scholarly journals Parking Functions of Types A and B

10.37236/1668 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
P. Biane
Keyword(s):  
Symmetric Group ◽  
Cayley Graph ◽  
Parking Functions ◽  
Type B ◽  
Noncrossing Partitions ◽  
Edge Labeling

The lattice of noncrossing partitions can be embedded into the Cayley graph of the symmetric group. This allows us to rederive connections between noncrossing partitions and parking functions. We use an analogous embedding for type B non-crossing partitions in order to answer a question raised by R. Stanley on the edge labeling of the type B non-crossing partitions lattice.

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 Minimal factorizations of a cycle: a multivariate generating function

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Philippe Biane ◽  
Matthieu Josuat-Vergès
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Simple Formula ◽  
Hook Length ◽  
Noncrossing Partitions ◽  
Long Cycle ◽  
Number Of Factors ◽  
Multivariate Generalization ◽  
International Audience

International audience It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.

scholarly journals Identities Derived from Noncrossing Partitions of Type $B$

10.37236/616 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
William Y. C. Chen ◽  
Andrew Y. Z. Wang ◽  
Alina F. Y. Zhao
Keyword(s):  
Type B ◽  
Noncrossing Partitions ◽  
Narayana Numbers ◽  
Combinatorial Constructions

Based on weighted noncrossing partitions of type $B$, we obtain type $B$ analogues of Coker's identities on the Narayana polynomials. A parity reversing involution is given for the alternating sum of Narayana numbers of type $B$. Moreover, we find type $B$ analogues of the refinements of Coker's identities due to Chen, Deutsch and Elizalde. By combinatorial constructions, we provide type $B$ analogues of three identities of Mansour and Sun also on the Narayana polynomials.

scholarly journals Cayley Graphs on the Symmetric Group Generated by Initial Reversals have Unit Spectral Gap

10.37236/267 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Filippo Cesi
Keyword(s):  
Irreducible Representation ◽  
Symmetric Group ◽  
Empirical Evidence ◽  
Cayley Graph ◽  
Simple Proof ◽  
Spectral Gap ◽  
Cayley Graphs ◽  
Complete Spectrum ◽  
Nonzero Eigenvalue ◽  
Direct Sum

In a recent paper Gunnells, Scott and Walden have determined the complete spectrum of the Schreier graph on the symmetric group corresponding to the Young subgroup $S_{n-2}\times S_2$ and generated by initial reversals. In particular they find that the first nonzero eigenvalue, or spectral gap, of the Laplacian is always 1, and report that "empirical evidence" suggests that this also holds for the corresponding Cayley graph. We provide a simple proof of this last assertion, based on the decomposition of the Laplacian of Cayley graphs, into a direct sum of irreducible representation matrices of the symmetric group.

Integral Cayley Graphs over Finite Groups

2020 ◽  
Vol 27 (01) ◽  
pp. 131-136
Author(s):  
Elena V. Konstantinova ◽  
Daria Lytkina
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Cyclic Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Generating Set ◽  
A Finite Group

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

and Spaces

2017 ◽  
Author(s):  
Matt Clay ◽  
Dan Margalit
Keyword(s):  
Group Theory ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Group Action ◽  
Metric Spaces ◽  
Geometric Group Theory ◽  
Geometric Object ◽  
Geometric Objects ◽  
Group Of Symmetries

This chapter discusses the notion of space, first by explaining what it means for a group to be a group of symmetries of a geometric object. This is the idea of group action, and some examples are given. The chapter proceeds by defining, for any group G, the Cayley graph of G and shows that the symmetric group of of this graph is precisely the group G. It then introduces metric spaces, which formalize the notion of a geometric object, and highlights numerous metric spaces that groups can act on. It also demonstrates that groups themselves are metric spaces; in other words, groups themselves can be thought of as geometric objects. The chapter concludes by using these ideas to frame the motivating questions of geometric group theory. Exercises relevant to each idea are included.

Fischer Matrices for Projective Representations of Generalized Symmetric Groups

2009 ◽  
Vol 16 (03) ◽  
pp. 449-462
Author(s):  
Mohammed S. Almestady ◽  
Alun O. Morris
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Symmetric Groups ◽  
Type B ◽  
Projective Representations ◽  
Ordinary Case ◽  
Covering Groups

The aim of this work is to calculate the Fischer matrices for the covering groups of the Weyl group of type Bn and the generalized symmetric group. It is shown that the Fischer matrices are the same as those in the ordinary case for the classes of Sn which correspond to partitions with all parts odd. For the classes of Sn which correspond to partitions in which no part is repeated more than m times, the Fischer matrices are shown to be different from the ordinary case.

scholarly journals Cayley graph on symmetric group generated by elements fixing k points

2015 ◽  
Vol 471 ◽  
pp. 405-426 ◽  
Author(s):  
Cheng Yeaw Ku ◽  
Terry Lau ◽  
Kok Bin Wong
Keyword(s):  
Symmetric Group ◽  
Cayley Graph

scholarly journals Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

2021 ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Symmetric Group ◽  
Partial Order ◽  
Structural Investigation ◽  
Canonical Representation ◽  
Weak Order ◽  
Noncrossing Partitions ◽  
Tamari Lattice ◽  
Semidistributive Lattice ◽  
Tamari Lattices

AbstractOrdering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.

scholarly journals Stability Analysis for $k$-wise Intersecting Families

10.37236/602 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vikram Kamat
Keyword(s):  
Stability Analysis ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Circle Method ◽  
Intersecting Family ◽  
Intersecting Families ◽  
Element Set

We consider the following generalization of the seminal Erdős–Ko–Rado theorem, due to Frankl. For some $k\geq 2$, let $\mathcal{F}$ be a $k$-wise intersecting family of $r$-subsets of an $n$ element set $X$, i.e. for any $F_1,\ldots,F_k\in \mathcal{F}$, $\cap_{i=1}^k F_i\neq \emptyset$. If $r\leq \dfrac{(k-1)n}{k}$, then $|\mathcal{F}|\leq {n-1 \choose r-1}$. We prove a stability version of this theorem, analogous to similar results of Dinur-Friedgut, Keevash-Mubayi and others for the EKR theorem. The technique we use is a generalization of Katona's circle method, initially employed by Keevash, which uses expansion properties of a particular Cayley graph of the symmetric group.

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