scholarly journals The $1/k$-Eulerian Polynomials of Type $B$

10.37236/9313 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Shi-Mei Ma ◽  
Jun Ma ◽  
Jean Yeh ◽  
Yeong-Nan Yeh
Keyword(s):  
Recurrence Relations ◽  
Type B ◽  
Eulerian Polynomials ◽  
Combinatorial Interpretations ◽  
Symmetric Decomposition

In this paper, we define the $1/k$-Eulerian polynomials of type $B$. Properties of these polynomials, including combinatorial interpretations, recurrence relations and $\gamma$-positivity are studied. In particular, we show that the $1/k$-Eulerian polynomials of type $B$ are $\gamma$-positive when $k>0$. Moreover, we define the $1/k$-derangement polynomials of type $B$, denoted $d_n^B(x;k)$. We show that the polynomials $d_n^B(x;k)$ are bi-$\gamma$-positive when $k\geq 1/2$. In particular, we get a symmetric decomposition of the polynomials $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.

scholarly journals Derangement Polynomials and Excedances of Type $B$

10.37236/81 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
William Y. C. Chen ◽  
Robert L. Tang ◽  
Alina F. Y. Zhao
Keyword(s):  
Normal Distribution ◽  
Generating Function ◽  
Type B ◽  
Eulerian Polynomials ◽  
Basic Properties ◽  
Asymptotic Normal Distribution ◽  
Asymptotic Normal

Based on the notion of excedances of type $B$ introduced by Brenti, we give a type $B$ analogue of the derangement polynomials. The connection between the derangement polynomials and Eulerian polynomials naturally extends to the type $B$ case. Using this relation, we derive some basic properties of the derangement polynomials of type $B$, including the generating function formula, the Sturm sequence property, and the asymptotic normal distribution. We also show that the derangement polynomials are almost symmetric in the sense that the coefficients possess the spiral property.

scholarly journals Gamma Positivity of the Descent Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups

10.37236/9037 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Hiranya Kishore Dey ◽  
Sivaramakrishnan Sivasubramanian
Keyword(s):  
Coxeter Groups ◽  
Weyl Groups ◽  
Alternating Group ◽  
Type B ◽  
Eulerian Polynomial ◽  
Eulerian Polynomials ◽  
Type D ◽  
Positive Elements

The Eulerian polynomial $A_n(t)$ enumerating descents in $\mathfrak{S}_n$ is known to be gamma positive for all $n$. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also known to be gamma positive for all $n$. We consider $A_n^+(t)$ and $A_n^-(t)$, the polynomials which enumerate descents in the alternating group $\mathcal{A}_n$ and in $\mathfrak{S}_n - \mathcal{A}_n$ respectively.  We show the following results about $A_n^+(t)$ and $A_n^-(t)$: both polynomials are gamma positive iff $n \equiv 0,1$ (mod 4). When $n \equiv 2,3$ (mod 4), both polynomials are not palindromic. When $n \equiv 2$ (mod 4), we show that {\sl two} gamma positive summands add up to give $A_n^+(t)$ and $A_n^-(t)$. When $n \equiv 3$ (mod 4), we show that {\sl three} gamma positive summands add up to give both $A_n^+(t)$ and $A_n^-(t)$.  We show similar gamma positivity results about the descent based type B and type D Eulerian polynomials when enumeration is done over the positive elements in the respective Coxeter groups. We also show that the polynomials considered in this work are unimodal.

scholarly journals Multifarious Results for q-Hermite Based Frobenius Type Eulerian Polynomials

2019 ◽  
Author(s):  
Waseem Khan ◽  
Idrees Ahmad Khan ◽  
Mehmet Acikgoz ◽  
Ugur Duran
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Series Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Eulerian Polynomials ◽  
New Class ◽  
Genocchi Polynomials

In this paper, a new class of q-Hermite based Frobenius type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.

scholarly journals Les polynômes eul\IeC èriens stables de type B

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Mirkó Visontai ◽  
Nathan Williams
Keyword(s):  
Linear Operator ◽  
Type B ◽  
Multivariate Polynomial ◽  
Eulerian Polynomial ◽  
Eulerian Polynomials ◽  
Signed Permutations ◽  
International Audience ◽  
Multiple Variables ◽  
Real Stability

International audience We give a multivariate analog of the type B Eulerian polynomial introduced by Brenti. We prove that this multivariate polynomial is stable generalizing Brenti's result that every root of the type B Eulerian polynomial is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability—a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Nous prèsentons un raffinement multivariè d'un polynôme eulèrien de type B dèfini par Brenti. En prouvant que ce polynôme est stable nous gènèralisons un rèsultat de Brenti selon laquel chaque racine du polynôme eulèrien de type B est rèelle. Notre preuve combine un raffinement de la statistique des descentes pour les permutations signèes avec la stabilitè—une gènèralisation de la propriètè d'avoir uniquement des racines rèelles aux polynômes en plusieurs variables. La connexion est que nos polynômes eulèriens raffinès satisfont une rècurrence donnèe par un opèrateur linèaire qui prèserve la stabilitè.

scholarly journals Permutations of Type $B$ with Fixed Number of Descents and Minus Signs

10.37236/7306 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Katarzyna Kril ◽  
Wojciech Młotkowski
Keyword(s):  
Three Dimensional ◽  
Fixed Number ◽  
Type B ◽  
Dimensional Array ◽  
Combinatorial Interpretations

We study three dimensional array of numbers $B(n,k,j)$, $0\le j,k\le n$, where $B(n,k,j)$ is the number of type $B$ permutations of order $n$ with $k$ descents and $j$ minus signs. We prove in particular, that $b(n,k,j):=B(n,k,j)/\binom{n}{j}$ is an integer and provide two combinatorial interpretations for these numbers.

UNIMODALITY AND COLOURED HOOK FACTORISATION

2015 ◽  
Vol 93 (1) ◽  
pp. 1-12
Author(s):  
ZHICONG LIN
Keyword(s):  
Fixed Points ◽  
Continued Fractions ◽  
Permutation Groups ◽  
Combinatorial Proof ◽  
Type B ◽  
Eulerian Polynomials ◽  
Recurrence Formulas ◽  
Major Index

We prove the unimodality of some coloured$q$-Eulerian polynomials, which involve the flag excedances, the major index and the fixed points on coloured permutation groups, via two recurrence formulas. In particular, we confirm a recent conjecture of Mongelli about the unimodality of the flag excedances over type B derangements. Furthermore, we find the coloured version of Gessel’s hook factorisation, which enables us to interpret these two recurrences combinatorially. We also provide a combinatorial proof of a symmetric and unimodal expansion for the coloured derangement polynomial, which was first established by Shin and Zeng using continued fractions.

scholarly journals The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler Theorem

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Arthur L.B. Yang ◽  
Philip B. Zhang
Keyword(s):  
Linear Operators ◽  
Type B ◽  
The Real ◽  
Eulerian Polynomials ◽  
Hurwitz Stability ◽  
Type D ◽  
International Audience

International audience Based on the Hermite–Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $s$-Eulerian polynomials. We also confirm Hyatt’s conjectures on the inter-lacing property of half Eulerian polynomials. Borcea and Brändén’s work on the characterization of linear operators preserving Hurwitz stability is critical to this approach. Basé sur le théorème de Hermite–Biehler, nous prouvons simultanément les polynômes eulériens de type $D$ et les polynômes eulériens affine de type $B$ ont seulement racines réelle, qui sont d’abord obtenue par Savage et Visontai en utilisant le théorie des polynômes $s$-eulériens. Nous confirmons aussi les conjectures de Hyatt sur la propriété entrelacement de polynômes mi-eulériens. Le travail de Borcea et Brändén sur la caractérisation des opérateurs linéaires préservant la stabilité Hurwitz est essentielle à cette approche.

scholarly journals Eulerian polynomials of type $D$ have only real roots

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Carla D. Savage ◽  
Mirkó Visontai
Keyword(s):  
Type B ◽  
Eulerian Polynomial ◽  
Eulerian Polynomials ◽  
Real Roots ◽  
Type D ◽  
International Audience

International audience We give an intrinsic proof of a conjecture of Brenti that all the roots of the Eulerian polynomial of type $D$ are real and a proof of a conjecture of Dilks, Petersen, and Stembridge that all the roots of the affine Eulerian polynomial of type $B$ are real, as well. Nous prouvons, de façon intrinsèque, une conjecture de Brenti affirmant que toutes les racines du polynôme eulérien de type $D$ sont réelles. Nous prouvons également une conjecture de Dilks, Petersen, et Stembridge que toutes les racines du polynôme eulérien affine de type $B$ sont réelles.

Sign in / Sign up

Export Citation Format

Share Document