scholarly journals Excedance Numbers for the Permutations of Type B

10.37236/2375 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Alina F.Y. Zhao
Keyword(s):  
Fixed Points ◽  
Generating Function ◽  
Type B ◽  
Euler Numbers ◽  
Combinatorial Construction

This work provides a study on the multidistribution of type $B$ excedances, fixed points and cycles on the permutations of type $B$. We derive the recurrences and closed formulas for the distribution of signed excedances on type $B$ permutations as well as derangements via combinatorial construction. Based on this result, we obtain the recurrence and generating function for the signed excedance polynomial and disclose some relationships with Euler numbers and Springer numbers, respectively.

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 Explicit Formulas Involving -Euler Numbers and Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Serkan Araci ◽  
Mehmet Acikgoz ◽  
Jong Jin Seo
Keyword(s):  
Generating Function ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Derivative Operator ◽  
Euler Numbers And Polynomials ◽  
Integer Ring

We deal with -Euler numbers and -Bernoulli numbers. We derive some interesting relations for -Euler numbers and polynomials by using their generating function and derivative operator. Also, we derive relations between the -Euler numbers and -Bernoulli numbers via the -adic -integral in the -adic integer ring.

scholarly journals Cyclic Sieving for Plane Partitions and Symmetry

2020 ◽  
Author(s):  
Sam Hopkins ◽  
Keyword(s):  
Fixed Points ◽  
Generating Function ◽  
Cyclic Group ◽  
Product Formula ◽  
Roots Of Unity ◽  
Plane Partitions ◽  
Cyclic Sieving Phenomenon ◽  
Combinatorial Set ◽  
Symmetric Plane ◽  
Cyclic Sieving

The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.

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 Some Properties of a Sequence Similar to Generalized Euler Numbers

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Haiqing Wang ◽  
Guodong Liu
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Central Factorial Numbers

We introduce the sequence {Un(x)} given by generating function (1/(et+e-t-1))x=∑n=0∞Un(x)(tn/n!)  (|t|<(1/3)π,1x:=1) and establish some explicit formulas for the sequence {Un(x)}. Several identities involving the sequence {Un(x)}, Stirling numbers, Euler polynomials, and the central factorial numbers are also presented.

scholarly journals On colored set partitions of type B n

2014 ◽  
Vol 12 (9) ◽  
Author(s):  
David Wang
Keyword(s):  
Generating Function ◽  
Asymptotic Expression ◽  
Type B ◽  
Set Partitions ◽  
Asymptotic Normal

AbstractGeneralizing Reiner’s notion of set partitions of type B n, we define colored B n-partitions by coloring the elements in and not in the zero-block respectively. Considering the generating function of colored B n-partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored B n-partition. We find an asymptotic expression of the total number of colored B n-partitions up to an error of O(n −1/2log7/2 n], and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored B n-partitions.

The case for a West Saxon minuscule

1997 ◽  
Vol 26 ◽  
pp. 63-79 ◽  
Author(s):  
Julia Crick
Keyword(s):  
Phase I ◽  
Fixed Points ◽  
Phase Ii ◽  
Type A ◽  
Viking Age ◽  
British Isles ◽  
Type B

Julian Brown's famous analysis of what he termed the Insular system of scripts marked out a number of routes, now well trodden, through the debris of undated and unlocalized manuscript material from the pre-Viking-Age British Isles. Ever since, the best hope for students of palaeography seeking to date and localize examples of early Insular minuscule has been to follow Brown's classification and identify them as Type A or B, Northumbrian or Southumbrian, and Phase I or II. Brown's schema, however, offered orientation rather than a map. As with any typology, it depends on a very few fixed points, themselves unusual because of their lack of anonymity: gospelbooks from Ireland and Northumbria dated by the survival of rare colophons, manuscripts connected with St Boniface which show the operation of a unique editorial mind. Although Brown's system has been successfully applied to the output of scriptoria whose influences, practices, connections, even locations remain mostly unknown, complications inevitably arise. This article concerns one of them, the recycling in Phase II of a type of minuscule displaying the cursiveness and capriciousness characteristic of Phase I: Type B minuscule as illustrated by the script of St Boniface.

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