scholarly journals A Decomposition of Parking Functions by Undesired Spaces

10.37236/5940 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Melody Bruce ◽  
Michael Dougherty ◽  
Max Hlavacek ◽  
Ryo Kudo ◽  
Ian Nicolas
Keyword(s):  
Parking Functions ◽  
Partition Lattice ◽  
Self Duality ◽  
Noncrossing Partition ◽  
Symmetric Chain Decomposition ◽  
Order Complexes ◽  
Maximal Chains ◽  
Symmetric Chain ◽  
Noncrossing Partition Lattice

There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable Möbius function, and a symmetric chain decomposition. We also explore connections with order complexes, labeled Dyck paths, and rooted forests.

scholarly journals Undesired Parking Spaces and Contractible Pieces of the Noncrossing Partition Link

10.37236/7212 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Michael Dougherty ◽  
Jon McCammond
Keyword(s):  
Simplicial Complexes ◽  
Order Complex ◽  
Parking Functions ◽  
Partition Lattice ◽  
Flag Complex ◽  
Noncrossing Partition ◽  
Noncrossing Partition Lattice

There are two natural simplicial complexes associated to the noncrossing partition lattice: the order complex of the full lattice and the order complex of the lattice with its bounding elements removed. The latter is a complex that we call the noncrossing partition link because it is the link of an edge in the former. The first author and his coauthors conjectured that various collections of simplices of the noncrossing partition link (determined by the undesired parking spaces in the corresponding parking functions) form contractible subcomplexes. In this article we prove their conjecture by combining the fact that the star of a simplex in a flag complex is contractible with the second author's theory of noncrossing hypertrees.

scholarly journals The Noncrossing Bond Poset of a Graph

10.37236/9253 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
C. Matthew Farmer ◽  
Joshua Hallam ◽  
Clifford Smyth
Keyword(s):  
Sufficient Conditions ◽  
Partition Lattice ◽  
Open Problems ◽  
Combinatorial Properties ◽  
Noncrossing Partition ◽  
Vertex Set ◽  
Connected Subgraphs ◽  
Necessary And Sufficient ◽  
Broken Circuit ◽  
Noncrossing Partition Lattice

The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of $G$. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the noncrossing partitions of $L_G$. Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that, for several families of graphs, the noncrossing bond poset also exhibits these properties. We present simple necessary and sufficient conditions on the graph to ensure the noncrossing bond poset is a lattice.  Additionally, for several families of graphs, we give combinatorial descriptions of the Möbius function and characteristic polynomial of the noncrossing bond poset. These descriptions are in terms of a noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be thought of as a noncrossing version of Whitney's NBC theorem for the chromatic polynomial. We also consider the shellability and supersolvability of the noncrossing bond poset, providing sufficient conditions for both. We end with some open problems. 

scholarly journals Submaximal factorizations of a Coxeter element in complex reflection groups

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Vivien Ripoll
Keyword(s):  
Reflection Group ◽  
Coxeter Element ◽  
Partition Lattice ◽  
Combinatorial Object ◽  
Complex Reflection Groups ◽  
Noncrossing Partition ◽  
Finite Reflection Group ◽  
International Audience ◽  
Cas A ◽  
Noncrossing Partition Lattice

International audience When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $NC(W)$ as a generalized Fuß-Catalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $NC(W)$ as fibers of a "Lyashko-Looijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$. Lorsque $W$ est un groupe de réflexion fini, le treillis $NC(W)$ des partitions non-croisées de type $W$ est un objet combinatoire très riche, qui généralise la notion de partitions non-croisées d'un $n$-gone. Une formule (seulement prouvée au cas par cas à l'heure actuelle) exprime le nombre de chaînes de longueur donnée dans $NC(W)$ sous la forme d'un nombre de Fuß-Catalan généralisé, qui dépend des degrés invariants de $W$. Nous décrivons une stratégie visant à comprendre certaines spécifications de cette formule de manière uniforme, en utilisant une interprétation des chaînes de $NC(W)$ comme fibres d'un "revêtement de Lyashko-Looijenga''. Ce revêtement est construit à partir de la géométrie de l'hypersurface du discriminant de $W$. Nous en déduisons de nouvelles formules de comptage pour certaines factorisations d'un élément de Coxeter de $W$.

scholarly journals Deformation of Chains via a Local Symmetric Group Action

10.37236/1459 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Patricia Hersh
Keyword(s):  
Symmetric Group ◽  
Group Action ◽  
Maximal Chain ◽  
Type B ◽  
Noncrossing Partition ◽  
Partition Lattices ◽  
Maximal Chains ◽  
Symmetric Chain

A symmetric group action on the maximal chains in a finite, ranked poset is local if the adjacent transpositions act in such a way that $(i,i+1)$ sends each maximal chain either to itself or to one differing only at rank $i$. We prove that when $S_n$ acts locally on a lattice, each orbit considered as a subposet is a product of chains. We also show that all posets with local actions induced by labellings known as $R^* S$-labellings have symmetric chain decompositions and provide $R^* S$-labellings for the type B and D noncrossing partition lattices, answering a question of Stanley.

scholarly journals Symmetric Chain Decompositions of Products of Posets with Long Chains

10.37236/7124 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Stefan David ◽  
Hunter Spink ◽  
Marius Tiba
Keyword(s):  
Minimal Element ◽  
Difficult Case ◽  
Chain Decomposition ◽  
Graded Poset ◽  
Canonical Bijection ◽  
Symmetric Chain Decomposition ◽  
Symmetric Chain ◽  
Long Chains

We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is taut, i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \cdots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of symmetric chain decompositions which induce orthogonal chain decompositions — the existence of the decompositions provided in this paper unexpectedly resolves the most difficult case of previous work by the second author on almost orthogonal symmetric chain decompositions (2017), making progress on a conjecture of Shearer and Kleitman (1979). In general, we show that for a finite graded poset $P$, there exists a canonical bijection between symmetric chain decompositions of $P \times m$ and $P \times n$ for $m, n\ge rk(P) + 1$, that preserves the existence of taut chains. If $P$ has a unique maximal and minimal element, then we also produce a canonical $(rk(P) +1)$ to $1$ surjection from symmetric chain decompositions of $P \times (rk(P) + 1)$ to symmetric chain decompositions of $P \times rk(P)$ which sends decompositions with taut chains to decompositions with taut chains.

scholarly journals Fixed Points of the Evacuation of Maximal Chains on Fuss Shapes

10.37236/6898 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Hsiang-Chun Hsu ◽  
Yu-Pei Huang
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Rectangular Shape ◽  
Explicit Characterization ◽  
Finite Poset ◽  
Maximal Chains

For a partition $\lambda$ of an integer, we associate $\lambda$ with a slender poset $P$ the Hasse diagram of which resembles the Ferrers diagram of $\lambda$. Let $X$ be the set of maximal chains of $P$. We consider Stanley's involution $\epsilon:X\rightarrow X$, which is extended from Schützenberger's evacuation on linear extensions of a finite poset. We present an explicit characterization of the fixed points of the map $\epsilon:X\rightarrow X$ when $\lambda$ is a stretched staircase or a rectangular shape. Unexpectedly, the fixed points have a nice structure, i.e., a fixed point can be decomposed in half into two chains such that the first half and the second half are the evacuation of each other. As a consequence, we prove anew Stembridge's $q=-1$ phenomenon for the maximal chains of $P$ under the involution $\epsilon$ for the restricted shapes.

scholarly journals Cubical Convex Ear Decompositions

10.37236/83 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Russ Woodroofe
Keyword(s):  
Finite Group ◽  
Necessary Conditions ◽  
Partition Lattice ◽  
Ear Decomposition ◽  
Interesting Special Case ◽  
Order Complexes ◽  
Special Case ◽  
The Way

We consider the problem of constructing a convex ear decomposition for a poset. The usual technique, introduced by Nyman and Swartz, starts with a $CL$-labeling and uses this to shell the 'ears' of the decomposition. We axiomatize the necessary conditions for this technique as a "$CL$-ced" or "$EL$-ced". We find an $EL$-ced of the $d$-divisible partition lattice, and a closely related convex ear decomposition of the coset lattice of a relatively complemented finite group. Along the way, we construct new $EL$-labelings of both lattices. The convex ear decompositions so constructed are formed by face lattices of hypercubes. We then proceed to show that if two posets $P_{1}$ and $P_{2}$ have convex ear decompositions ($CL$-ceds), then their products $P_{1}\times P_{2}$, $P_{1}\check{\times} P_{2}$, and $P_{1}\hat{\times} P_{2}$ also have convex ear decompositions ($CL$-ceds). An interesting special case is: if $P_{1}$ and $P_{2}$ have polytopal order complexes, then so do their products.

scholarly journals Semigroups with atomistic congruence lattices

1992 ◽  
Vol 52 (1) ◽  
pp. 88-102 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Congruence Lattice ◽  
Disjoint Union ◽  
Partition Lattice ◽  
Congruence Lattices

AbstractIn this note a characterization of semigroups with atomistic consruence lattices, given for weakly reductive semigroups, is generalized to arbitrary semigroups. Also, it is shown that there is a complete congruence on the congruence lattice of such a semigroup that decomposes it into a disjoint union of intervals of the partition lattice.

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