scholarly journals The Cyclic Sieving Phenomenon on Circular Dyck Paths

10.37236/8720 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Per Alexandersson ◽  
Svante Linusson ◽  
Samu Potka
Keyword(s):  
Cyclic Group ◽  
Catalan Numbers ◽  
Natural Action ◽  
Cyclic Sieving Phenomenon ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
Cyclic Sieving

We give a $q$-enumeration of circular Dyck paths, which is a superset of the classical Dyck paths enumerated by the Catalan numbers. These objects have recently been studied by Alexandersson and Panova. Furthermore, we show that this $q$-analogue exhibits the cyclic sieving phenomenon under a natural action of the cyclic group. The enumeration and cyclic sieving is generalized to Möbius paths. We also discuss properties of a generalization of cyclic sieving, which we call subset cyclic sieving, and introduce the notion of Lyndon-like cyclic sieving that concerns special recursive properties of combinatorial objects exhibiting the cyclic sieving phenomenon.

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 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 Maximal Fillings of Moon Polyominoes, Simplicial Complexes, and Schubert Polynomials

10.37236/1167 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Luis Serrano ◽  
Christian Stump
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Canonical Connection ◽  
Cyclic Sieving Phenomenon ◽  
North East ◽  
Schubert Polynomials ◽  
Dyck Paths ◽  
Vertex Decomposable ◽  
Cyclic Sieving

We exhibit a canonical connection between maximal $(0,1)$-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we  show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials. Moreover, for Ferrers shapes we construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between $k$-triangulations of the $n$-gon and $k$-fans of Dyck paths of length $2(n-2k)$. Using this, we translate a conjectured cyclic sieving phenomenon for $k$-triangulations with rotation to the language of $k$-flagged tableaux with promotion.

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.

scholarly journals Generalized triangulations, pipe dreams, and simplicial spheres

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Luis Serrano ◽  
Christian Stump
Keyword(s):  
Simplicial Complex ◽  
Canonical Connection ◽  
Cyclic Sieving Phenomenon ◽  
North East ◽  
Schubert Polynomials ◽  
Dyck Paths ◽  
Nous Montrons ◽  
International Audience ◽  
Vertex Decomposable ◽  
Cyclic Sieving

International audience We exhibit a canonical connection between maximal $(0,1)$-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable and thus a shellable sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between $k$-triangulations of the $n$-gon and $k$-fans of Dyck paths. Using this, we translate a conjectured cyclic sieving phenomenon for $k$-triangulations with rotation to $k$-flagged tableaux with promotion. Nous décrivons un lien canonique entre les $(0,1)$-remplissages maximaux d'un polyomino-lune évitant les chaînes Nord-Est d'une longueur donnée, et les "pipe dreams'' réduits d'une certaine permutation. En suivant cette approche nous montrons que le complexe simplicial de tels remplissages maximaux est une sphère "vertex-decomposable'' et donc "shellable''. En particulier, cela entraîne un résultat de positivité sur les polynômes de Schubert. De plus, nous construisons, dans le cas des diagrammes de Ferrers, une bijection vers les remplissages maximaux évitant les chaînes Sud-Est de même longueur, qui se spécialise en une bijection entre les $k$-triangulations d'un $n$-gone et les $k$-faisceaux de chemins de Dyck. A l'aide de celle-ci, nous traduisons une instance conjecturale du phénomène de tamis cyclique pour les $k$-triangulations avec rotation dans le cadre des tableaux $k$-marqués avec promotion.

scholarly journals Invariant Tensors and the Cyclic Sieving Phenomenon

10.37236/4569 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Bruce W. Westbury
Keyword(s):  
Lie Algebras ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Natural Action ◽  
Rectangular Shape ◽  
Cyclic Sieving Phenomenon ◽  
Finite Dimensional ◽  
Cyclic Sieving ◽  
Fake Degree ◽  
Invariant Tensors

We construct a large class of examples of the cyclic sieving phenomenon by exploiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the associated Kashiwara crystal. For $r\ge 0$, the triple $(X,c,P)$ which exhibits the cyclic sieving phenomenon is constructed as follows: the set $X$ is the set of isolated vertices in the crystal $\otimes^rB$; the map $c\colon X\rightarrow X$ is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial $P$ is the fake degree of the Frobenius character of a representation of $\mathfrak{S}_r$ related to the natural action of $\mathfrak{S}_r$ on the subspace of invariant tensors in $\otimes^rM$. Taking $M$ to be the defining representation of $\mathrm{SL}(n)$ gives the cyclic sieving phenomenon for rectangular tableaux.

scholarly journals Crystals, Semistandard Tableaux and Cyclic Sieving Phenomenon

10.37236/8802 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Young-Tak Oh ◽  
Euiyong Park
Keyword(s):  
Crystal Structure ◽  
Cyclic Group ◽  
Young Diagram ◽  
Young Tableaux ◽  
Cyclic Action ◽  
Cyclic Sieving Phenomenon ◽  
Cyclic Sieving

In this paper, we study a new cyclic sieving phenomenon on the set $\mathsf{SST}_n(\lambda)$ of semistandard Young tableaux with the cyclic action $\mathsf{c}$ arising from its $U_q(\mathfrak{sl}_n)$-crystal structure. We prove that if $\lambda$ is a Young diagram with $\ell(\lambda) < n$ and $\gcd( n, |\lambda| )=1$, then the triple $\left( \mathsf{SST}_n(\lambda), \mathsf{C}, q^{- \kappa(\lambda)} s_\lambda(1,q, \ldots, q^{n-1}) \right) $ exhibits the cyclic sieving phenomenon, where $\mathsf{C}$ is the cyclic group generated by $\mathsf{c}$. We further investigate a connection between $\mathsf{c}$ and the promotion $\mathsf{pr}$ and show the bicyclic sieving phenomenon given by $\mathsf{c}$ and $\mathsf{pr}^n$ for hook shape.

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 The cyclic sieving phenomenon: a survey

2011 ◽  
pp. 183-234 ◽  
Author(s):  
Bruce Sagan
Keyword(s):  
Cyclic Sieving Phenomenon ◽  
Cyclic Sieving
Sign in / Sign up

Export Citation Format

Share Document