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 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 On the Catalan Numbers and Some of Their Identities

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 62 ◽  
Author(s):  
Wenpeng Zhang ◽  
Li Chen
Keyword(s):  
Catalan Numbers ◽  
Recursive Sequences ◽  
Combinatorial Methods

The main purpose of this paper is using the elementary and combinatorial methods to study the properties of the Catalan numbers, and give two new identities for them. In order to do this, we first introduce two new recursive sequences, then with the help of these sequences, we obtained the identities for the convolution involving the Catalan numbers.

scholarly journals Pattern avoidance in inversion sequences

2015 ◽  
Vol 25 (2) ◽  
pp. 157-176 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Functional Equation ◽  
Generating Function ◽  
Generating Functions ◽  
Kernel Method ◽  
Fibonacci Numbers ◽  
Pattern Avoidance ◽  
Explicit Formulas ◽  
Schröder Numbers ◽  
Single Pattern ◽  
Combinatorial Methods

Abstract A permutation of length n may be represented, equivalently, by a sequence a1a2 • • • an satisfying 0 < ai < i for all z, which is called an inversion sequence. In analogy to the usual case for permutations, the pattern avoidance question is addressed for inversion sequences. In particular, explicit formulas and/or generating functions are derived which count the inversion sequences of a given length that avoid a single pattern of length three. Among the sequences encountered are the Fibonacci numbers, the Schröder numbers, and entry A200753 in OEIS. We make use of both algebraic and combinatorial methods to establish our results. An explicit Injection is given between two of the avoidance classes, and in three cases, the kernel method is used to solve a functional equation satisfied by the generating function enumerating the class in question.

scholarly journals Ascent sequences and Fibonacci numbers

Filomat ◽  
2015 ◽  
Vol 29 (4) ◽  
pp. 703-712
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Functional Equation ◽  
Kernel Method ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Pattern Avoidance ◽  
Combinatorial Structures ◽  
Ascent Sequences ◽  
Pattern Avoiding Permutations ◽  
Difficult Cases ◽  
Combinatorial Methods

An ascent sequence is one consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Ascent sequences have recently been shown to be related to (2+2)-free posets and a variety of other combinatorial structures. Let Fn denote the Fibonacci sequence given by the recurrence Fn = Fn-1 + Fn-2 if n ? 2, with F0 = 0 and F1 = 1. In this paper, we draw connections between ascent sequences and the Fibonacci numbers by showing that several pattern-avoidance classes of ascent sequences are enumerated by either Fn+1 or F2n-1. We make use of both algebraic and combinatorial methods to establish our results. In one of the apparently more difficult cases, we make use of the kernel method to solve a functional equation and thus determine the distribution of some statistics on the avoidance class in question. In two other cases, we adapt the scanning-elements algorithm, a technique which has been used in the enumeration of certain classes of pattern-avoiding permutations, to the comparable problem concerning pattern-avoiding ascent sequences.

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 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 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 The largest singletons in weighted set partitions and its applications

2011 ◽  
Vol Vol. 13 no. 3 (Combinatorics) ◽  
Author(s):  
Yidong Sun ◽  
Yanjie Xu
Keyword(s):  
Fixed Points ◽  
Combinatorial Identities ◽  
Total Weight ◽  
Explicit Formulas ◽  
Set Partitions ◽  
International Audience ◽  
Combinatorial Methods

Combinatorics International audience Recently, Deutsch and Elizalde studied the largest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let A (n,k) (t) denote the total weight of partitions on [n + 1] = \1,2,..., n + 1\ with the largest singleton \k + 1\. In this paper, explicit formulas for A (n,k) (t) and many combinatorial identities involving A (n,k) (t) are obtained by umbral operators and combinatorial methods. In particular, the permutation case leads to an identity related to tree enumerations, namely, [GRAPHICS] where D-k is the number of permutations of [k] with no fixed points.

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.

Online Prediction Model Based on New Kernel Method

2014 ◽  
Vol 7 (1) ◽  
pp. 107
Author(s):  
Ilyes Elaissi ◽  
Okba Taouali ◽  
Messaoud Hassani
Keyword(s):  
Prediction Model ◽  
Kernel Method ◽  
Model Based ◽  
Online Prediction
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