On the Foundation of Combinatorial Theory. X. A Categorical Setting for Symmetric Functions

International audience We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (\emphi.e. Baxter permutations, pairs of twin binary trees, \emphetc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like insertion algorithm. The algebraic properties of this Hopf algebra are studied. This Hopf algebra appeared for the first time in the work of Reading [Lattice congruences, fans and Hopf algebras, \textitJournal of Combinatorial Theory Series A, 110:237–273, 2005]. Nous proposons une nouvelle construction d'une sous-algèbre de Hopf de l'algèbre de Hopf des fonctions quasi-symétriques libres dont les bases sont indexées par les objets de la famille combinatoire de Baxter (\emphi.e. permutations de Baxter, couples d'arbres binaires jumeaux, \emphetc.). Cette construction repose sur la définition du mono\"ıde de Baxter, analogue du mono\"ıde plaxique et du mono\"ıde sylvestre, et d'un algorithme d'insertion analogue à l'algorithme de Robinson-Schensted. Les propriétés algébriques de cette algèbre de Hopf sont étudiées. Cette algèbre de Hopf est apparue pour la première fois dans le travail de Reading [Lattice congruences, fans and Hopf algebras, \textitJournal of Combinatorial Theory Series A, 110:237–273, 2005].

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CLASSES OF WEIGHTED SYMMETRIC FUNCTIONS

Real Analysis Exchange ◽

10.2307/44151958 ◽

1988 ◽

Vol 14 (2) ◽

pp. 429

Author(s):

Tran

Keyword(s):

Symmetric Functions

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WEIGHTED SYMMETRIC FUNCTIONS

Real Analysis Exchange ◽

10.2307/44152009 ◽

1989 ◽

Vol 15 (1) ◽

pp. 313

Author(s):

Tran

Keyword(s):

Symmetric Functions

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Applying Ateb–Gabor Filters to Biometric Imaging Problems

Symmetry ◽

10.3390/sym13040717 ◽

2021 ◽

Vol 13 (4) ◽

pp. 717

Author(s):

Mariia Nazarkevych ◽

Natalia Kryvinska ◽

Yaroslav Voznyi

Keyword(s):

Wavelet Transformation ◽

Symmetric Functions ◽

Gabor Filter ◽

Signal To Noise Ratio ◽

The Other ◽

Mean Square ◽

Image Transformation ◽

Gabor Filtering ◽

Signal To Noise ◽

Filtration Method

This article presents a new method of image filtering based on a new kind of image processing transformation, particularly the wavelet-Ateb–Gabor transformation, that is a wider basis for Gabor functions. Ateb functions are symmetric functions. The developed type of filtering makes it possible to perform image transformation and to obtain better biometric image recognition results than traditional filters allow. These results are possible due to the construction of various forms and sizes of the curves of the developed functions. Further, the wavelet transformation of Gabor filtering is investigated, and the time spent by the system on the operation is substantiated. The filtration is based on the images taken from NIST Special Database 302, that is publicly available. The reliability of the proposed method of wavelet-Ateb–Gabor filtering is proved by calculating and comparing the values of peak signal-to-noise ratio (PSNR) and mean square error (MSE) between two biometric images, one of which is filtered by the developed filtration method, and the other by the Gabor filter. The time characteristics of this filtering process are studied as well.

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A symmetric Bloch–Okounkov theorem

Research in the Mathematical Sciences ◽

10.1007/s40687-021-00253-8 ◽

2021 ◽

Vol 8 (2) ◽

Author(s):

Jan-Willem M. van Ittersum

Keyword(s):

Symmetric Functions ◽

Graded Algebra ◽

Symmetric Polynomials ◽

Generating Series

AbstractThe algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.

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Mathematical Properties of Variable Topological Indices

Symmetry ◽

10.3390/sym13010043 ◽

2020 ◽

Vol 13 (1) ◽

pp. 43

Author(s):

José M. Sigarreta

Keyword(s):

Real Number ◽

Large Class ◽

Symmetric Functions ◽

Topological Indices ◽

Current Interest ◽

Zagreb Indices ◽

Mathematical Properties ◽

Connectivity Indices ◽

Optimal Bounds ◽

New Framework

A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and/or other indices. In this paper we study two general topological indices Aα and Bα, defined for each graph H=(V(H),E(H)) by Aα(H)=∑ij∈E(H)f(di,dj)α and Bα(H)=∑i∈V(H)h(di)α, where di denotes the degree of the vertex i and α is any real number. Many important topological indices can be obtained from Aα and Bα by choosing appropriate symmetric functions and values of α. This new framework provides new tools that allow to obtain in a unified way inequalities involving many different topological indices. In particular, we obtain new optimal bounds on the variable Zagreb indices, the variable sum-connectivity index, the variable geometric-arithmetic index and the variable inverse sum indeg index. Thus, our approach provides both new tools for the study of topological indices and new bounds for a large class of topological indices. We obtain several optimal bounds of Aα (respectively, Bα) involving Aβ (respectively, Bβ). Moreover, we provide several bounds of the variable geometric-arithmetic index in terms of the variable inverse sum indeg index, and two bounds of the variable inverse sum indeg index in terms of the variable second Zagreb and the variable sum-connectivity indices.

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