scholarly journals Some Combinatorial Interpretations and Applications of Fuss-Catalan Numbers

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Chin-Hung Lin
Keyword(s):  
Catalan Numbers ◽  
Catalan Number ◽  
Basic Properties ◽  
Generalized Catalan Numbers ◽  
Combinatorial Interpretations

Fuss-Catalan number is a family of generalized Catalan numbers. We begin by two definitions of Fuss-Catalan numbers and some basic properties. And we give some combinatorial interpretations different from original Catalan numbers. Finally we generalize the Jonah's theorem as its applications.

scholarly journals Restricted ascent sequences and Catalan numbers

2014 ◽  
Vol 8 (2) ◽  
pp. 288-303 ◽  
Author(s):  
David Callan ◽  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Recent Work ◽  
Equivalence Class ◽  
Catalan Numbers ◽  
Catalan Number ◽  
Combinatorial Interpretations ◽  
Single Pattern ◽  
Ascent Sequences ◽  
Wilf Equivalence

In this paper, we identify all members of the (4,4)-Wilf equivalence class for ascent sequences corresponding to the Catalan number Cn = 1 / n+1 (2n/n). This extends recent work concerning avoidance of a single pattern and provides apparently new combinatorial interpretations for Cn. In several cases, the subset of the class consisting of those members having exactly m ascents is given by the Narayana number Nn;m+1 = 1 / n (n/m+1)(n/m). We conclude by considering a further refinement in the case of avoiding 021.

scholarly journals Generalized Catalan Numbers from Hypergraphs

10.37236/8733 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Paul E. Gunnells
Keyword(s):  
Matrix Models ◽  
Associative Algebra ◽  
Matrix Model ◽  
Catalan Numbers ◽  
Set Partitions ◽  
Generalized Catalan Numbers ◽  
Combinatorial Interpretations ◽  
Plane Trees ◽  
Infinite Collection

The Catalan numbers $C_{n} \in \{1,1,2,5,14,42,\dots \}$ form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting rooted plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we define a generalization of the Catalan numbers. In fact we actually define an infinite collection of generalizations $C_{n}^{(m)}$, $m\geq 1$, with $C_{n}^{(1)}$ equal to the usual Catalans $C_{n}$; the sequence $C_{n}^{(m)}$ comes from studying certain matrix models attached to hypergraphs. We also give some combinatorial interpretations of these numbers.

Analysis of random point images with the use of symbolic computation codes and generalized Catalan numbers

2016 ◽  
Vol 52 (6) ◽  
pp. 529-536
Author(s):  
A. L. Reznik ◽  
A. V. Tuzikov ◽  
A. A. Solov’ev ◽  
A. V. Torgov
Keyword(s):  
Symbolic Computation ◽  
Catalan Numbers ◽  
Random Point ◽  
Generalized Catalan Numbers ◽  
Computation Codes

scholarly journals On Pattern-Avoiding Partitions

10.37236/763 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Vít Jelínek ◽  
Toufik Mansour
Keyword(s):  
Stirling Numbers ◽  
Equivalence Classes ◽  
Catalan Numbers ◽  
Set Partition ◽  
Combinatorial Interpretations

A set partition of size $n$ is a collection of disjoint blocks $B_1,B_2,\ldots$, $B_d$ whose union is the set $[n]=\{1,2,\ldots,n\}$. We choose the ordering of the blocks so that they satisfy $\min B_1 < \min B_2 < \cdots < \min B_d$. We represent such a set partition by a canonical sequence $\pi_1,\pi_2,\ldots,\pi_n$, with $\pi_i=j$ if $i\in B_j$. We say that a partition $\pi$ contains a partition $\sigma$ if the canonical sequence of $\pi$ contains a subsequence that is order-isomorphic to the canonical sequence of $\sigma$. Two partitions $\sigma$ and $\sigma'$ are equivalent, if there is a size-preserving bijection between $\sigma$-avoiding and $\sigma'$-avoiding partitions. We determine all the equivalence classes of partitions of size at most $7$. This extends previous work of Sagan, who described the equivalence classes of partitions of size at most $3$. Our classification is largely based on several new infinite families of pairs of equivalent patterns. For instance, we prove that there is a bijection between $k$-noncrossing and $k$-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence. Our results also yield new combinatorial interpretations of the Catalan numbers and the Stirling numbers.

scholarly journals On a refinement of the generalized Catalan numbers for Weyl groups

2004 ◽  
Vol 357 (1) ◽  
pp. 179-196 ◽  
Author(s):  
Christos A. Athanasiadis
Keyword(s):  
Catalan Numbers ◽  
Weyl Groups ◽  
Generalized Catalan Numbers

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 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 Divisibility of generalized Catalan numbers

2007 ◽  
Vol 114 (6) ◽  
pp. 1089-1100 ◽  
Author(s):  
Matjaž Konvalinka
Keyword(s):  
Catalan Numbers ◽  
Generalized Catalan Numbers

scholarly journals Bijections between noncrossing and nonnesting partitions for classical reflection groups

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Alex Fink ◽  
Benjamin Iriarte Giraldo
Keyword(s):  
Type A ◽  
Reflection Group ◽  
Catalan Numbers ◽  
Reflection Groups ◽  
Generalized Catalan Numbers ◽  
Elementary Methods ◽  
International Audience ◽  
Type Sets

International audience We present $\textit{type preserving}$ bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner. The bijections for the abstract Coxeter types $B$, $C$ and $D$ are new in the literature. To find them we define, for every type, sets of statistics that are in bijection with noncrossing and nonnesting partitions, and this correspondence is established by means of elementary methods in all cases. The statistics can be then seen to be counted by the generalized Catalan numbers Cat$(W)$ when $W$ is a classical reflection group. In particular, the statistics of type $A$ appear as a new explicit example of objects that are counted by the classical Catalan numbers.

scholarly journals On Teaching of Generalized Catalan Numbers with the Maple´s Help

2018 ◽  
Vol 11 (1) ◽  
pp. 25-40
Author(s):  
Francisco Regis Vieira Alves ◽  
Keyword(s):  
Catalan Numbers ◽  
Generalized Catalan Numbers
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