A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations

Order ◽  
2005 ◽  
Vol 22 (4) ◽  
pp. 311-328 ◽  
Author(s):  
Elena Barcucci ◽  
Antonio Bernini ◽  
Luca Ferrari ◽  
Maddalena Poneti
Keyword(s):  
Distributive Lattice ◽  
Lattice Structure ◽  
Noncrossing Partitions ◽  
Dyck Paths

scholarly journals Partitions of an Integer into Powers

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Matthieu Latapy
Keyword(s):  
Distributive Lattice ◽  
Lattice Structure ◽  
Tree Structure ◽  
Dynamical Model ◽  
International Audience ◽  
Partitions Of Integers ◽  
Natural Way

International audience In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.

scholarly journals Pairs of noncrossing free Dyck paths and noncrossing partitions

2009 ◽  
Vol 309 (9) ◽  
pp. 2834-2838 ◽  
Author(s):  
William Y.C. Chen ◽  
Sabrina X.M. Pang ◽  
Ellen X.Y. Qu ◽  
Richard P. Stanley
Keyword(s):  
Noncrossing Partitions ◽  
Dyck Paths

Generalized intervals in partially ordered groups

1959 ◽  
Vol 55 (2) ◽  
pp. 165-171 ◽  
Author(s):  
D. C. J. Burgess
Keyword(s):  
Distributive Lattice ◽  
Lattice Structure ◽  
Ordered Group ◽  
Partially Ordered ◽  
Partially Ordered Group ◽  
Ordered Groups ◽  
Normal Sense

1. Introduction. The present paper is chiefly concerned with a generalization, to be known as a ‘D-interval’, of the notion of interval or segment in an arbitrary partially ordered group. This idea is originally due to Duthie (2), but was developed by him only in a lattice. In analogy with the use of the interval in the normal sense, notions of ‘D-distributivity’ and ‘D-modularity’ are defined in terms of the D-interval, and analogues of known properties of lattice-groups or ‘l– groups’ can be formulated which might be valid when a lattice structure is no longer assumed to exist; in particular, an attempt is made to provide such a generalization of the result of Freudenthal (3) that every Z-group is a distributive lattice, but, for an arbitrary partially ordered group, it is shown that only an ‘approximation’ (in terms of non-Archimedean elements) to the desired result actually holds, although any Archimedean partially ordered group is necessarily D-distributive.

Lattice Structure of D, T, and SR Fuzzy Flip-Flops Under Max-Min Logic

2005 ◽  
Vol 9 (6) ◽  
pp. 661-668 ◽  
Author(s):  
Shinichi Yoshida ◽  
◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Logic ◽  
Distributive Lattice ◽  
System Design ◽  
Lattice Structure ◽  
Design Method ◽  
Boolean Lattice ◽  
Lattice Structures ◽  
Flip Flop ◽  
Different Types ◽  
Fuzzy Logical

Lattice structures of fuzzy flip-flops are described. A binary flip-flop (e.g. D, T, set-type SR, or reset-type SR flip-flop) can be extended to a fuzzy flip-flop in various ways. Under max-min fuzzy logic, there are 4 types of D fuzzy flip-flops extended from a binary D flip-flop, 136 types of SR fuzzy flip-flops extended from a binary SR flip-flop, and only one T fuzzy flip-flop. There is a lattice structure among different types of fuzzy flip-flops extended from a same binary flip-flop in terms of the order of ambiguity and the order of fuzzy logical value. These results show that fuzzy flip-flops under max-min fuzzy logic construct distributive lattice structures. Moreover D and T fuzzy flip-flops constructs Boolean lattice. And there exists a order monotone between two lattices of same fuzzy flip-flop under the order of ambiguity and the order of fuzzy logical value. Proposed analysis and results have potential to establish a fuzzy sequential system design method.

scholarly journals Restricted Dumont permutations, Dyck paths, and noncrossing partitions

2006 ◽  
Vol 306 (22) ◽  
pp. 2851-2869 ◽  
Author(s):  
Alexander Burstein ◽  
Sergi Elizalde ◽  
Toufik Mansour
Keyword(s):  
Noncrossing Partitions ◽  
Dyck Paths ◽  
Dumont Permutations

scholarly journals Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs

2017 ◽  
Vol Vol. 18 no. 3 (Graph Theory) ◽  
Author(s):  
Stefan Felsner ◽  
Daniel Heldt
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Distributive Lattice ◽  
Maximum Degree ◽  
Lattice Structure ◽  
Mixing Time ◽  
Plane Graph ◽  
Negative Results ◽  
Slow Mixing ◽  
Constant Maximum

We study Markov chains for $\alpha$-orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function $\alpha$. The set of $\alpha$-orientations of a plane graph has a natural distributive lattice structure. The moves of the up-down Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the $\alpha$-orientation. We have a positive and several negative results regarding the mixing time of such Markov chains. A 2-orientation of a plane quadrangulation is an orientation where every inner vertex has outdegree 2. We show that there is a class of plane quadrangulations such that the up-down Markov chain on the 2-orientations of these quadrangulations is slowly mixing. On the other hand the chain is rapidly mixing on 2-orientations of quadrangulations with maximum degree at most 4. Regarding examples for slow mixing we also revisit the case of 3-orientations of triangulations which has been studied before by Miracle et al.. Our examples for slow mixing are simpler and have a smaller maximum degree, Finally we present the first example of a function $\alpha$ and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the $\alpha$-orientations of these graphs is slowly mixing.

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 Lattice Structures from Planar Graphs

10.37236/1768 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Stefan Felsner
Keyword(s):  
Distributive Lattice ◽  
Planar Graph ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Lattice Structure ◽  
General Theorem ◽  
Lattice Structures ◽  
Schnyder Woods

The set of all orientations of a planar graph with prescribed outdegrees carries the structure of a distributive lattice. This general theorem is proven in the first part of the paper. In the second part the theorem is applied to show that interesting combinatorial sets related to a planar graph have lattice structure: Eulerian orientations, spanning trees and Schnyder woods. For the Schnyder wood application some additional theory has to be developed. In particular it is shown that a Schnyder wood for a planar graph induces a Schnyder wood for the dual.

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