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 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 Two Descent Statistics over 321-Avoiding Centrosymmetric Involutions

10.37236/5531 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Marilena Barnabei ◽  
Flavio Bonetti ◽  
Sergi Elizalde ◽  
Matteo Silimbani
Keyword(s):  
Symmetric Group ◽  
Binomial Coefficients ◽  
Young Diagrams ◽  
Hyperoctahedral Group ◽  
Major Index

Centrosymmetric involutions in the symmetric group ${\mathcal S}_{2n}$ are permutations $\pi$ such that $\pi=\pi^{-1}$ and $\pi(i)+\pi(2n+1-i)=2n+1$ for all $i$, and they are in bijection with involutions of the hyperoctahedral group. We describe the distribution of some natural descent statistics on $321$-avoiding centrosymmetric involutions, including the number of descents in the first half of the involution, and the sum of the positions of these descents. Our results are based on two new bijections, one betweencentrosymmetric involutions in ${\mathcal S}_{2n}$ and subsets of $\{1,\dots,n\}$, and another one showing that certain statistics on Young diagrams that fit inside a rectangle are equidistributed. We also use the latter bijection to refine a known result stating that the distribution of the major index on $321$-avoiding involutions is given by the $q$-analogue of the central binomial coefficients.

A Class of Trinomials with Galois Group Sn

2012 ◽  
Vol 19 (spec01) ◽  
pp. 905-911 ◽  
Author(s):  
Anuj Bishnoi ◽  
Sudesh K. Khanduja
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Galois Group ◽  
Rational Numbers ◽  
Alternating Group ◽  
Polynomial Formula

A well known result of Schur states that if n is a positive integer and a0, a1,…,an are arbitrary integers with a0an coprime to n!, then the polynomial [Formula: see text] is irreducible over the field ℚ of rational numbers. In case each ai = 1, it is known that the Galois group of fn(x) over ℚ contains An, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials fn(x) which leads to the construction of trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters.

scholarly journals Equivalences for pattern avoiding involutions and classification

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Mark Dukes ◽  
Vít Jelínek ◽  
Toufik Mansour ◽  
Astrid Reifegerste
Keyword(s):  
Symmetric Group ◽  
Signed Permutations ◽  
International Audience

International audience We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of $S_5$, $S_6$, and $S_7$ for both permutations and involutions. Nous complétons la classification de Wilf des motifs signés de longueur 5 à la fois pour les permutations signées et les involutions signées. Nous donnons de nouvelles équivalences générales de motifs qui prouvent les conjectures de Jaggard concernant les involutions dans le groupe symétrique évitant certains motifs de longueur 5 et 6. De cette manière nous complétons également la classification de Wilf de $S_5$, $S_6$ et $S_7$ à la fois pour les permutations et les involutions.

scholarly journals A Basis for the Diagonally Signed-Symmetric Polynomials

10.37236/3224 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
José Manuel Gómez
Keyword(s):  
Polynomial Ring ◽  
Symmetric Polynomials ◽  
Hyperoctahedral Group ◽  
Free Basis ◽  
Signed Permutations ◽  
Invariant Ring

Let $n\ge 1$ be an integer and let $B_{n}$ denote the hyperoctahedral group of rank $n$. The group $B_{n}$ acts on the polynomial ring $Q[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]$ by signed permutations simultaneously on both of the sets of variables $x_{1},\dots,x_{n}$ and $y_{1},\dots,y_{n}.$ The invariant ring $M^{B_{n}}:=Q[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]^{B_{n}}$  is the ring of diagonally signed-symmetric polynomials. In this article, we provide an explicit free basis of $M^{B_{n}}$ as a module over the ring of symmetric polynomials on both of the sets of variables $x_{1}^{2},\dots, x^{2}_{n}$ and  $y_{1}^{2},\dots, y^{2}_{n}$ using signed descent monomials.

scholarly journals Cyclic sieving for longest reduced words in the hyperoctahedral group

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
T. K. Petersen ◽  
L. Serrano
Keyword(s):  
Generating Function ◽  
Hyperoctahedral Group ◽  
Cyclic Action ◽  
Orbit Structure ◽  
Cyclic Sieving Phenomenon ◽  
Major Index ◽  
Nous Montrons ◽  
International Audience ◽  
Cyclic Sieving ◽  
Cette Action

International audience We show that the set $R(w_0)$ of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon. More specifically, $R(w_0)$ possesses a natural cyclic action given by moving the first letter of a word to the end, and we show that the orbit structure of this action is encoded by the generating function for the major index on $R(w_0)$. Nous montrons que l'ensemble $R(w_0)$ des expressions réduites pour l'élément le plus long du groupe hyperoctaédral présente le phénomène cyclique de tamisage. Plus précisément, $R(w_0)$ possède une action naturelle cyclique donnée par le déplacement de la première lettre d'un mot vers la fin, et nous montrons que la structure d'orbite de cette action est codée par la fonction génératrice pour l'indice majeur sur $R(w_0)$.

scholarly journals Cyclic Sieving for Longest Reduced Words in the Hyperoctahedral Group

10.37236/339 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
T. Kyle Petersen ◽  
Luis Serrano
Keyword(s):  
Generating Function ◽  
Hyperoctahedral Group ◽  
Cyclic Action ◽  
Orbit Structure ◽  
Cyclic Sieving Phenomenon ◽  
Major Index ◽  
Cyclic Sieving ◽  
Reduced Words

We show that the set $R(w_0)$ of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon. More specifically, $R(w_0)$ possesses a natural cyclic action given by moving the first letter of a word to the end, and we show that the orbit structure of this action is encoded by the generating function for the major index on $R(w_0)$.

Integral Cayley Graphs over Finite Groups

2020 ◽  
Vol 27 (01) ◽  
pp. 131-136
Author(s):  
Elena V. Konstantinova ◽  
Daria Lytkina
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Cyclic Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Generating Set ◽  
A Finite Group

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

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