scholarly journals Catalan numbers and pattern restricted set partitions

2012 ◽  
Vol 312 (20) ◽  
pp. 2979-2991
Author(s):  
Toufik Mansour ◽  
Matthias Schork ◽  
Mark Shattuck
Keyword(s):  
Catalan Numbers ◽  
Set Partitions

scholarly journals The dual of number sequences, Riordan polynomials, and Sheffer polynomials

2021 ◽  
Vol 10 (1) ◽  
pp. 153-165
Author(s):  
Tian-Xiao He ◽  
José L. Ramírez
Keyword(s):  
Catalan Numbers ◽  
Combinatorial Identities ◽  
Bell Numbers ◽  
Set Partitions ◽  
Polynomial Sequences ◽  
Number Sequences ◽  
Sheffer Polynomials ◽  
Sheffer Polynomial

Abstract In this paper we introduce different families of numerical and polynomial sequences by using Riordan pseudo involutions and Sheffer polynomial sequences. Many examples are given including dual of Hermite numbers and polynomials, dual of Bell numbers and polynomials, among other. The coefficients of some of these polynomials are related to the counting of different families of set partitions and permutations. We also studied the dual of Catalan numbers and dual of Fuss-Catalan numbers, giving several combinatorial identities.

scholarly journals Pattern-Avoiding Set Partitions and Catalan Numbers

10.37236/2048 ◽  
2012 ◽  
Vol 18 (2) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Kernel Method ◽  
Catalan Numbers ◽  
Set Partitions ◽  
Narayana Numbers ◽  
Combinatorial Methods

We identify several subsets of the partitions of $[n]$, each characterized by the avoidance of a pair of patterns, respectively of lengths four and five.  Each of the classes we consider is enumerated by the Catalan numbers.  Furthermore, the members of each class having a prescribed number of blocks are enumerated by the Narayana numbers.  We use both algebraic and combinatorial methods to establish our results.  In some of the cases, we make use of the kernel method to solve the recurrence arising when a further statistic is considered.  In other cases, we define bijections with previously enumerated classes which preserve the number of blocks.  Two of our bijections are of an algorithmic nature and systematically replace the occurrences of one pattern with those of another having the same length.

scholarly journals The Delta Conjecture

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
James Haglund ◽  
Jeffrey B. Remmel ◽  
Andrew Timothy Wilson
Keyword(s):  
Symmetric Function ◽  
Ordered Set ◽  
Macdonald Polynomials ◽  
Catalan Numbers ◽  
Set Partitions ◽  
Combinatorial Interpretations ◽  
Llt Polynomials ◽  
Ordered Set Partitions ◽  
International Audience ◽  
Tesler Matrices

International audience We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.

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.

scholarly journals ON DIVISIBILITY OF BINOMIAL COEFFICIENTS

2012 ◽  
Vol 93 (1-2) ◽  
pp. 189-201 ◽  
Author(s):  
ZHI-WEI SUN
Keyword(s):  
Higher Order ◽  
Catalan Numbers ◽  
Binomial Coefficients

AbstractIn this paper, motivated by Catalan numbers and higher-order Catalan numbers, we study factors of products of at most two binomial coefficients.

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