scholarly journals Catalan words avoiding pairs of length three patterns

2021 ◽  
Vol vol. 22 no. 2, Permutation... (Special issues) ◽  
Author(s):  
Jean-Luc Baril ◽  
Carine Khalil ◽  
Vincent Vajnovszki
Keyword(s):  
Structural Properties ◽  
Generating Functions ◽  
Recursive Decomposition ◽  
Integer Sequence ◽  
Combinatorial Interpretations

Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.

A general construction of mutually orthogonal latin squares of prime order

2018 ◽  
Vol 7 (1-2) ◽  
pp. 77-93
Author(s):  
J. A. Saka ◽  
O. O. Oyadare
Keyword(s):  
Structural Properties ◽  
Generating Function ◽  
Generating Functions ◽  
Prime Order ◽  
Latin Squares ◽  
General Construction ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Complete Set ◽  
General Method

This paper presents a general method of constructing a complete set of Mutually Orthogonal Latin Squares (MOLS) of the order of any prime, via the use of generating functions dened on the nite eld of this order. Apart from using the generating function to get a complete set of Mutually Orthogonal Latin Squares, the studies of the generating functions opens up the possibility of getting at the deep structural properties of MOLS. Copious examples were given for detailed illustrations.

scholarly journals On BC-Subtrees in Multi-Fan and Multi-Wheel Graphs

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 36
Author(s):  
Yu Yang ◽  
Long Li ◽  
Wenhu Wang ◽  
Hua Wang
Keyword(s):  
Structural Properties ◽  
Generating Functions ◽  
Topological Index ◽  
Extremal Problems ◽  
Distance Constraint ◽  
Wheel Graphs ◽  
Recursive Formulas ◽  
Cyclic Graphs ◽  
New Perspective

The BC-subtree (a subtree in which any two leaves are at even distance apart) number index is the total number of non-empty BC-subtrees of a graph, and is defined as a counting-based topological index that incorporates the leaf distance constraint. In this paper, we provide recursive formulas for computing the BC-subtree generating functions of multi-fan and multi-wheel graphs. As an application, we obtain the BC-subtree numbers of multi-fan graphs, r multi-fan graphs, multi-wheel (wheel) graphs, and discuss the change of the BC-subtree numbers between different multi-fan or multi-wheel graphs. We also consider the behavior of the BC-subtree number in these structures through the study of extremal problems and BC-subtree density. Our study offers a new perspective on understanding new structural properties of cyclic graphs.

scholarly journals A Triple Lacunary Generating Function for Hermite Polynomials

10.37236/1927 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ira M. Gessel ◽  
Pallavi Jayawant
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function ◽  
Generating Functions ◽  
Hermite Polynomials ◽  
Combinatorial Proof ◽  
Classical Orthogonal Polynomials ◽  
Combinatorial Objects ◽  
Combinatorial Interpretations

Some of the classical orthogonal polynomials such as Hermite, Laguerre, Charlier, etc. have been shown to be the generating polynomials for certain combinatorial objects. These combinatorial interpretations are used to prove new identities and generating functions involving these polynomials. In this paper we apply Foata's approach to generating functions for the Hermite polynomials to obtain a triple lacunary generating function. We define renormalized Hermite polynomials $h_n(u)$ by $$\sum_{n= 0}^\infty h_n(u) {z^n\over n!}=e^{uz+{z^2\!/2}}.$$ and give a combinatorial proof of the following generating function: $$ \sum_{n= 0}^\infty h_{3n}(u) {{z^n\over n!}}= {e^{(w-u)(3u-w)/6}\over\sqrt{1-6wz}} \sum_{n= 0}^\infty {{(6n)!\over 2^{3n}(3n)!(1-6wz)^{3n}} {z^{2n}\over(2n)!}}, $$ where $w=(1-\sqrt{1-12uz})/6z=uC(3uz)$ and $C(x)=(1-\sqrt{1-4x})/(2x)$ is the Catalan generating function. We also give an umbral proof of this generating function.

scholarly journals Moments of Catalan Triangle Numbers

2020 ◽  
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Generating Functions ◽  
Catalan Numbers ◽  
Integer Sequences ◽  
Combinatorial Interpretations ◽  
Sums Of Powers ◽  
Alternating Sums

In this chapter, we consider the Catalan numbers, C n = 1 n + 1 2 n n , and two of their generalizations, Catalan triangle numbers, B n , k and A n , k , for n , k ∈ N . They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of these Catalan triangle numbers, i.e., with the following sums: ∑ k = 1 n k m B n , k j , ∑ k = 1 n + 1 2 k − 1 m A n , k j , for j , n ∈ N and m ∈ N ∪ 0 . We present their closed expressions for some values of m and j . Alternating sums are also considered for particular powers. Other famous integer sequences are studied in Section 3, and its connection with Catalan triangle numbers are given in Section 4. Finally we conjecture some properties of divisibility of moments and alternating sums of powers in the last section.

scholarly journals On Subtree Number Index of Generalized Book Graphs, Fan Graphs, and Wheel Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Daoqiang Sun ◽  
Zhengying Zhao ◽  
Xiaoxiao Li ◽  
Jiayi Cao ◽  
Yu Yang
Keyword(s):  
Structural Analysis ◽  
Structural Properties ◽  
Generating Function ◽  
Generating Functions ◽  
Asymptotic Properties ◽  
Wheel Graphs

With generating function and structural analysis, this paper presents the subtree generating functions and the subtree number index of generalized book graphs, generalized fan graphs, and generalized wheel graphs, respectively. As an application, this paper also briefly studies the subtree number index and the asymptotic properties of the subtree densities in regular book graphs, regular fan graphs, and regular wheel graphs. The results provide the basis for studying novel structural properties of the graphs generated by generalized book graphs, fan graphs, and wheel graphs from the perspective of the subtree number index.

scholarly journals Automatic Classification of Restricted Lattice Walks

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Alin Bostan ◽  
Manuel Kauers
Keyword(s):  
Structural Properties ◽  
Generating Functions ◽  
Automatic Classification ◽  
Experimental Mathematics ◽  
Lattice Walks ◽  
International Audience

International audience We propose an $\textit{experimental mathematics approach}$ leading to the computer-driven $\textit{discovery}$ of various conjectures about structural properties of generating functions coming from enumeration of restricted lattice walks in 2D and in 3D.

scholarly journals On the Positive Moments of Ranks of Partitions

10.37236/3852 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
William Y. C. Chen ◽  
Kathy Q. Ji ◽  
Erin Y. Y. Shen
Keyword(s):  
Generating Functions ◽  
Combinatorial Interpretation ◽  
Combinatorial Interpretations

By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of the $2k$-th symmetrized moment $\eta_{2k}(n)$ of ranks of partitions of $n$ in terms of $(k+1)$-marked Durfee symbols of $n$. In this paper, we consider the $k$-th symmetrized positive moment $\bar{\eta}_k(n)$ of ranks of partitions of $n$ which is defined as the truncated sum over positive ranks of partitions of $n$. As combinatorial interpretations of $\bar{\eta}_{2k}(n)$ and $\bar{\eta}_{2k-1}(n)$, we show that for given $k$ and $i$ with $1\leq i\leq k+1$, $\bar{\eta}_{2k-1}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being zero and $\bar{\eta}_{2k}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being positive. The interpretations of $\bar{\eta}_{2k-1}(n)$ and $\bar{\eta}_{2k}(n)$ are independent of $i$, and they imply the interpretation of $\eta_{2k}(n)$ given by Andrews since $\eta_{2k}(n)$ equals $\bar{\eta}_{2k-1}(n)$ plus twice of $\bar{\eta}_{2k}(n)$. Moreover, we obtain the generating functions for $\bar{\eta}_{2k}(n)$ and $\bar{\eta}_{2k-1}(n)$.

Newborn's Preference for Faces

1996 ◽  
Vol 1 (3) ◽  
pp. 200-205 ◽  
Author(s):  
Carlo Umiltà ◽  
Francesca Simion ◽  
Eloisa Valenza
Keyword(s):  
Visual System ◽  
Structural Properties ◽  
Sensory Properties ◽  
Human Face ◽  
System Experiment ◽  
Face Experiment

Four experiments were aimed at elucidating some aspects of the preference for facelike patterns in newborns. Experiment 1 showed a preference for a stimulus whose components were located in the correct arrangement for a human face. Experiment 2 showed a preference for stimuli that had optimal sensory properties for the newborn visual system. Experiment 3 showed that babies directed their attention to a facelike pattern even when it was presented simultaneously with a non-facelike stimulus with optimal sensory properties. Experiment 4 showed the preference for facelike patterns in the temporal hemifield but not in the nasal hemifield. It was concluded that newborns' preference for facelike patterns reflects the activity of a subcortical system which is sensitive to the structural properties of the stimulus.

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