$-triangulations and pairs of non-crossing Dyck paths

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Carlos M. Nicolás
Keyword(s):  
Catalan Numbers ◽  
Hankel Determinant ◽  
Dyck Paths ◽  
International Audience

International audience A $k$-triangulation of the $n$-gon is a maximal set of diagonals of the $n$-gon containing no subset of $k+1$ mutually crossing diagonals. The number of $k$-triangulations of the $n$-gon, determined by Jakob Jonsson, is equal to a $k \times k$ Hankel determinant of Catalan numbers. This determinant is also equal to the number of $k$ non-crossing Dyck paths of semi-length $n-2k$. This brings up the problem of finding a combinatorial bijection between these two sets. In FPSAC 2007, Elizalde presented such a bijection for the case $k=2$. We construct another bijection for this case that is stronger and simpler that Elizalde's. The bijection preserves two sets of parameters, degrees and generalized returns. As a corollary, we generalize Jonsson's formula for $k=2$ by counting the number of $2$-triangulations of the $n$-gon with a given degree at a fixed vertex. Une $k$-triangulation du $n$-gon est un ensemble maximal de diagonales du $n$-gon ne contenant pas de sous-ensemble de $k+1$ diagonales mutuellement croisant. Le nombre de $k$-triangulations du $n$-gon, déterminé par Jakob Jonsson, est égal à un déterminant de Hankel $k \times k$ de nombres de Catalan. Ce déterminant est aussi égal au nombre de $k$ chemins de Dyck de largo $n-2k$ que ne pas se croiser. Cela porte le problème de trouver une bijection de type combinatoire entre ces deux ensembles. À la FPSAC 2007, Elizalde a présenté une telle bijection pour le cas $k = 2$. Nous construisons une autre bijection pour ce cas qui est plus forte et plus simple que de l'Elizalde. La bijection conserve deux ensembles de paramètres, les degré et les retours généralisée. De ce, nous généralisons la formule de Jonsson pour $k = 2$ en comptant le nombre de $2$-triangulations du $n$-gon avec un degré à un vertex fixe.

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 CUNTZ–KRIEGER ALGEBRAS AND A GENERALIZATION OF CATALAN NUMBERS

2013 ◽  
Vol 24 (05) ◽  
pp. 1350040 ◽  
Author(s):  
KENGO MATSUMOTO
Keyword(s):  
Directed Graph ◽  
Generating Functions ◽  
Transition Matrix ◽  
Catalan Numbers ◽  
Partial Isometries ◽  
Dyck Paths ◽  
Generalized Catalan Numbers

For a directed graph G, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz–Krieger algebra [Formula: see text] for the transition matrix AGof the directed edges of G. The generalized Catalan numbers [Formula: see text] enumerate the number of Dyck paths for the graph G. Its generating functions will be studied.

EXACT p-ADIC ORDERS FOR DIFFERENCES OF MOTZKIN NUMBERS

2014 ◽  
Vol 10 (03) ◽  
pp. 653-667 ◽  
Author(s):  
TAMÁS LENGYEL
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Motzkin Numbers ◽  
Modulo P ◽  
The Difference ◽  
Covering Systems

For any prime p, we establish congruences modulo pn+1 for the difference of the pn+1 th and pn th Motzkin numbers and determine the p-adic order of the difference. The results confirm recent conjectures on the order. The applied techniques involve the use of congruences for the differences of certain Catalan numbers and binomial coefficients, congruential identities for sums of Catalan numbers, central binomial and trinomial coefficients, infinite incongruent disjoint covering systems and the solution of congruential recurrences.

scholarly journals Motzkin Paths, Motzkin Polynomials and Recurrence Relations

10.37236/4781 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Roy Oste ◽  
Joris Van der Jeugt
Keyword(s):  
Recurrence Relations ◽  
Catalan Numbers ◽  
Tridiagonal Matrix ◽  
Efficient Computation ◽  
Combinatorial Objects ◽  
Motzkin Numbers ◽  
The Matrix ◽  
Motzkin Paths

We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers.  Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial "counting" all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials) have been studied before, but here we deduce some properties based on recurrence relations. The recurrence relations proved here also allow an efficient computation of the Motzkin polynomials. Finally, we show that the matrix entries of powers of an arbitrary tridiagonal matrix are essentially given by Motzkin polynomials, a property commonly known but usually stated without proof.

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.

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