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 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 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.

A NEW CONGRUENCE MODULO 25 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

2014 ◽  
Vol 91 (1) ◽  
pp. 41-46 ◽  
Author(s):  
ERNEST X. W. XIA
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Plane Partitions ◽  
Arithmetic Properties ◽  
Symmetric Plane ◽  
Congruence Modulo

AbstractFor any positive integer $n$, let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of $n$. Recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.89 (2014), 473–478] and Yao [‘New infinite families of congruences modulo 4 and 8 for 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.90 (2014), 37–46] proved a number of congruences satisfied by $f(n)$. In particular, Hirschhorn and Sellers proved that $f(10n+5)\equiv 0\ (\text{mod}\ 5)$. In this paper, we establish the generating function of $f(30n+25)$ and prove that $f(250n+125)\equiv 0\ (\text{mod\ 25}).$

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 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.

scholarly journals Cyclic Sieving for Longest Reduced Words in the Hyperoctahedral Group

10.37236/339 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
T. Kyle Petersen ◽  
Luis Serrano
Keyword(s):  
Generating Function ◽  
Hyperoctahedral Group ◽  
Cyclic Action ◽  
Orbit Structure ◽  
Cyclic Sieving Phenomenon ◽  
Major Index ◽  
Cyclic Sieving ◽  
Reduced Words

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 $R(w_0)$.

scholarly journals Cyclic Sieving Phenomenon in Non-Crossing Connected Graphs

10.37236/496 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Alan Guo
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Connected Graphs ◽  
Cyclic Sieving Phenomenon ◽  
Cyclic Sieving

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.

scholarly journals ARITHMETIC PROPERTIES OF 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

2013 ◽  
Vol 89 (3) ◽  
pp. 473-478 ◽  
Author(s):  
MICHAEL D. HIRSCHHORN ◽  
JAMES A. SELLERS
Keyword(s):  
Generating Function ◽  
Main Diagonal ◽  
Plane Partitions ◽  
Combinatorial Objects ◽  
Arithmetic Properties ◽  
Symmetric Plane ◽  
Image Position

AbstractBlecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’,Util. Math. 88(2012), 223–235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight$n$. He also proved that the generating function for$f(n), $the number of 1-shell totally symmetric plane partitions of weight$n$, is given by$$\begin{eqnarray*}\displaystyle \sum _{n\geq 0}f(n){q}^{n} = 1+ \sum _{n\geq 1}{q}^{3n- 2} \prod _{i= 0}^{n- 2} (1+ {q}^{6i+ 3} ).\end{eqnarray*}$$In this brief note, we prove a number of arithmetic properties satisfied by$f(n)$using elementary generating function manipulations and well-known results of Ramanujan and Watson.

scholarly journals The poset perspective on alternating sign matrices

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jessica Striker
Keyword(s):  
Generating Function ◽  
Young Tableaux ◽  
Alternating Sign Matrices ◽  
Plane Partitions ◽  
Combinatorial Objects ◽  
Symmetric Plane ◽  
International Audience ◽  
Order Ideals ◽  
Nous Utilisons

International audience Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

Sign in / Sign up

Export Citation Format

Share Document