scholarly journals The Cyclic Sieving Phenomenon for Faces of Cyclic Polytopes

10.37236/319 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Yeh-Jong Pan
Keyword(s):  
Convex Hull ◽  
Group Action ◽  
Convex Polytope ◽  
Moment Curve ◽  
Cyclic Polytope ◽  
Cyclic Polytopes ◽  
Cyclic Sieving Phenomenon ◽  
Cyclic Sieving

A cyclic polytope of dimension $d$ with $n$ vertices is a convex polytope combinatorially equivalent to the convex hull of $n$ distinct points on a moment curve in ${\Bbb R}^d$. In this paper, we prove the cyclic sieving phenomenon, introduced by Reiner-Stanton-White, for faces of an even-dimensional cyclic polytope, under a group action that cyclically translates the vertices. For odd-dimensional cyclic polytopes, we enumerate the faces that are invariant under an automorphism that reverses the order of the vertices and an automorphism that interchanges the two end vertices, according to the order on the curve. In particular, for $n=d+2$, we give instances of the phenomenon under the groups that cyclically translate the odd-positioned and even-positioned vertices, respectively.

scholarly journals Calculus rules for combinations of ellipsoids and applications

1993 ◽  
Vol 47 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alberto Seeger
Keyword(s):  
Convex Hull ◽  
Convex Polytope ◽  
Finite Family ◽  
Minkowski Sum ◽  
Calculus Rules ◽  
Ellipsoidal Approximations

We derive formulas for the Minkowski sum, the convex hull, the intersection, and the inverse sum of a finite family of ellipsoids. We show how these formulas can be used to obtain inner and outer ellipsoidal approximations of a convex polytope.

scholarly journals The Cyclic Sieving Phenomenon for Non-Crossing Forests

10.37236/2419 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Stefan Kluge
Keyword(s):  
Cyclic Group ◽  
Cyclic Sieving Phenomenon ◽  
Cyclic Sieving

In this paper we prove that the set of non-crossing forests together with a cyclic group acting on it by rotation and a natural q-analogue of the formula for their number exhibits the cyclic sieving phenomenon, as conjectured by Alan Guo.

scholarly journals NORMAL CYCLIC POLYTOPES AND CYCLIC POLYTOPES THAT ARE NOT VERY AMPLE

2013 ◽  
Vol 96 (1) ◽  
pp. 61-77 ◽  
Author(s):  
TAKAYUKI HIBI ◽  
AKIHIRO HIGASHITANI ◽  
LUKAS KATTHÄN ◽  
RYOTA OKAZAKI
Keyword(s):  
Cyclic Polytope ◽  
Cyclic Polytopes ◽  
Positive Integers

AbstractLet $d$ and $n$ be positive integers such that $n\geq d+ 1$ and ${\tau }_{1} , \ldots , {\tau }_{n} $ integers such that ${\tau }_{1} \lt \cdots \lt {\tau }_{n} $. Let ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )\subset { \mathbb{R} }^{d} $ denote the cyclic polytope of dimension $d$ with $n$ vertices $({\tau }_{1} , { \tau }_{1}^{2} , \ldots , { \tau }_{1}^{d} ), \ldots , ({\tau }_{n} , { \tau }_{n}^{2} , \ldots , { \tau }_{n}^{d} )$. We are interested in finding the smallest integer ${\gamma }_{d} $ such that if ${\tau }_{i+ 1} - {\tau }_{i} \geq {\gamma }_{d} $ for $1\leq i\lt n$, then ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )$ is normal. One of the known results is ${\gamma }_{d} \leq d(d+ 1)$. In the present paper a new inequality ${\gamma }_{d} \leq {d}^{2} - 1$ is proved. Moreover, it is shown that if $d\geq 4$ with ${\tau }_{3} - {\tau }_{2} = 1$, then ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )$ is not very ample.

scholarly journals The cyclic sieving phenomenon

2004 ◽  
Vol 108 (1) ◽  
pp. 17-50 ◽  
Author(s):  
V. Reiner ◽  
D. Stanton ◽  
D. White
Keyword(s):  
Cyclic Sieving Phenomenon ◽  
Cyclic Sieving

scholarly journals q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

2019 ◽  
Vol 15 (09) ◽  
pp. 1919-1968 ◽  
Author(s):  
Ofir Gorodetsky
Keyword(s):  
Binomial Coefficients ◽  
Roots Of Unity ◽  
General Criterion ◽  
Lucas Numbers ◽  
Cyclic Sieving Phenomenon ◽  
Similar Formula ◽  
Apéry Numbers ◽  
Higher Derivatives ◽  
Cyclic Sieving

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding [Formula: see text]-supercongruence. Similar [Formula: see text]-supercongruences are established for binomial coefficients and the Apéry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the [Formula: see text]-Lucas numbers.

scholarly journals Cyclic sieving for two families of non-crossing graphs

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Svetlana Poznanović
Keyword(s):  
Cyclic Group ◽  
Orbit Structure ◽  
Cyclic Sieving Phenomenon ◽  
Nous Montrons ◽  
International Audience ◽  
Cyclic Sieving ◽  
Cette Action

International audience We prove the cyclic sieving phenomenon for non-crossing forests and non-crossing graphs. More precisely, the cyclic group acts on these graphs naturally by rotation and we show that the orbit structure of this action is encoded by certain polynomials. Our results confirm two conjectures of Alan Guo. Nous prouvons le phénomène de crible cyclique pour les forêts et les graphes sans croisement. Plus précisément, le groupe cyclique agit sur ces graphes naturellement par rotation et nous montrons que la structure d'orbite de cette action est codée par certains polynômes. Nos résultats confirment deux conjectures de Alan Guo.

scholarly journals The Negative $q$-Binomial

10.37236/2040 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Shishuo Fu ◽  
V. Reiner ◽  
Dennis Stanton ◽  
Nathaniel Thiem
Keyword(s):  
Binomial Coefficient ◽  
Cyclic Sieving Phenomenon ◽  
Cyclic Sieving

Interpretations for the $q$-binomial coefficient evaluated at $-q$ are discussed.  A $(q,t)$-version is established, including an instance of a cyclic sieving phenomenon involving unitary spaces.

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 phenomenon in non-crossing connected graphs

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Alan Guo
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Connected Graphs ◽  
Cyclic Sieving Phenomenon ◽  
International Audience ◽  
Cyclic Sieving

International audience A non-crossing connected graph is a connected graph on vertices arranged in a circle such that its edges do not cross. The count for such graphs can be made naturally into a q-binomial generating function. We prove that this generating function exhibits the cyclic sieving phenomenon, as conjectured by S.-P. Eu. Un graphe connexe dont les sommets sont disposés sur un cercle est sans croisement si ses arêtes ne se croisent pas. Nous démontrons une conjecture de S.-P. Eu affirmant que la fonction génératrice q-binomiale dénombrant de tels graphes exhibe le phénomène du crible cyclique.

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