Towards m-Cambrian Lattices

For positive integers $m$ and $k$, we introduce a family of lattices $\mathcal{C}_{k}^{(m)}$ associated to the Cambrian lattice $\mathcal{C}_{k}$ of the dihedral group $I_{2}(k)$. We show that $\mathcal{C}_{k}^{(m)}$ satisfies some basic properties of a Fuss-Catalan generalization of $\mathcal{C}_{k}$, namely that $\mathcal{C}_{k}^{(1)}=\mathcal{C}_{k}$ and \bigl\lvert\mathcal{C}_{k}^{(m)}\bigr\rvert=\mbox{Cat}^{(m)}\bigl(I_{2}(k)\bigr). Subsequently, we prove some structural and topological properties of these lattices---namely that they are trim and EL-shellable---which were known for $\mathcal{C}_{k}$ before. Remarkably, our construction coincides in the case $k=3$ with the $m$-Tamari lattice of parameter 3 due to Bergeron and Pr{\'e}ville-Ratelle. Eventually, we investigate this construction in the context of other Coxeter groups, in particular we conjecture that the lattice completion of the analogous construction for the symmetric group $\mathfrak{S}_{n}$ and the long cycle $(1\;2\;\ldots\;n)$ is isomorphic to the $m$-Tamari lattice of parameter $n$.Comment: 20 pages, 13 figures. The results of this paper are subsumed by arXiv:1312.2520, and it will therefore not be publishe

Generating Random Elements of Finite Distributive Lattices

This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using ``coupling from the past'' to remove initialization bias from Monte Carlo randomization. The article describes several applications to specific kinds of combinatorial objects such as tilings, constrained lattice paths, and alternating-sign matrices.Comment: 13 page

Three Fuss-Catalan posets in interaction and their associative algebras

We introduce $\delta$-cliffs, a generalization of permutations and increasing trees depending on a range map $\delta$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we describe sufficient conditions to have EL-shellable posets, lattices with algorithms to compute the meet and the join of two elements, and lattices constructible by interval doubling. Some of these subposets admit natural geometric realizations. Then, we introduce three families of subposets which, for some maps $\delta$, have underlying sets enumerated by the Fuss-Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. These three families of posets fit into a chain for the order extension relation and they share some properties. Finally, in the same way as the product of the Malvenuto-Reutenauer algebra forms intervals of the right weak order of permutations, we construct algebras whose products form intervals of the lattices of $\delta$-cliff. We provide necessary and sufficient conditions on $\delta$ to have associative, finitely presented, or free algebras. We end this work by using the previous Fuss-Catalan posets to define quotients of our algebras of $\delta$-cliffs. In particular, one is a generalization of the Loday-Ronco algebra and we get new generalizations of this structure.Comment: 63 page

Lattices related to conway's construction

Em 2002 e 2012 foram provados alguns resultados sobre a estrutura de reticulado associada à construção de Conway. Foi também mostrado que o conjunto dos jogos nascidos até ao dia n é um reticulado distributivo completo e que essa estrutura é mantida considerando um conjunto inicial não vazio, desde que seja auto-gerado. Neste trabalho é aprofundada a condição suficiente de distributividade e é dado o primeiro exemplo conhecido de reticulado modular não distributivo proveniente de uma construção tipo Conway. O principal resultado é o Teorema de Representação com Jogos que estabelece que reticulados completos, finitos e infinitos, podem emergir no primeiro dia de uma construção de Conway para certo conjunto inicial. Finalmente, é analisada a construção transfinita: é provado um Teorema de Convergência para a construção de Conway, e é apresentada uma condição que estabelece se o conjunto dos jogos nascidos em dias anteriores a um certo ordinal é um reticulado; ABSTRACT: In 2002 and 2012 some results on the lattice structure associated with Conway’s construction were proved. It was also shown that the set of games born by day n is a complete distributive lattice and that this structure is maintained with a not-empty initial set, provided that it is self-generated. This work deepens the sufficient condition for distributivity. The first known example of non-distributive modular lattice from a Conway’s construction is given. The main result is the Representation Theorem with Games, which states that complete lattices, finite and infinite, can emerge on the first day of a Conway’s construction for some initial set. Finally, the transfinite construction is analyzed: a Convergence Theorem for Conway’s construction is proved, and a condition that establishes whether the class of games born in the days before a certain ordinal is a lattice is presented

Toward the Enumeration of Maximal Chains in the Tamari Lattices

abstract: The Tamari lattices have been intensely studied since they first appeared in Dov Tamari’s thesis around 1952. He defined the n-th Tamari lattice T(n) on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Despite their interesting aspects and the attention they have received, a formula for the number of maximal chains in the Tamari lattices is still unknown. The purpose of this thesis is to convey my results on progress toward the solution of this problem and to discuss future work. A few years ago, Bergeron and Préville-Ratelle generalized the Tamari lattices to the m-Tamari lattices. The original Tamari lattices T(n) are the case m=1. I establish a bijection between maximum length chains in the m-Tamari lattices and standard m-shifted Young tableaux. Using Thrall’s formula, I thus derive the formula for the number of maximum length chains in T(n). For each i greater or equal to -1 and for all n greater or equal to 1, I define C(i,n) to be the set of maximal chains of length n+i in T(n). I establish several properties of maximal chains (treated as tableaux) and identify a particularly special property: each maximal chain may or may not possess a plus-full-set. I show, surprisingly, that for all n greater or equal to 2i+4, each member of C(i,n) contains a plus-full-set. Utilizing this fact and a collection of maps, I obtain a recursion for the number of elements in C(i,n) and an explicit formula based on predetermined initial values. The formula is a polynomial in n of degree 3i+3. For example, the number of maximal chains of length n in T(n) is n choose 3. I discuss current work and future plans involving certain equivalence classes of maximal chains in the Tamari lattices. If a maximal chain may be obtained from another by swapping a pair of consecutive edges with another pair in the Hasse diagram, the two maximal chains are said to differ by a square move. Two maximal chains are said to be in the same equivalence class if one may be obtained from the other by making a set of square moves.Dissertation/ThesisDoctoral Dissertation Mathematics 201

Discovery of the D-basis in binary tables based on hypergraph dualization

Discovery of (strong) association rules, or implications, is an important task in data management, and it nds application in arti cial intelligence, data mining and the semantic web. We introduce a novel approach for the discovery of a speci c set of implications, called the D-basis, that provides a representation for a reduced binary table, based on the structure of its Galois lattice. At the core of the method are the D-relation de ned in the lattice theory framework, and the hypergraph dualization algorithm that allows us to e ectively produce the set of transversals for a given Sperner hypergraph. The latter algorithm, rst developed by specialists from Rutgers Center for Operations Research, has already found numerous applications in solving optimization problems in data base theory, arti cial intelligence and game theory. One application of the method is for analysis of gene expression data related to a particular phenotypic variable, and some initial testing is done for the data provided by the University of Hawaii Cancer Cente

The mm-Cover Posets and the Strip-Decomposition of mm-Dyck Paths

In the first part of this article we present a realization of the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$ in terms of $m$-tuples of Dyck paths of height $n$, equipped with componentwise rotation order. For that, we define the $m$-cover poset $\mathcal{P}^{\langle m \rangle}$ of an arbitrary bounded poset $\mathcal{P}$, and show that the smallest lattice completion of the $m$-cover poset of the Tamari lattice $\mathcal{T}_n$ is isomorphic to the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$. A crucial tool for the proof of this isomorphism is a decomposition of $m$-Dyck paths into $m$-tuples of classical Dyck paths, which we call the strip-decomposition. Subsequently, we characterize the cases where the $m$-cover poset of an arbitrary poset is a lattice. Finally, we show that the $m$-cover poset of the Cambrian lattice of the dihedral group is a trim lattice with cardinality equal to the generalized Fuss-Catalan number of the dihedral group