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.

scholarly journals Area of Brownian Motion with Generatingfunctionology

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Michel Nguyên Thê
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Generating Functions ◽  
Limit Distributions ◽  
Dyck Paths ◽  
Different Types ◽  
International Audience

International audience This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

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.

A direct derivation of the Catalan formula

2013 ◽  
Vol 97 (538) ◽  
pp. 53-60 ◽  
Author(s):  
Gerry Leversha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Reflection Principle ◽  
Catalan Numbers ◽  
Equal Size ◽  
Alternative Form ◽  
Direct Derivation ◽  
Image Position

Many readers will be familiar with the sequence of Catalan numbers {Cn: n ≥ 0} and the formulawith its alternative formThese can be proved by using recurrence relations, generating functions or André's reflection principle. A good reference for all of these methods is Martin Griffiths' book [1].However, none of these approaches strikes me as being naturally combinatorial. A formula such as (1) is often derived by making a list of all the ways of doing something, and then subdividing this list into classes of equal size, so that either one class consists entirely of ‘valid’ cases or there is exactly one ‘valid’ case in each list.

scholarly journals Markov chains with quasitoeplitz transition matrix: first zero hitting

1989 ◽  
Vol 2 (3) ◽  
pp. 205-216
Author(s):  
Alexander M. Dukhovny
Keyword(s):  
Markov Chains ◽  
Boundary Value Problems ◽  
Unit Circle ◽  
Generating Functions ◽  
Transition Matrix ◽  
Boundary Value ◽  
Riemann Boundary ◽  
Hitting Probabilities

This paper continues the investigation of Markov Chains with a quasitoeplitz transition matrix. Generating functions of first zero hitting probabilities and mean times are found by the solution of special Riemann boundary value problems on the unit circle. Duality is discussed.

Generalized Catalan numbers in problems of processing of random discrete images

2011 ◽  
Vol 47 (6) ◽  
pp. 533-536
Author(s):  
A. L. Reznik ◽  
V. M. Efimov ◽  
A. A. Solov’ev ◽  
A. V. Torgov
Keyword(s):  
Catalan Numbers ◽  
Generalized Catalan Numbers

scholarly journals Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Cyclic Groups ◽  
Bijective Proof ◽  
Avoidance Condition

We continue the study of the generalized pattern avoidance condition for Ck≀Sn, the wreath product of the cyclic group Ck with the symmetric group Sn, initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers.

scholarly journals Variations on Descents and Inversions in Permutations

10.37236/856 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Denis Chebikin
Keyword(s):  
Generating Functions ◽  
Euler Number ◽  
Weighted Sum ◽  
Eulerian Polynomials ◽  
Joint Distributions ◽  
Dyck Paths ◽  
Sigma 1 ◽  
Descent Set ◽  
New Statistics ◽  
Factor Set

We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation $\sigma = \sigma_1\sigma_2\cdots\sigma_n$ defined as the set of indices $i$ such that either $i$ is odd and $\sigma_i > \sigma_{i+1}$, or $i$ is even and $\sigma_i < \sigma_{i+1}$. We show that this statistic is equidistributed with the odd $3$-factor set statistic on permutations $\tilde{\sigma} = \sigma_1\sigma_2\cdots\sigma_{n+1}$ with $\sigma_1=1$, defined to be the set of indices $i$ such that the triple $\sigma_i \sigma_{i+1} \sigma_{i+2}$ forms an odd permutation of size $3$. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same, establishing a connection to two classical Mahonian statistics, maj and stat, along the way. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials $\sum_{\sigma\in\mathcal{S}_n} t^{{\rm des}(\sigma)+1}$ using alternating descents. For the alternating descent set statistic, we define the generating polynomial in two non-commutative variables by analogy with the $ab$-index of the Boolean algebra $B_n$, providing a link to permutations without consecutive descents. By looking at the number of alternating inversions, which we define in the paper, in alternating (down-up) permutations, we obtain a new $q$-analog of the Euler number $E_n$ and show how it emerges in a $q$-analog of an identity expressing $E_n$ as a weighted sum of Dyck paths.

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.

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