scholarly journals Signed Words and Permutations II; The Euler-Mahonian Polynomials

10.37236/1879 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Dominique Foata ◽  
Guo-Niu Han
Keyword(s):  
Symmetric Group ◽  
Generating Functions ◽  
Statistical Study ◽  
Record Values ◽  
The Other ◽  
Length Function ◽  
Major Index ◽  
Signed Permutations ◽  
Inversion Number ◽  
The One

As for the symmetric group of ordinary permutations there is also a statistical study of the group of signed permutations, that consists of calculating multivariable generating functions for this group by statistics involving record values and the length function. Two approaches are here systematically explored, using the flag-major index on the one hand, and the flag-inversion number on the other hand. The MacMahon Verfahren appears as a powerful tool throughout.

Induced Representations and Invariants

1950 ◽  
Vol 2 ◽  
pp. 334-343 ◽  
Author(s):  
G. DE B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Explicit Formula ◽  
Linear Group ◽  
Schur Functions ◽  
The Other ◽  
Induced Representations ◽  
Direct Sum ◽  
Invariant Matrices ◽  
The One

1. Introduction. The problem of the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices is intimately associated with the representation theory of the full linear group on the one hand and with the representation theory of the symmetric group on the other. In a previous paper the author gave an explicit formula for this reduction in terms of characters of the symmetric group. Later J. A. Todd derived the same formula using Schur functions, i.e. characters of representations of the full linear group.

scholarly journals Odd and Even Major Indices and One-Dimensional Characters for Classical Weyl Groups

2020 ◽  
Vol 24 (4) ◽  
pp. 809-835
Author(s):  
Francesco Brenti ◽  
Paolo Sentinelli
Keyword(s):  
Generating Functions ◽  
Weyl Groups ◽  
One Dimensional ◽  
Major Index ◽  
The One

Abstract We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz’s identity, of the Gessel–Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs.

scholarly journals 0-Hecke algebra actions on coinvariants and flags

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Generating Functions ◽  
Hecke Algebra ◽  
Flag Variety ◽  
Major Index ◽  
Inversion Number ◽  
International Audience ◽  
Descent Set ◽  
Complete Flag ◽  
Coinvariant Algebra

International audience By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. En étudiant l'action de l'algèbre de 0-Hecke sur l'algèbre coinvariante et la variété de drapeaux complète, nous interprétons les fonctions génératrices qui comptent les permutations avec un ensemble inverse de descentes fixé, selon leur nombre d'inversions et leur "major index''.

Marine storms in coastal tourist areas of the Canary Islands

2021 ◽  
Author(s):  
Amalia Yanes Luque ◽  
José Ángel Rodríguez Báez ◽  
Pablo Máyer Suárez ◽  
Pedro Dorta Antequera ◽  
Abel López Díez ◽  
...  
Keyword(s):  
Canary Islands ◽  
Significant Wave Height ◽  
Statistical Study ◽  
The Other ◽  
Small Island ◽  
Economic Sector ◽  
Gran Canaria ◽  
The One ◽  
Transport Mobility ◽  
Degree Of Severity

Abstract The small island areas oriented to coastal tourism are highly vulnerable to marine storms, given the fluctuation of this economic sector and the increase of its exposure due to the increasing anthropogenic presence on the coastline. In this context, the storms between 1958 and 2017 are studied here in the most important tourist enclaves of Tenerife and Gran Canaria, located in the municipalities of Arona and Adeje (SW of Tenerife) and in San Bartolomé de Tirajana and Mogán (S-SW of Gran Canaria). In order to perform the analysis, wave data from State Ports (Spain) (SIMAR-44 nodes, Ministry of Transport, Mobility and Urban Agenda of Spain) are used. A statistical study of the phenomena can help establish a threshold for the identification of a marine storm based on a significant wave height (Hs) of 2.7 meters in Arona and Adeje and 1.7 meters in San Bartolomé de Tirajana and Mogán. The application of this threshold identified 144 episodes in the SW of Tenerife and 154 in the S-SW of Gran Canaria. Their behavior is examined from, on the one hand, the duration, seasonality, type of waves and degree of severity that characterizes them; and, on the other, from the state of the atmosphere that concurs in its genesis by means of the Wetterzentrale synoptic maps. Finally, a first assessment of the effects of these stormy phenomena is made using local press reports on drowning, damage to urban furniture and port infrastructures and loss of sand on the beaches.

scholarly journals MacMahon-type Identities for Signed Even Permutations

10.37236/1836 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Dan Bernstein
Keyword(s):  
Symmetric Group ◽  
Alternating Group ◽  
Hyperoctahedral Group ◽  
Major Index ◽  
Signed Permutations ◽  
Natural Analogues ◽  
Classic Theorem

MacMahon's classic theorem states that the length and major index statistics are equidistributed on the symmetric group $S_n$. By defining natural analogues or generalizations of those statistics, similar equidistribution results have been obtained for the alternating group $A_n$ by Regev and Roichman, for the hyperoctahedral group $B_n$ by Adin, Brenti and Roichman, and for the group of even-signed permutations $D_n$ by Biagioli. We prove analogues of MacMahon's equidistribution theorem for the group of signed even permutations and for its subgroup of even-signed even permutations.

Modular Representations of the Symmetric Group

1951 ◽  
Vol 3 ◽  
pp. 309-327 ◽  
Author(s):  
J. H. Chung
Keyword(s):  
Symmetric Group ◽  
The Other ◽  
Modular Representations ◽  
Young Diagrams ◽  
Starting Point ◽  
The One ◽  
The Relationship

The theory of modular representations of the symmetric group was studied first by Nakayama (5, 6), and later by Thrall and Nesbitt (11) and Robinson (7, 8, 9). Nakayama built up his elaborate theory of hooks for the express purpose of studying this problem, while Robinson's extensive work on the various phases of the relationship between Young diagrams, skew diagrams and star diagrams on the one hand, and representations of the symmetric group on the other, culminating in a set of relations among the degrees of the representations, serves as a starting point for this paper.

Statistical Correlation Between the Thermal Transition Stress of a Specimen and its Resistance to Alternating Torsional Fatigue

1983 ◽  
Vol 105 (2) ◽  
pp. 200-205
Author(s):  
R. Harry ◽  
F. Bastenaire ◽  
F. Joubert ◽  
L. Regnier
Keyword(s):  
Nondestructive Test ◽  
Statistical Study ◽  
Endurance Limit ◽  
Close Correlation ◽  
Statistical Correlation ◽  
The Other ◽  
Thermal Transition ◽  
Transition Stress ◽  
Torsional Fatigue ◽  
The One

The conventional difficulty encountered in fatigue studies is the impossibility of performing more than one fatigue test on a specimen. The purpose of this paper is to present a statistical study in order to: 1) confirm the close correlation between the transition stress determined for each specimen following a rapid thermal nondestructive test and on the one hand, the lifetime N, when subjected to a fatigue treatment at a given level τ, in the elastic area; and on the other hand, the actual endurance limit specific to the specimen; and 2) provide an evaluation of the results obtained.

scholarly journals Inversion Generating Functions for Signed Pattern Avoiding Permutations

10.37236/6545 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Naiomi T. Cameron ◽  
Kendra Killpatrick
Keyword(s):  
Generating Functions ◽  
Combinatorial Proof ◽  
Hyperoctahedral Group ◽  
Major Index ◽  
Signed Permutations ◽  
Mahonian Statistics ◽  
Almost All ◽  
Open Question ◽  
Pattern Avoiding Permutations

We consider the classical Mahonian statistics on the set $B_n(\Sigma)$ of signed permutations in the hyperoctahedral group $B_n$ which avoid all patterns in $\Sigma$, where $\Sigma$ is a set of patterns of length two.  In 2000, Simion gave the cardinality of $B_n(\Sigma)$ in the cases where $\Sigma$ contains either one or two patterns of length two and showed that $\left|B_n(\Sigma)\right|$ is constant whenever $\left|\Sigma\right|=1$, whereas in most but not all instances where $\left|\Sigma\right|=2$, $\left|B_n(\Sigma)\right|=(n+1)!$.  We answer an open question of Simion by providing bijections from $B_n(\Sigma)$ to $S_{n+1}$ in these cases where $\left|B_n(\Sigma)\right|=(n+1)!$.  In addition, we extend Simion's work by providing a combinatorial proof in the language of signed permutations for the major index on $B_n(21, \bar{2}\bar{1})$ and by giving the major index on $D_n(\Sigma)$ for $\Sigma =\{21, \bar{2}\bar{1}\}$ and $\Sigma=\{12,21\}$.  The main result of this paper is to give the inversion generating functions for $B_n(\Sigma)$ for almost all sets $\Sigma$ with $\left|\Sigma\right|\leq2.$

scholarly journals Marine storms in coastal tourist areas of the Canary Islands

2021 ◽  
Author(s):  
Amalia Yanes Luque ◽  
José A. Rodríguez-Báez ◽  
Pablo Máyer Suárez ◽  
Pedro Dorta Antequera ◽  
Abel López-Díez ◽  
...  
Keyword(s):  
Canary Islands ◽  
Significant Wave Height ◽  
Statistical Study ◽  
The Other ◽  
Small Island ◽  
Economic Sector ◽  
Gran Canaria ◽  
The One ◽  
Transport Mobility ◽  
Degree Of Severity

AbstractThe small island areas oriented to coastal tourism are highly vulnerable to marine storms, given the fluctuation of this economic sector and the increase in its exposure due to the increasing anthropogenic presence on the coastline. In this context, the storms between 1958 and 2017 are studied here in the most important tourist enclaves of Tenerife and Gran Canaria, located in the municipalities of Arona and Adeje (SW of Tenerife) and in San Bartolomé de Tirajana and Mogán (S-SW of Gran Canaria). In order to perform the analysis, wave data from State Ports (Spain) (SIMAR-44 nodes, Ministry of Transport, Mobility and Urban Agenda of Spain) are used. A statistical study of the phenomena can help establish a threshold for the identification of a marine storm based on a significant wave height (Hs) of 2.7 m in Arona and Adeje and 1.7 m in San Bartolomé de Tirajana and Mogán. The application of this threshold identified 144 episodes in the SW of Tenerife and 154 in the S-SW of Gran Canaria. Their behavior is examined from, on the one hand, the duration, seasonality, type of waves and degree of severity that characterizes them; and, on the other, from the state of the atmosphere that concurs in its genesis by means of the Wetterzentrale synoptic maps. Finally, a first assessment of the effects of these stormy phenomena is made using local press reports on drowning, damage to urban furniture and port infrastructures and loss of sand on the beaches.

Note on Selection of Coordinate Systems for Geodynamics

1975 ◽  
Vol 26 ◽  
pp. 395-407
Author(s):  
S. Henriksen
Keyword(s):  
Sea Level ◽  
The Other ◽  
Coordinate Systems ◽  
Mean Sea Level ◽  
The Earth ◽  
The One ◽  
Static Systems ◽  
Selection Of

The first question to be answered, in seeking coordinate systems for geodynamics, is: what is geodynamics? The answer is, of course, that geodynamics is that part of geophysics which is concerned with movements of the Earth, as opposed to geostatics which is the physics of the stationary Earth. But as far as we know, there is no stationary Earth – epur sic monere. So geodynamics is actually coextensive with geophysics, and coordinate systems suitable for the one should be suitable for the other. At the present time, there are not many coordinate systems, if any, that can be identified with a static Earth. Certainly the only coordinate of aeronomic (atmospheric) interest is the height, and this is usually either as geodynamic height or as pressure. In oceanology, the most important coordinate is depth, and this, like heights in the atmosphere, is expressed as metric depth from mean sea level, as geodynamic depth, or as pressure. Only for the earth do we find “static” systems in use, ana even here there is real question as to whether the systems are dynamic or static. So it would seem that our answer to the question, of what kind, of coordinate systems are we seeking, must be that we are looking for the same systems as are used in geophysics, and these systems are dynamic in nature already – that is, their definition involvestime.

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