scholarly journals On some Euler-Mahonian Distributions

10.37236/6702 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Angela Carnevale
Keyword(s):  
Symmetric Group ◽  
Joint Distribution ◽  
Definition Of

We prove that the pair of statistics (des,maj) on multiset permutations is equidistributed with the pair (stc,inv) on certain quotients of the symmetric group. We define an analogue of the statistic stc on multiset permutations whose joint distribution with the inversions equals that of (des,maj). We extend the definition of the statistic stc to hyperoctahedral and even hyperoctahedral groups. These functions, together with the Coxeter length, are equidistributed with (ndes,nmaj) and (ddes,dmaj), respectively.

scholarly journals Modular Representations of Sn

1964 ◽  
Vol 16 ◽  
pp. 191-203 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Modular Representations ◽  
Modular Theory ◽  
Definition Of ◽  
Image Position

The purpose of this paper is to clarify and sharpen the argument in the last two chapters of the author's Representation theory of the symmetric group(3). When these chapters were written the peculiar properties of the case p = 2 were not fully appreciated. No difficulty arises in the definition of the block in terms of the p-core, or in the application of the general modular theory based on the formula

scholarly journals A Unified Definition of Mutual Information with Applications in Machine Learning

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Guoping Zeng
Keyword(s):  
Machine Learning ◽  
Mutual Information ◽  
Joint Distribution ◽  
Credit Scoring ◽  
Random Variables ◽  
Joint Probability ◽  
Class 1 ◽  
Probability Spaces ◽  
Definition Of ◽  
Joint Probabilities

There are various definitions of mutual information. Essentially, these definitions can be divided into two classes: (1) definitions with random variables and (2) definitions with ensembles. However, there are some mathematical flaws in these definitions. For instance, Class 1 definitions either neglect the probability spaces or assume the two random variables have the same probability space. Class 2 definitions redefine marginal probabilities from the joint probabilities. In fact, the marginal probabilities are given from the ensembles and should not be redefined from the joint probabilities. Both Class 1 and Class 2 definitions assume a joint distribution exists. Yet, they all ignore an important fact that the joint or the joint probability measure is not unique. In this paper, we first present a new unified definition of mutual information to cover all the various definitions and to fix their mathematical flaws. Our idea is to define the joint distribution of two random variables by taking the marginal probabilities into consideration. Next, we establish some properties of the newly defined mutual information. We then propose a method to calculate mutual information in machine learning. Finally, we apply our newly defined mutual information to credit scoring.

scholarly journals Systems with Failure-Dependent Lifetimes of Components

2009 ◽  
Vol 46 (04) ◽  
pp. 1052-1072 ◽  
Author(s):  
M. Burkschat
Keyword(s):  
Order Statistics ◽  
Joint Distribution ◽  
Coherent System ◽  
Coherent Systems ◽  
Sequential Order ◽  
Failure Times ◽  
Sequential Order Statistics ◽  
Definition Of ◽  
General Component

A model for describing the lifetimes of coherent systems, where the failures of components may have an impact on the lifetimes of the remaining components, is proposed. The model is motivated by the definition of sequential order statistics (cf. Kamps (1995)). Sequential order statistics describe the successive failure times in a sequentialk-out-of-nsystem, where the distribution of the remaining components' lifetimes is allowed to change after every failure of a component. In the present paper, general component lifetimes which can be influenced by failures are considered. The ordered failure times of these components can be used to extend the concept of sequential order statistics. In particular, a definition of sequential order statistics based on exchangeable components is proposed. By utilizing the system signature (cf. Samaniego (2007)), the distribution of the lifetime of a coherent system with failure-dependent exchangeable component lifetimes is shown to be given by a mixture of the distributions of sequential order statistics. Furthermore, some results on the joint distribution of sequential order statistics based on exchangeable components are given.

scholarly journals Pattern Avoidance in Permutations: Linear and Cyclic Orders

10.37236/1690 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Antoine Vella
Keyword(s):  
Symmetric Group ◽  
Joint Distribution ◽  
Pattern Avoidance ◽  
Horizontal Edge ◽  
Ordered Sets ◽  
Dyck Paths ◽  
Euler Totient Function ◽  
Enumeration Problem ◽  
Counting Lemma ◽  
Horizontal And Vertical Edges

We generalize the notion of pattern avoidance to arbitrary functions on ordered sets, and consider specifically three scenarios for permutations: linear, cyclic and hybrid, the first one corresponding to classical permutation avoidance. The cyclic modification allows for circular shifts in the entries. Using two bijections, both ascribable to both Deutsch and Krattenthaler independently, we single out two geometrically significant classes of Dyck paths that correspond to two instances of simultaneous avoidance in the purely linear case, and to two distinct patterns in the hybrid case: non-decreasing Dyck paths (first considered by Barcucci et al.), and Dyck paths with at most one long vertical or horizontal edge. We derive a generating function counting Dyck paths by their number of low and high peaks, long horizontal and vertical edges, and what we call sinking steps. This translates into the joint distribution of fixed points, excedances, deficiencies, descents and inverse descents over 321-avoiding permutations. In particular we give an explicit formula for the number of 321-avoiding permutations with precisely $k$ descents, a problem recently brought up by Reifegerste. In both the hybrid and purely cyclic scenarios, we deal with the avoidance enumeration problem for all patterns of length up to 4. Simple Dyck paths also have a connection to the purely cyclic case; here the orbit-counting lemma gives a formula involving the Euler totient function and leads us to consider an interesting subgroup of the symmetric group.

On a conjecture of Carter concerning irreducible Specht modules

1978 ◽  
Vol 83 (1) ◽  
pp. 11-17 ◽  
Author(s):  
G. D. James
Keyword(s):  
Irreducible Representation ◽  
Symmetric Group ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Specht Module ◽  
Specht Modules ◽  
Definition Of

We study the question: Which ordinary irreducible representations of the symmetric group remain irreducible modulo a prime p?Let Sλ be the Specht module corresponding to the partition λ of n. The definition of Sλ is ‘independent of the field we are working over’. When the field has characteristic zero, Sλ is irreducible, and gives the ordinary irreducible representation of corresponding to the partition λ. Thus we are interested in the problem of whether or not Sλ is irreducible over a field of characteristic p.

scholarly journals On the Joint Distribution of Descents and Signs of Permutations

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Jason Fulman ◽  
Gene B. Kim ◽  
Sangchul Lee ◽  
T. Kyle Petersen
Keyword(s):  
Symmetric Group ◽  
Central Limit ◽  
Generating Functions ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Coxeter Groups ◽  
Central Limit Theorems ◽  
Hyperoctahedral Group ◽  
Eulerian Numbers

We study the joint distribution of descents and sign for elements of the symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and $B$). For both groups, this has an application to riffle shuffling: for large decks of cards the sign is close to random after a single shuffle. In both groups, we derive generating functions for the Eulerian distribution refined according to sign, and use them to give two proofs of central limit theorems for positive and negative Eulerian numbers.

scholarly journals Rational valued and real valued projective characters of finite groups

1980 ◽  
Vol 21 (1) ◽  
pp. 23-28 ◽  
Author(s):  
J. F. Humphreys
Keyword(s):  
Symmetric Group ◽  
Finite Groups ◽  
General Information ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complex Character ◽  
Definition Of ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Spin Characters

It is well-known [3; V.13.7] that each irreducible complex character of a finite group G is rational valued if and only if for each integer m coprime to the order of G and each g ∈ G, g is conjugate to gm. In particular, for each positive integer n, the symmetric group on n symbols, S(n), has all its irreducible characters rational valued. The situation for projective characters is quite different. In [5], Morris gives tables of the spin characters of S(n) for n ≤ 13 as well as general information about the values of these characters for any symmetric group. It can be seen from these results that in no case are all the spin characters of S(n) rational valued and, indeed, for n ≥ 6 these characters are not even all real valued. In section 2 of this note, we obtain a necessary and sufficient condition for each irreducible character of a group G associated with a 2-cocycle α to be rational valued. A corresponding result for real valued projective characters is discussed in section 3. Section 1 contains preliminary definitions and notation, including the definition of projective characters given in [2].

scholarly journals Systems with Failure-Dependent Lifetimes of Components

2009 ◽  
Vol 46 (4) ◽  
pp. 1052-1072 ◽  
Author(s):  
M. Burkschat
Keyword(s):  
Order Statistics ◽  
Joint Distribution ◽  
Coherent System ◽  
Coherent Systems ◽  
Sequential Order ◽  
Failure Times ◽  
Sequential Order Statistics ◽  
Definition Of ◽  
General Component

A model for describing the lifetimes of coherent systems, where the failures of components may have an impact on the lifetimes of the remaining components, is proposed. The model is motivated by the definition of sequential order statistics (cf. Kamps (1995)). Sequential order statistics describe the successive failure times in a sequential k-out-of-n system, where the distribution of the remaining components' lifetimes is allowed to change after every failure of a component. In the present paper, general component lifetimes which can be influenced by failures are considered. The ordered failure times of these components can be used to extend the concept of sequential order statistics. In particular, a definition of sequential order statistics based on exchangeable components is proposed. By utilizing the system signature (cf. Samaniego (2007)), the distribution of the lifetime of a coherent system with failure-dependent exchangeable component lifetimes is shown to be given by a mixture of the distributions of sequential order statistics. Furthermore, some results on the joint distribution of sequential order statistics based on exchangeable components are given.

scholarly journals Universal induced characters and restriction rules for the classical groups

1990 ◽  
Vol 117 ◽  
pp. 173-205 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Linear Group ◽  
General Linear Group ◽  
Hilbert Series ◽  
Point Of View ◽  
Classical Groups ◽  
Starting Point ◽  
Definition Of ◽  
Matrix Invariants

The purpose of this paper is the study of some basic properties of universal induced characters and their applications to the representation theory of the classical groups (for the definition of a universal induced character, see § 3).The starting point was the paper [F] by E. Formanek on matrix invariants. In his paper [F], Formanek has investigated the Hilbert series for the ring of matrix invariants from the point of view of the representation theory of the general linear group and the symmetric group. In this paper we shall study polynomial concomitants of a group from the same point of view.

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