scholarly journals Extensions of Partial Cyclic Orders, Euler Numbers and Multidimensional Boustrophedons

10.37236/7145 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sanjay Ramassamy
Keyword(s):  
Cyclic Order ◽  
Euler Numbers ◽  
Entringer Numbers

We enumerate total cyclic orders on $\left\{x_1,\ldots,x_n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(x_i,x_{i+1},x_{i+2})$, with indices taken modulo $n$. In some cases, the problem reduces to the enumeration of descent classes of permutations, which is done via the boustrophedon construction. In other cases, we solve the question by introducing multidimensional versions of the boustrophedon. In particular we find new interpretations for the Euler up/down numbers and the Entringer numbers.

scholarly journals The Complete cd-Index of Boolean Lattices

10.37236/5100 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Neil J.Y. Fan ◽  
Liao He
Keyword(s):  
Coxeter Group ◽  
Nonnegative Integer ◽  
Boolean Lattice ◽  
Combinatorial Interpretation ◽  
Euler Numbers ◽  
Combinatorial Interpretations ◽  
Bruhat Graph ◽  
Boolean Lattices ◽  
One To One ◽  
Entringer Numbers

Let $[u,v]$ be a Bruhat interval of a Coxeter group such that the Bruhat graph $BG(u,v)$ of $[u,v]$ is isomorphic to a Boolean lattice. In this paper, we provide a combinatorial explanation for the coefficients of the complete cd-index of $[u,v]$. Since in this case the complete cd-index and the cd-index of $[u,v]$ coincide, we also obtain a new combinatorial interpretation for the coefficients of the cd-index of Boolean lattices. To this end, we label an edge in $BG(u,v)$ by a pair of nonnegative integers and show that there is a one-to-one correspondence between such sequences of nonnegative integer pairs and Bruhat paths in $BG(u,v)$. Based on this labeling, we construct a flip $\mathcal{F}$ on the set of Bruhat paths in $BG(u,v)$, which is an involution that changes the ascent-descent sequence of a path. Then we show that the flip $\mathcal{F}$ is compatible with any given reflection order and also satisfies the flip condition for any cd-monomial $M$. Thus by results of Karu, the coefficient of $M$ enumerates certain Bruhat paths in $BG(u,v)$, and so can be interpreted as the number of certain sequences of nonnegative integer pairs. Moreover, we give two applications of the flip $\mathcal{F}$. We enumerate the number of cd-monomials in the complete cd-index of $[u,v]$ in terms of Entringer numbers, which are refined enumerations of Euler numbers. We also give a refined enumeration of the coefficient of d${}^n$ in terms of Poupard numbers, and so obtain new combinatorial interpretations for Poupard numbers and reduced tangent numbers.

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

scholarly journals EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

2021 ◽  
pp. 1-66
Author(s):  
Tom Bachmann ◽  
Kirsten Wickelgren
Keyword(s):  
Commutative Algebra ◽  
Vector Bundles ◽  
Euler Class ◽  
Complete Intersections ◽  
Bilinear Forms ◽  
Euler Numbers ◽  
Coherent Sheaves ◽  
Linear Subspaces ◽  
Algebraic Vector

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

An Algorithm to Slice 3-D Shapes for Reconstruction in Prototyping Systems

1991 ◽  
Author(s):  
K. L. Chalasani ◽  
B. Grogan ◽  
A. Bagchi ◽  
C. C. Jara-Almonte ◽  
A. A. Ogale ◽  
...  
Keyword(s):  
Data Structure ◽  
Rapid Prototyping ◽  
Surface Finish ◽  
Cyclic Order ◽  
Contour Tracing ◽  
Building Process ◽  
Dense Grid ◽  
A Chain ◽  
Definition Of

Abstract Rapid Prototyping (RP) processes reduce the time consumed in the manufacture of a prototype by producing parts directly from a CAD representation, without tooling. The StereoLithography Apparatus (SLA), and most other recent RP processes build a 3-D object from 2.5-D layers. Slicing is the process of defining layers to be built by the system. In this paper a framework is proposed for the development of algorithms for the representation and definition of layers for use in the SLA, with a view to determine if the slicing algorithms will affect surface finish in any significant manner. Currently, it is not possible to automatically vary slice thicknesses within the same object, using the existent algorithm. Also, it would be useful to use a dense grid for hatching or skin filling any given layer, or to change the hatch-pattern if desired. In addition, simulation of the layered building process would be helpful, so that the user can prespecify parameters that need to be varied during the process. The proposed framework incorporates these and other features. Two approaches for determining contours on each slice are suggested and their implementation is discussed. In the first, the layers are defined by the intersections of a plane with the surfaces defining the object. The plane is moved up from the base of the object as it is being built in increments. All intersections found are stored in a data structure, and sorted in head to tail fashion to define a contour for all closed areas on a layer. The second approach uses a scanline-type search to look for an intersection that will trigger a contour-tracing procedure. The contour-tracer is invoked whenever an unused edge is found in the search. This saves storage and sorting times, because the contour is determined as a chain of edges, in cyclic order. It is envisaged that results of this work on the SLA can be applied to other RP processes entailing layered building.

scholarly journals Some Identities Involving the Fubini Polynomials and Euler Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 300 ◽  
Author(s):  
Guohui Chen ◽  
Li Chen
Keyword(s):  
Power Series ◽  
Second Order ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Non Linear ◽  
Combinatorial Methods

In this paper, we first introduce a new second-order non-linear recursive polynomials U h , i ( x ) , and then use these recursive polynomials, the properties of the power series and the combinatorial methods to prove some identities involving the Fubini polynomials, Euler polynomials and Euler numbers.

scholarly journals Some Symmetric Identities Involving Fubini Polynomials and Euler Numbers

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 303 ◽  
Author(s):  
Zhao Jianhong ◽  
Chen Zhuoyu
Keyword(s):  
Calculation Method ◽  
Computational Formula ◽  
Euler Numbers ◽  
Special Sequence ◽  
Computational Problem ◽  
Elementary Methods ◽  
Symmetric Identities ◽  
Recursive Calculation

The aim of this paper is to use elementary methods and the recursive properties of a special sequence to study the computational problem of one kind symmetric sums involving Fubini polynomials and Euler numbers, and give an interesting computational formula for it. At the same time, we also give a recursive calculation method for the general case.

Sign in / Sign up

Export Citation Format

Share Document