scholarly journals Signed Excedance Enumeration in the Hyperoctahedral group

10.37236/3544 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Sivaramakrishnan Sivasubramanian
Keyword(s):  
Symmetric Group ◽  
Generating Functions ◽  
Hyperoctahedral Group

Several signed excedance-like statistics have nice formulae or generating functions when summed over the symmetric group and over its subset of derangements.  We give counterparts of some of these results when we sum over the hyperoctahedral group and its subset of derangements.  Our results motivate us to define and derive attractive bivariate formulae which generalise some of these results for the symmetric group.

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.

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 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.

scholarly journals Generating Functions for Permutations which Contain a Given Descent Set

10.37236/299 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Generating Functions ◽  
Symmetric Function ◽  
Schur Functions ◽  
Permutation Statistics ◽  
Finite Set ◽  
Set Of Permutations ◽  
Descent Set ◽  
Positive Integers

A large number of generating functions for permutation statistics can be obtained by applying homomorphisms to simple symmetric function identities. In particular, a large number of generating functions involving the number of descents of a permutation $\sigma$, $des(\sigma)$, arise in this way. For any given finite set $S$ of positive integers, we develop a method to produce similar generating functions for the set of permutations of the symmetric group $S_n$ whose descent set contains $S$. Our method will be to apply certain homomorphisms to symmetric function identities involving ribbon Schur functions.

scholarly journals Combinatorial Gelfand Models

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Ron M. Adin ◽  
Alex Postnikov ◽  
Yuval Roichman
Keyword(s):  
Symmetric Group ◽  
Hecke Algebra ◽  
Hyperoctahedral Group ◽  
International Audience ◽  
Combinatorial Construction

International audience A combinatorial construction of Gelfand models for the symmetric group, for its Iwahori-Hecke algebra and for the hyperoctahedral group is presented.

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.

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.

scholarly journals Transitive Factorizations in the Hyperoctahedral Group

2008 ◽  
Vol 60 (2) ◽  
pp. 297-312
Author(s):  
G. Bini ◽  
I. P. Goulden ◽  
D. M. Jackson
Keyword(s):  
Symmetric Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Point Of View ◽  
Moduli Space Of Curves ◽  
Reflection Groups ◽  
Hurwitz Numbers ◽  
Hyperoctahedral Group ◽  
Geometric Point ◽  
Enumeration Problem

AbstractThe classical Hurwitz enumeration problem has a presentation in terms of transitive factorizations in the symmetric group. This presentation suggests a generalization from type A to other finite reflection groups and, in particular, to type B. We study this generalization both from a combinatorial and a geometric point of view, with the prospect of providing a means of understanding more of the structure of the moduli spaces of maps with an S2-symmetry. The type A case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting for the type B case that is studied here.

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