scholarly journals Cyclic Sieving and Rational Catalan Theory

10.37236/5681 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Michelle Bodnar ◽  
Brendon Rhoades
Keyword(s):  
Parking Functions ◽  
Noncrossing Partitions ◽  
Cyclic Sieving Phenomenon ◽  
Cyclic Sieving ◽  
Positive Integers ◽  
Coprime Positive Integers

Let $a < b$ be coprime positive integers. Armstrong, Rhoades, and Williams (2013) defined a set NC(a,b) of `rational noncrossing partitions', which form a subset of the ordinary noncrossing partitions of $\{1, 2, \dots, b-1\}$.  Confirming a conjecture of Armstrong et. al., we prove that NC(a,b) is closed under rotation and prove an instance of the cyclic sieving phenomenon for this rotational action.  We also define a rational generalization of the $\mathfrak{S}_a$-noncrossing parking functions of Armstrong, Reiner, and Rhoades.

scholarly journals Parking Functions and Noncrossing Partitions

10.37236/1335 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Symmetric Function ◽  
Local Action ◽  
Parking Functions ◽  
Noncrossing Partitions ◽  
Parking Function ◽  
Noncrossing Partition ◽  
Increasing Rearrangement ◽  
Positive Integers

A parking function is a sequence $(a_1,\dots,a_n)$ of positive integers such that, if $b_1\leq b_2\leq \cdots\leq b_n$ is the increasing rearrangement of the sequence $(a_1,\dots, a_n),$ then $b_i\leq i$. A noncrossing partition of the set $[n]=\{1,2,\dots,n\}$ is a partition $\pi$ of the set $[n]$ with the property that if $a < b < c < d$ and some block $B$ of $\pi$ contains both $a$ and $c$, while some block $B'$ of $\pi$ contains both $b$ and $d$, then $B=B'$. We establish some connections between parking functions and noncrossing partitions. A generating function for the flag $f$-vector of the lattice NC$_{n+1}$ of noncrossing partitions of $[{\scriptstyle n+1}]$ is shown to coincide (up to the involution $\omega$ on symmetric function) with Haiman's parking function symmetric function. We construct an edge labeling of NC$_{n+1}$ whose chain labels are the set of all parking functions of length $n$. This leads to a local action of the symmetric group ${S}_n$ on NC$_{n+1}$.

scholarly journals Rational Associahedra and Noncrossing Partitions

10.37236/3432 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Drew Armstrong ◽  
Brendon Rhoades ◽  
Nathan Williams
Keyword(s):  
Simplicial Complex ◽  
Rational Number ◽  
Perfect Matchings ◽  
Lattice Paths ◽  
Noncrossing Partitions ◽  
Positive Rational Number ◽  
Dyck Paths ◽  
Narayana Numbers ◽  
Positive Integers ◽  
Coprime Positive Integers

Each positive rational number $x>0$ can be written uniquely as $x=a/(b-a)$ for coprime positive integers $0<a<b$. We will identify $x$ with the pair $(a,b)$. In this paper we define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass}(x)=\mathsf{Ass}(a,b)$ called the rational associahedron.  It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the rational Catalan number $$\mathsf{Cat}(x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)!}{a!\,b!}.$$The cases $(a,b)=(n,n+1)$ and $(a,b)=(n,kn+1)$ recover the classical associahedron and its "Fuss-Catalan" generalization studied by Athanasiadis-Tzanaki and Fomin-Reading.  We prove that $\mathsf{Ass}(a,b)$ is shellable and give nice product formulas for its $h$-vector (the rational Narayana numbers) and $f$-vector (the rational Kirkman numbers).  We define $\mathsf{Ass}(a,b)$ via rational Dyck paths: lattice paths from $(0,0)$ to $(b,a)$ staying above the line $y = \frac{a}{b}x$.  We also use rational Dyck paths to define a rational generalization of noncrossing perfect matchings of $[2n]$.  In the case $(a,b) = (n, mn+1)$, our construction produces the noncrossing partitions of $[(m+1)n]$ in which each block has size $m+1$.

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.

Classification of irreducible weak modules over Virasoro vertex operator algebras

2019 ◽  
Vol 30 (11) ◽  
pp. 1950054
Author(s):  
Guobo Chen ◽  
Dejia Cheng ◽  
Jianzhi Han ◽  
Yucai Su
Keyword(s):  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Vertex Operator Algebras ◽  
Positive Integers ◽  
Coprime Positive Integers

The classification of irreducible weak modules over the Virasoro vertex operator algebra [Formula: see text] is obtained in this paper. As one of the main results, we also classify all irreducible weak modules over the simple Virasoro vertex operator algebras [Formula: see text] for [Formula: see text] [Formula: see text], where [Formula: see text] are coprime positive integers.

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

A NEW CHARACTERIZATION OF ALMOST SPORADIC GROUPS

2002 ◽  
Vol 01 (03) ◽  
pp. 267-279 ◽  
Author(s):  
AMIR KHOSRAVI ◽  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Sporadic Groups ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Order Components ◽  
Positive Integers ◽  
Coprime Positive Integers

Let G be a finite group. Based on the prime graph of G, the order of G can be divided into a product of coprime positive integers. These integers are called order components of G and the set of order components is denoted by OC(G). Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper we prove that almost sporadic simple groups, except Aut (J2) and Aut (McL), and the automorphism group of PSL(2, 2n) where n=2sare also uniquely determined by their order components. Also we discuss about the characterizability of Aut (PSL(2, q)). As corollaries of these results, we generalize a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.

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 A further correspondence between $(bc,\bar{b})$-parking functions and $(bc,\bar{b})$-forests

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Heesung Shin ◽  
Jiang Zeng
Keyword(s):  
The Other ◽  
Parking Functions ◽  
Fixed Sequence ◽  
Parking Function ◽  
International Audience ◽  
Positive Integers ◽  
Appl Math

International audience For a fixed sequence of $n$ positive integers $(a,\bar{b}) := (a, b, b,\ldots, b)$, an $(a,\bar{b})$-parking function of length $n$ is a sequence $(p_1, p_2, \ldots, p_n)$ of positive integers whose nondecreasing rearrangement $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfies $q_i \leq a+(i-1)b$ for any $i=1,\ldots, n$. A $(a,\bar{b})$-forest on $n$-set is a rooted vertex-colored forests on $n$-set whose roots are colored with the colors $0, 1, \ldots, a-1$ and the other vertices are colored with the colors $0, 1, \ldots, b-1$. In this paper, we construct a bijection between $(bc,\bar{b})$-parking functions of length $n$ and $(bc,\bar{b})$-forests on $n$-set with some interesting properties. As applications, we obtain a generalization of Gessel and Seo's result about $(c,\bar{1})$-parking functions [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] and a refinement of Yan's identity [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] between an inversion enumerator for $(bc,\bar{b})$-forests and a complement enumerator for $(bc,\bar{b})$-parking functions. Soit $(a,\bar{b}) := (a, b, b,\ldots, b)$ une suite d'entiers positifs. Une $(a,\bar{b})$-fonction de parking est une suite $(p_1, p_2, \ldots, p_n)$ d'entiers positives telle que son réarrangement non décroissant $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfait $q_i \leq a+(i-1)b$ pour tout $i=1,\ldots, n$. Une $(a,\bar{b})$-forêt enracinée sur un $n$-ensemble est une forêt enracinée dont les racines sont colorées avec les couleurs $0, 1, \ldots, a-1$ et les autres sommets sont colorés avec les couleurs $0, 1, \ldots, b-1$. Dans cet article, on construit une bijection entre $(bc,\bar{b})$-fonctions de parking et $(bc,\bar{b})$-forêts avec des des propriétés intéressantes. Comme applications, on obtient une généralisation d'un résultat de Gessel-Seo sur $(c,\bar{1})$-fonctions de parking [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] et une extension de l'identité de Yan [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] entre l'énumérateur d'inversion de $(bc,\bar{b})$-forêts et l'énumérateur complémentaire de $(bc,\bar{b})$-fonctions de parking.

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