scholarly journals Equipopularity Classes of 132-Avoiding Permutations

10.37236/3675 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Lynn Chua ◽  
Krishanu Roy Sankar
Keyword(s):  
Generating Functions ◽  
Complete Classification ◽  
Connected Components ◽  
Plane Tree ◽  
Plane Trees ◽  
Set Of Permutations ◽  
Spine Structure

The popularity of a pattern $p$ in a set of permutations is the sum of the number of copies of $p$ in each permutation of the set. We study pattern popularity in the set of 132-avoiding permutations. Two patterns are equipopular if, for all $n$, they have the same popularity in the set of length-$n$ 132-avoiding permutations. There is a well-known bijection between 132-avoiding permutations and binary plane trees. The spines of a binary plane tree are defined as the connected components when all edges connecting left children to their parents are deleted, and the spine structure is the sorted sequence of lengths of the spines. Rudolph shows that patterns of the same length are equipopular if their associated binary plane trees have the same spine structure. We prove the converse of this result using the method of generating functions, which gives a complete classification of 132-avoiding permutations into equipopularity classes.

scholarly journals Finite Eulerian posets which are binomial or Sheffer

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Hoda Bidkhori
Keyword(s):  
Generating Functions ◽  
Complete Classification ◽  
International Audience ◽  
Original Question ◽  
Eulerian Posets

International audience In this paper we study finite Eulerian posets which are binomial or Sheffer. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows: (1) We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; (2) We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; (3) In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the \emphboolean lattice by looking at smaller intervals. Nous étudions les ensembles partiellement ordonnés finis (EPO) qui sont soit binomiaux soit de type Sheffer (deux notions reliées aux séries génératrices et à la géométrie). Nos résultats sont les suivants: (1) nous déterminons la structure des EPO Euleriens et binomiaux; nous classifions ainsi les fonctions factorielles de tous ces EPO; (2) nous donnons une classification presque complète des fonctions factorielles des EPO Euleriens de type Sheffer; (3) dans la plupart de ces cas, nous déterminons complètement la structure des EPO Euleriens et Sheffer, ce qui est plus fort que classifier leurs fonctions factorielles. Nous étudions aussi les EPO Euleriens triangulaires. Cet article répond à des questions de R. Ehrenborg and M. Readdy. Il est aussi motivé par le travail de R. Stanley sur la reconnaissance du treillis booléen via l'étude des petits intervalles.

scholarly journals Connected Components of the Hurwitz Space for the Symmetric Group of Degree 7

2020 ◽  
Vol 24 (2) ◽  
pp. 129
Author(s):  
Haval M. Mohammed Salih
Keyword(s):  
Symmetric Group ◽  
Monodromy Group ◽  
Riemann Sphere ◽  
Complete Classification ◽  
Connected Components ◽  
Branch Points ◽  
Genus Zero ◽  
Computational Tools ◽  
Hurwitz Space

The Hurwitz space  is the space of genus  covers of the Riemann sphere  with branch points and the monodromy group . Let be the symmetric group . In this paper, we enumerate the connected components of . Our approach uses computational tools, relying on the computer algebra system GAP and the MAPCLASS package, to find the connected components of . This work gives us the complete classification of  primitive genus zero symmetric group of degree seven. 

scholarly journals Protection number in plane trees

2017 ◽  
Vol 11 (2) ◽  
pp. 314-326
Author(s):  
Clemens Heuberger ◽  
Helmut Prodinger
Keyword(s):  
Generating Functions ◽  
Singularity Analysis ◽  
Random Tree ◽  
Minimal Distance ◽  
Plane Tree ◽  
Plane Trees

The protection number of a plane tree is the minimal distance of the root to a leaf; this definition carries over to an arbitrary node in a plane tree by considering the maximal subtree having this node as a root. We study the the protection number of a uniformly chosen random tree of size n and also the protection number of a uniformly chosen node in a uniformly chosen random tree of size n. The method is to apply singularity analysis to appropriate generating functions. Additional results are provided as well.

scholarly journals Thin Lehman Matrices and Their Graphs

10.37236/437 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jonathan Wang
Keyword(s):  
Positive Integer ◽  
Maximum Clique ◽  
Circulant Matrix ◽  
Complete Classification ◽  
Connected Components ◽  
Connected Component ◽  
Johnson Graph ◽  
Induced Subgraph ◽  
The Matrix

Two square $0,1$ matrices $A,B$ are a pair of Lehman matrices if $AB^T = J+dI$, where $J$ is the matrix of all $1$s and $d$ is a positive integer. It is known that there are infinitely many such matrices when $d=1$, and these matrices are called thin Lehman matrices. An induced subgraph of the Johnson graph may be defined given any Lehman matrix, where the vertices of the graph correspond to rows of the matrix. These graphs are used to study thin Lehman matrices. We show that any connected component of such a graph determines the corresponding rows of the matrix up to permutations of the columns. We also provide a sharp bound on the maximum clique size of such graphs and give a complete classification of Lehman matrices whose graphs have at most two connected components. Some constraints on when a circulant matrix can be Lehman are also provided. Many general classes of thin Lehman matrices are constructed in the paper.

scholarly journals Rigidity of maximal holomorphic representations of Kähler groups

2015 ◽  
Vol 26 (14) ◽  
pp. 1550113 ◽  
Author(s):  
Marco Spinaci
Keyword(s):  
Riemann Surface ◽  
Lie Group ◽  
Complete Classification ◽  
Schwarz Lemma ◽  
Connected Components ◽  
Equivariant Map ◽  
Kähler Groups ◽  
Diagonal Embedding

We investigate representations of Kähler groups [Formula: see text] to a semisimple non-compact Hermitian Lie group [Formula: see text] that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor–Wood inequality similar to those found by Burger–Iozzi and Koziarz–Maubon. Thanks to the study of the case of equality in Royden’s version of the Ahlfors–Schwarz lemma, we can completely describe the case of maximal holomorphic representations. If [Formula: see text], these appear if and only if [Formula: see text] is a ball quotient, and essentially reduce to the diagonal embedding [Formula: see text]. If [Formula: see text] is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, which thus appear as preferred elements of the respective maximal connected components.

scholarly journals Pattern Popularity in 132-Avoiding Permutations

10.37236/2634 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Kate Rudolph
Keyword(s):  
Complete Classification ◽  
Specific Form ◽  
Plane Trees

The popularity of a pattern $p$ is the total number of copies of $p$ within all permutations of a set. We address popularity in the set of $132$-avoidng permutations. Bóna showed that in this set, all other non-monotone length-$3$ patterns are equipopular, and proved equipopularity relations between some length-$k$ patterns of a specific form. We prove equipopularity relations between general length-$k$ patterns, based on the structure of their corresponding binary plane trees. Our result explains all equipopularity relations for patterns of length up to $7$, and we conjecture that it provides a complete classification of equipopularity in $132$-avoiding permutations.

Liapunov stability and adding machines

1995 ◽  
Vol 15 (2) ◽  
pp. 271-290 ◽  
Author(s):  
Jorge Buescu ◽  
Ian Stewart
Keyword(s):  
Complete Classification ◽  
Cantor Set ◽  
Connected Components ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Continuous Maps ◽  
Liapunov Stable ◽  
Transitive Set ◽  
Adding Machines

AbstractLet X be a locally connected locally compact metric space and f: X → X a continuous map. Let A be a compact transitive set under f. If A is asymptotically stable, then it has finitely many connected components, which are cyclically permuted. If it is Liapunov stable, then A may have infinitely many connected components. Our main result states that these form a Cantor set on which f is topologically conjugate to an adding machine. A number of consequences are derived, including a complete classification of compact transitive sets for continuous maps of the interval and the Liapunov instability of the invariant Cantor set of Denjoy maps of the circle.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

scholarly journals Characteristic cohomology and observables in higher spin gravity

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Alexey Sharapov ◽  
Evgeny Skvortsov
Keyword(s):  
Higher Spin Gravity ◽  
Complete Classification ◽  
Gravity Models ◽  
Simons Theory ◽  
Surface Charges ◽  
Higher Spin ◽  
Dynamical Invariants ◽  
Chern Simons ◽  
Conserved Currents

Abstract We give a complete classification of dynamical invariants in 3d and 4d Higher Spin Gravity models, with some comments on arbitrary d. These include holographic correlation functions, interaction vertices, on-shell actions, conserved currents, surface charges, and some others. Surprisingly, there are a good many conserved p-form currents with various p. The last fact, being in tension with ‘no nontrivial conserved currents in quantum gravity’ and similar statements, gives an indication of hidden integrability of the models. Our results rely on a systematic computation of Hochschild, cyclic, and Chevalley-Eilenberg cohomology for the corresponding higher spin algebras. A new invariant in Chern-Simons theory with the Weyl algebra as gauge algebra is also presented.

scholarly journals Nil-clean quadratic elements

2017 ◽  
Vol 16 (10) ◽  
pp. 1750197 ◽  
Author(s):  
Janez Šter
Keyword(s):  
Complete Classification ◽  
Strong Condition ◽  
Clean Rings

We provide a strong condition holding for nil-clean quadratic elements in any ring. In particular, our result implies that every nil-clean involution in a ring is unipotent. As a consequence, we give a complete classification of weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl. 15 (2016) 1650148, doi: 10.1142/S0219498816501486].

Sign in / Sign up

Export Citation Format

Share Document