scholarly journals Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Cyclic Groups ◽  
Bijective Proof ◽  
Avoidance Condition

We continue the study of the generalized pattern avoidance condition for Ck≀Sn, the wreath product of the cyclic group Ck with the symmetric group Sn, initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers.

Characteristics of GWrSn

1976 ◽  
Vol 79 (3) ◽  
pp. 433-441
Author(s):  
A. G. Williams
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Wreath Product ◽  
Generating Functions ◽  
Block Structure ◽  
Clifford Theory ◽  
A Finite Group ◽  
The Relationship

The ‘characteristics’ of the wreath product GWrSn, where G is a finite group, are certain polynomials (to be defined in section 2) which are generating functions for the simple characters of GWrSn. Schur (8) first used characteristics of the symmetric group. Specht (9) defined characteristics for GWrSn and found a relation between the characteristics of GWrSn and those of Sn which determined the simple characters of GWrSn. The object of this paper is to describe the p-block structure of GWrSn in the case where p is not a factor of the order of G. We use the relationship between the characteristics of GWrSn and those of Sn, which we deduce from a knowledge of the simple characters of GWrSn (these can be determined, independently of Specht's work, by using Clifford theory).

COUNTING ELEMENTS IN THE SYMMETRIC GROUP

2009 ◽  
Vol 19 (03) ◽  
pp. 305-313 ◽  
Author(s):  
DAVID EL-CHAI BEN-EZRA
Keyword(s):  
Symmetric Group ◽  
Finite Groups ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Combinatorial Identities ◽  
Subgroup Growth

By using simple ideas from subgroup growth of pro-finite groups we deduce some combinatorial identities on generating functions counting various elements in symmetric groups.

scholarly journals Pattern Avoidance in Matchings and Partitions

10.37236/2976 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Jonathan Bloom ◽  
Sergi Elizalde
Keyword(s):  
Fixed Points ◽  
Generating Functions ◽  
Direct Proof ◽  
Pattern Avoidance ◽  
Equivalence Classes ◽  
Set Partitions ◽  
Bijective Proof ◽  
Rook Placements ◽  
Wilf Equivalence ◽  
Diagram Representation

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize $3$-crossings and $3$-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards.We enumerate $312$-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the $321$-avoiding (i.e., $3$-noncrossing) case. Our approach provides a more direct proof of a formula of Bóna for the number of $1342$-avoiding permutations. We also give a bijective proof of the shape-Wilf-equivalence of the patterns $321$ and $213$ which greatly simplifies existing proofs by Backelin-West-Xin and Jelínek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.

scholarly journals Cyclic Derangements

10.37236/435 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Sami H. Assaf
Keyword(s):  
Fixed Point ◽  
Symmetric Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Enumerative Combinatorics ◽  
Finite Cyclic Group ◽  
Classic Problem

A classic problem in enumerative combinatorics is to count the number of derangements, that is, permutations with no fixed point. Inspired by a recent generalization to facet derangements of the hypercube by Gordon and McMahon, we generalize this problem to enumerating derangements in the wreath product of any finite cyclic group with the symmetric group. We also give $q$- and $(q,t)$-analogs for cyclic derangements, generalizing results of Gessel, Brenti and Chow.

scholarly journals Gelfand Pairs Involving the Wreath Product of Finite Abelian Groups with Symmetric Groups

2020 ◽  
pp. 1-7
Author(s):  
Omar Tout
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Wreath Product ◽  
Abelian Groups ◽  
Symmetric Groups ◽  
Gelfand Pair ◽  
Gelfand Pairs ◽  
Finite Abelian Groups ◽  
A Finite Group

Abstract It is well known that the pair $(\mathcal {S}_n,\mathcal {S}_{n-1})$ is a Gelfand pair where $\mathcal {S}_n$ is the symmetric group on n elements. In this paper, we prove that if G is a finite group then $(G\wr \mathcal {S}_n, G\wr \mathcal {S}_{n-1}),$ where $G\wr \mathcal {S}_n$ is the wreath product of G by $\mathcal {S}_n,$ is a Gelfand pair if and only if G is abelian.

scholarly journals $\delta$-Connectivity in Random Lifts of Graphs

10.37236/6639 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Shashwat Silas
Keyword(s):  
Lower Bound ◽  
Symmetric Group ◽  
Wreath Product ◽  
Connected Graph ◽  
Minimum Degree ◽  
Symmetric Groups ◽  
Gamma Delta ◽  
Random Generators

Amit and Linial have shown that a random lift of a connected graph with minimum degree $\delta\ge3$ is asymptotically almost surely (a.a.s.) $\delta$-connected and mentioned the problem of estimating this probability as a function of the degree of the lift. Using a connection between a random $n$-lift of a graph and a randomly generated subgroup of the symmetric group on $n$-elements, we show that this probability is at least  $1 - O\left(\frac{1}{n^{\gamma(\delta)}}\right)$ where $\gamma(\delta)>0$ for $\delta\ge 5$ and it is strictly increasing with $\delta$. We extend this to show that one may allow $\delta$ to grow slowly as a function of the degree of the lift and the number of vertices and still obtain that random lifts are a.a.s. $\delta$-connected. We also simplify a later result showing a lower bound on the edge expansion of random lifts. On a related note, we calculate the probability that a subgroup of a wreath product of symmetric groups generated by random generators is transitive, extending a well known result of Dixon which covers the case for subgroups of the symmetric group.

scholarly journals BEING CAYLEY AUTOMATIC IS CLOSED UNDER TAKING WREATH PRODUCT WITH VIRTUALLY CYCLIC GROUPS

2021 ◽  
pp. 1-11
Author(s):  
DMITRY BERDINSKY ◽  
MURRAY ELDER ◽  
JENNIFER TABACK
Keyword(s):  
Computer Science ◽  
Wreath Product ◽  
Cyclic Group ◽  
Wreath Products ◽  
Infinite Cyclic Group ◽  
Cyclic Groups ◽  
Automatic Groups

Abstract We extend work of Berdinsky and Khoussainov [‘Cayley automatic representations of wreath products’, International Journal of Foundations of Computer Science27(2) (2016), 147–159] to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.

scholarly journals A Novel Technique for the Construction of Safe Substitution Boxes Based on Cyclic and Symmetric Groups

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Abdul Razaq ◽  
Hanan A. Al-Olayan ◽  
Atta Ullah ◽  
Arshad Riaz ◽  
Adil Waheed
Keyword(s):  
Symmetric Group ◽  
Cyclic Group ◽  
Symmetric Groups ◽  
Galois Field ◽  
Substitution Box ◽  
Novel Technique ◽  
Initial Sequence ◽  
Substitution Boxes ◽  
The Matrix ◽  
Better Than

In the literature, different algebraic techniques have been applied on Galois field GF(28) to construct substitution boxes. In this paper, instead of Galois field GF(28), we use a cyclic group C255 in the formation of proposed substitution box. The construction proposed S-box involves three simple steps. In the first step, we introduce a special type of transformation T of order 255 to generate C255. Next, we adjoin 0 to C255 and write the elements of C255∪0 in 16×16 matrix to destroy the initial sequence 0,1,2,…,255. In the 2nd step, the randomness in the data is increased by applying certain permutations of the symmetric group S16 on rows and columns of the matrix. In the last step we consider the symmetric group S256, and positions of the elements of the matrix obtained in step 2 are changed by its certain permutations to construct the suggested S-box. The strength of our S-box to work against cryptanalysis is checked through various tests. The results are then compared with the famous S-boxes. The comparison shows that the ability of our S-box to create confusion is better than most of the famous S-boxes.

scholarly journals The center of the wreath product of symmetric group algebras

2021 ◽  
Vol 31 (2) ◽  
pp. 302-322
Author(s):  
O. Tout ◽  
Keyword(s):  
Symmetric Group ◽  
Recent Result ◽  
Wreath Product ◽  
Group Algebra ◽  
Symmetric Groups ◽  
Conjugacy Classes ◽  
Group Algebras ◽  
Hyperoctahedral Group ◽  
Symmetric Group Algebras

We consider the wreath product of two symmetric groups as a group of blocks permutations and we study its conjugacy classes. We give a polynomiality property for the structure coefficients of the center of the wreath product of symmetric group algebras. This allows us to recover an old result of Farahat and Higman about the polynomiality of the structure coefficients of the center of the symmetric group algebra and to generalize our recent result about the polynomiality property of the structure coefficients of the center of the hyperoctahedral group algebra.

Unitary Representations of Generalized Symmetric Groups

1969 ◽  
Vol 21 ◽  
pp. 28-38 ◽  
Author(s):  
B. M. Puttaswamaiah
Keyword(s):  
Normal Subgroup ◽  
Symmetric Group ◽  
Permutation Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Symmetric Groups ◽  
Unitary Representations ◽  
Complex Field ◽  
Permutation Matrices ◽  
Image Position

In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute withObviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) andthen {c1, c2, …, cn} generate a normal subgroup Q(n) of order mn and {s1, s2, …, sn…1} generate a subgroup S(n) isomorphic to S(n, 1).

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