On intervals in some posets of forests

Linear Intervals in the Tamari and the Dyck Lattices and in the alt-Tamari Posets

We count the number of linear intervals in the Tamari and the Dyck lattices according to their height, using generating series and Lagrange inversion. Surprisingly, these numbers are the same in both lattices. We define a new family of posets on Dyck paths, which we call alt-Tamari posets. Each alt-Tamari poset depends on the choice of an increment function delta in {0,1}^n. We recover the Tamari and the Dyck lattices as extreme cases with delta = 1 and delta = 0, respectively. We prove that all the alt-Tamari posets have the same number of linear intervals of any given height.

Enumerative combinatorics of intervals in the Dyck pattern poset

We initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

Exceptional and modern intervals of the Tamari lattice

In this article we use the theory of interval-posets recently introduced by Ch{â}tel and Pons in order to describe some interesting families of intervals in the Tamari lattices. These families are defined as interval-posets avoiding specific configurations. At first, we consider what we call exceptional interval-posets and show that they correspond to the intervals which are obtained as images of noncrossing trees in the Dendriform operad. We also show that the exceptional intervals are exactly the intervals of the Tamari lattice induced by intervals in the poset of noncrossing partitions. In …

Posets related to the connectivity set of Coxeter groups

We define the notion of connectivity set for elements of any finitely generated Coxeter group. Then we define an order related to this new statistic and show that the poset is graded and each interval is a shellable lattice. This implies that any interval is Cohen-Macauley. We also give a Galois connection between intervals in this poset and a boolean poset. This allows us to compute the Mobius function for any interval.

Conjugacy and Iteration of Standard Interval Rank in Finite Ordered Sets

In order theory, a rank function measures the vertical "level" of a poset element. It is an integer-valued function on a poset which increments with the covering relation, and is only available on a graded poset. Defining a vertical measure to an arbitrary finite poset can be accomplished by extending a rank function to be interval-valued. This establishes an order homomorphism from a base poset to a poset over real intervals, and a standard (canonical) specific interval rank function is available as an extreme case. Various ordering relations are available over intervals, and we begin in this…

电网面偏序集的可壳性与CW偏序集性质

Shellability of face posets of electrical networks and the CW poset property

本文证明了Thomas Lam关于平面电阻网络面偏序集可壳性的猜想，结合其欧拉性质得出其为CW偏序集，并为系统发育树边积空间的CW分解正则性提供了显式证明。

从分解树视角看待排列的区间偏序集

The interval posets of permutations seen from the decomposition tree perspective

本文从排列分解树视角研究区间偏序集，给出从分解树构造该偏序集的方法，并用符号与解析组合学解决相关开放计数问题，同时计算其 Möbius 函数。

The structure of the consecutive pattern poset

The consecutive pattern poset is the infinite partially ordered set of all permutations where $σ\leτ$ if $τ$ has a subsequence of adjacent entries in the same relative order as the entries of $σ$. We study the structure of the intervals in this poset from topological, poset-theoretic, and enumerative perspectives. In particular, we prove that all intervals are rank-unimodal and strongly Sperner, and we characterize disconnected and shellable intervals. We also show that most intervals are not shellable and have Möbius function equal to zero.

Unit Interval Orders of Open and Closed Intervals

A poset $P = (X,\prec)$ is a unit OC interval order if there exists a representation that assigns an open or closed real interval $I(x)$ of unit length to each $x \in P$ so that $x \prec y$ in $P$ precisely when each point of $I(x)$ is less than each point in $I(y)$. In this paper we give a forbidden poset characterization of the class of unit OC interval orders and an efficient algorithm for recognizing the class. The algorithm takes a poset $P $ as input and either produces a representation or returns a forbidden poset induced in $P$.

On the topology of the permutation pattern poset

The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those interval…

Interval Orders with Two Interval Lengths

A poset $P = (X,\prec)$ has an interval representation if each $x \in X$ can be assigned a real interval $I_x$ so that $x \prec y$ in $P$ if and only if $I_x$ lies completely to the left of $I_y$. Such orders are called \emph{interval orders}. In this paper we give a surprisingly simple forbidden poset characterization of those posets that have an interval representation in which each interval length is either 0 or 1. In addition, for posets $(X,\prec)$ with a weight of 1 or 2 assigned to each point, we characterize those that have an interval representation in which for each $x \in X$ the len…

单位区间偏序集上一个猜想的双射证明

Bijective proof of a conjecture on unit interval posets

本文利用左对齐着色树重新表述 zeta 映射，为单位区间偏序集两种表示之间的关系猜想提供了一个双射证明。

On derived equivalences for categories of generalized intervals of a finite poset

We study two constructions related to the intervals of finite posets. The first one is a poset. The second one is more complicated. Loosely speaking it can be seen as a poset with some extra zero-relations. As main result, we show that these two constructions are equivalent at the level of derived categories.

Families of Subsets Without a Given Poset in the Interval Chains

For two posets $P$ and $Q$, we say $Q$ is $P$-free if there does not exist any order-preserving injection from $P$ to $Q$. The speical case for $Q$ being the Boolean lattice $B_n$ is well-studied, and the optiamal value is denoted as $\lanp$. Let us define $\La(Q,P)$ to be the largest size of any $P$-free subposet of $Q$. In this paper, we give an upper bound for $\La(Q,P)$ when $Q$ is a double chain and $P$ is any graded poset, which is better than the previous known upper bound, by means of finding the indpendence number of an auxiliary graph related to $P$. For the auxiliary graph, we can f…

The Poset of Mesh Patterns

We introduce the poset of mesh patterns, which generalises the permutation pattern poset. We fully classify the mesh patterns for which the interval [1^\emptyset,m] is non-pure, where 1^\emptyset is the unshaded singleton mesh pattern. We present some results on the Möbius function of the poset, and show that μ(1^\emptyset,m) is almost always zero. Finally, we introduce a class of disconnected and non-shellable intervals by generalising the direct product operation from permutations to mesh patterns.

区间闭集上的切换、行运动与同稳性

Toggling, rowmotion, and homomesy on interval-closed sets

本文从枚举与动力学角度研究有限偏序集的区间闭集，用广义切换群定义行运动并给出全局刻画，同时针对反链序和与链乘积偏序集证明了枚举及同稳性结果。

偏序集的广义区间多项式

The Generalized Interval Polynomial of a Poset

本文针对有限偏序集定义了可计数上集、下集及其交集的广义区间多项式，研究其在偏序集运算下的性质，证明其可区分串并联偏序集，并探讨其与量子引力因果集方法的关联。

格点拓扑、移动和布雷哈特区间多面体格子

Poset topology, moves, and Bruhat interval polytope lattices

本文研究了方向简单多面体1骨架定向所形成的格点拓扑，证明了布雷哈特区间多面体格子中秩序复形的同调性质，并展示了饱和链在特定移动下的连通性，尤其在格拉姆斯曼排列情况下加强了波斯诺夫定理的限制。

Intervals of Permutations with a Fixed Number of Descents are Shellable

The set of all permutations, ordered by pattern containment, is a poset. We present an order isomorphism from the poset of permutations with a fixed number of descents to a certain poset of words with subword order. We use this bijection to show that intervals of permutations with a fixed number of descents are shellable, and we present a formula for the Möbius function of these intervals. We present an alternative proof for a result on the Möbius function of intervals $[1,π]$ such that $π$ has exactly one descent. We prove that if $π$ has exactly one descent and avoids 456123 and 356124, then…

Poset limits can be totally ordered

S.Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529--563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that each poset limit can be represented as a kernel on the unit interval with the standard order, thus answering an open question of Janson. We provide two proofs: real-analytic and combinatorial. The combinatorial proof is based on a Szemeredi-type Regularity Lemma for posets which may be of independent interest. Also, as a by-product of the analytic proof, we show that every atomless ordered probabili…

The poset of graphs ordered by induced containment

We study the poset $\mathcal{G}$ of all unlabelled graphs, up to isomorphism, with $H\le G$ if $H$ occurs as an induced subgraph in $G$. We present some general results on the Möbius function of intervals of $\mathcal{G}$ and some results for specific classes of graphs. This includes a case where the Möbius function is given by the Catalan numbers, which we prove using discrete Morse theory, and another case where it equals the Fibonacci numbers, therefore showing that the Möbius function is unbounded. A classification of the disconnected intervals of $\mathcal{G}$ is presented, which gives a …

Catalan Intervals and Uniquely Sorted Permutations

For each positive integer $k$, we consider five well-studied posets defined on the set of Dyck paths of semilength $k$. We prove that uniquely sorted permutations avoiding various patterns are equinumerous with intervals in these posets. While most of our proofs are bijective, some use generating trees and generating functions. We end with several conjectures.

SB-labelings and posets with each interval homotopy equivalent to a sphere or a ball

We introduce a new class of poset edge labelings for locally finite lattices which we call $SB$-labelings. We prove for finite lattices which admit an $SB$-labeling that each open interval has the homotopy type of a ball or of a sphere of some dimension. Natural examples include the weak order, the Tamari lattice, and the finite distributive lattices.

通过Motzkin路径和四分仪游走对区间闭集进行枚举

Enumeration of interval-closed sets via Motzkin paths and quarter-plane walks

本文通过构造双射分别将两条链之积偏序集的区间闭集映射到双色Motzkin路径，将截断矩形偏序集的区间闭集映射到四分仪游走，从而求得了对应的生成函数与函数方程。

Interval-Valued Rank in Finite Ordered Sets

We consider the concept of rank as a measure of the vertical levels and positions of elements of partially ordered sets (posets). We are motivated by the need for algorithmic measures on large, real-world hierarchically-structured data objects like the semantic hierarchies of ontological databases. These rarely satisfy the strong property of gradedness, which is required for traditional rank functions to exist. Representing such semantic hierarchies as finite, bounded posets, we recognize the duality of ordered structures to motivate rank functions which respect verticality both from the botto…

Motzkin Intervals and Valid Hook Configurations

We define a new natural partial order on Motzkin paths that serves as an intermediate step between two previously-studied partial orders. We provide a bijection between valid hook configurations of $312$-avoiding permutations and intervals in these new posets. We also show that valid hook configurations of permutations avoiding $132$ (or equivalently, $231$) are counted by the same numbers that count intervals in the Motzkin-Tamari posets that Fang recently introduced, and we give an asymptotic formula for these numbers. We then proceed to enumerate valid hook configurations of permutations av…

Geometric realizations of Tamari interval lattices via cubic coordinates

We introduce cubic coordinates, which are integer words encoding intervals in the Tamari lattices. Cubic coordinates are in bijection with interval-posets, themselves known to be in bijection with Tamari intervals. We show that in each degree the set of cubic coordinates forms a lattice, isomorphic to the lattice of Tamari intervals. Geometric realizations are naturally obtained by placing cubic coordinates in space, highlighting some of their properties. We consider the cellular structure of these realizations. Finally, we show that the poset of cubic coordinates is shellable.

Bruhat区间多面体的1-骨架偏序集

One-skeleton posets of Bruhat interval polytopes

该研究针对Bruhat区间多面体的1-骨架偏序集展开，证明其为格，完成该多面体为简单多面体的情形分类，解决了相关公开问题与猜想，还分类了Schubert簇中一般环面轨道闭包光滑的情形。

Poset topology of $s$-weak order via SB-labelings

Ceballos and Pons generalized weak order on permutations to a partial order on certain labeled trees, thereby introducing a new class of lattices called $s$-weak order. They also generalized the Tamari lattice by defining a particular sublattice of $s$-weak order called the $s$-Tamari lattice. We prove that the homotopy type of each open interval in $s$-weak order and in the $s$-Tamari lattice is either a ball or sphere. We do this by giving $s$-weak order and the $s$-Tamari lattice a type of edge labeling known as an SB-labeling. We characterize which intervals are homotopy equivalent to sphe…

A Formula for the Möbius function of the Permutation Poset Based on a Topological Decomposition

We present a two term formula for the Möbius function of intervals in the poset of all permutations, ordered by pattern containment. The first term in this formula is the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the Möbius function of this and other posets, but simpler than most of them. The second term in the formula is complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the sec…

两类 level Eulerian 偏序集

Two classes of level Eulerian posets

本文构造两类 level Eulerian 偏序集，刻画其秩 k+1 区间的 cd-index，并证明第一类区间的 order complex 同胚于球面。

A Simple Proof Characterizing Interval Orders with Interval Lengths between 1 and $k$

A poset $P= (X, \prec)$ has an interval representation if each $x \in X$ can be assigned a real interval $I_x$ so that $x \prec y$ in $P$ if and only if $I_x$ lies completely to the left of $I_y$. Such orders are called \emph{interval orders}. Fishburn proved that for any positive integer $k$, an interval order has a representation in which all interval lengths are between $1$ and $k$ if and only if the order does not contain $\mathbf{(k+2)+1}$ as an induced poset. In this paper, we give a simple proof of this result using a digraph model.

Blockwise simple permutations

A permutation is called {\it {block-wise simple}} if it contains no interval of the form $p_1\oplus p_2$ or $p_1 \ominus p_2$. We present this new set of permutations and explore some of its combinatorial properties. We present a generating function for this set, as well as a recursive formula for counting block-wise simple permutations. Following Tenner, who founded the notion of interval posets, we characterize and count the interval posets corresponding to block-wise simple permutations. We also present a bijection between these interval posets and certain tiling's of the $n$-gon. Finally, …

有限区间多重集与偏序集表示的 universal 分类

Multisets of finite intervals and a universal category of poset representations

本文提出一个通用构造，将任意偏序集分配为一个有限余完备的加法范畴，并探讨其在有限偏序集下的阿贝尔性质。

区间超图格

Interval hypergraphic lattices

本文研究了区间超图的格性质，确定了哪些区间超图对应的偏序集是格、分配格、半分配格或弱序的格商。

Three Fuss-Catalan posets in interaction and their associative algebras

We introduce $δ$-cliffs, a generalization of permutations and increasing trees depending on a range map $δ$. 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 $δ$, have underlying sets enumerated by the Fuss-Catala…

On the (co)homology of the poset of weighted partitions

We consider the poset of weighted partitions $Π_n^w$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $Π_n^w$ provide a generalization of the lattice $Π_n$ of partitions, which we show possesses many of the well-known properties of $Π_n$. In particular, we prove these intervals are EL-shellable, we show that the Möbius invariant of each maximal interval is given up to sign by the number of rooted trees on on node set $\{1,2,\dots,n\}$ having a fixed number of descents, we find combinatorial bases for homology and cohomology, and …

贪婪Tamari区间与二部平面地图的分解

Decomposition of Greedy Tamari Intervals and Bipartite Planar Maps

通过建立贪婪Tamari区间与二部平面地图的同构递归分解，给出了贪婪1-Tamari区间与平面2-星座等计数的组合证明，并提出了更一般的猜想。

Dyck algebras, interval temporal logic and posets of intervals

We investigate a natural Heyting algebra structure on the set of Dyck paths of the same length. We provide a geometrical description of the operations of pseudocomplement and relative pseudocomplement, as well as of regular elements. We also find a logic-theoretic interpretation of such Heyting algebras, which we call Dyck algebras, by showing that they are the algebraic counterpart of a certain fragment of a classical interval temporal logic (also known as Halpern-Shoham logic). Finally, we propose a generalization of our approach, suggesting a similar study of the Heyting algebra arising fro…

Homotopy type of intervals of the second higher Bruhat orders

The higher Bruhat order is a poset of cubical tilings of a cyclic zonotope whose covering relations are cubical flips. For a 2-dimensional zonotope, the higher Bruhat order is isomorphic to a poset on commutation classes of reduced words for the longest element of a type A Coxeter system. For this case, we prove that the noncontractible intervals are in natural correspondence with the zonogonal tilings of a zonogon. Our proof uses some tools developed by Felsner and Weil to show that the two standard orderings on the rhombic tilings of a zonogon are identical.

The Betti poset in monomial resolutions

Let $P$ be a finite partially ordered set with unique minimal element $\hat{0}$. We study the Betti poset of $P$, created by deleting elements $q\in P$ for which the open interval $(\hat{0}, q)$ is acyclic. Using basic simplicial topology, we demonstrate an isomorphism in homology between open intervals of the form $(\hat{0},p)\subset P$ and corresponding open intervals in the Betti poset. Our motivating application is that the Betti poset of a monomial ideal's lcm-lattice encodes both its $\mathbb{Z}^{d}$-graded Betti numbers and the structure of its minimal free resolution. In the case of ri…

CAT-generation of ideals

We consider the problem of generating all ideals of a poset. It is a long standing open problem, whether or not the ideals of any poset can be generated in constant amortized time, CAT for short. We refine the tree traversal, a method introduced by Pruesse and Ruskey in 1993, to obtain a CAT-generator for two large classes of posets: posets of interval dimension at most two and so called locally planar posets. This includes all posets for which a CAT-generator was known before. Posets of interval dimension at most two generalize both, interval orders and 2-dimensional posets. Locally planar po…

On the Möbius Function of Permutations With One Descent

The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the Möbius function of intervals $[1,π]$ in this poset, for any permutation $π$ with at most one descent. We compute the Möbius function as a function of the number and positions of pairs of consecutive letters in $π$ that are consecutive in value. As a result of this we show that the Möbius function is unbounded on the poset of all permutations. We show that the Möbius function is zero on any interval $[1,π]$ where $π$ has a triple of consecutive letters whose values are consecutive and monotone. We…

Applications of Quotient Posets

In this paper we consider the characteristic polynomial of not necessarily ranked posets. We do so by allowing the rank to be an arbitrary function from the poset to the nonnegative integers. We will prove two results showing that the characteristic polynomial of a poset has nonnegative integral roots. Our factorization theorems will then be used to show that any interval of the Tamari lattice has a characteristic polynomial which factors in this way. Blass and Sagan's result about LL lattices will also be shown to be a consequence of our factorization theorems. Finally we will use quotient po…

Generating geometry axioms from poset axioms

Two axioms of order geoemtry are the poset axioms of transitivity and antisymmetry of the relation "is in front of" when looking from a point. From these axioms, by looking from an interval instead of a point, further well-known axioms of order geometry are generated in the following sense: Transitivity when looking from an interval is equivalent to Assioma XIII of paragraph 10 in G. Peano, I principii di geometria logacimente exposti. Assuming this axiom, antisymmetry when looking from an interval is equivalent VIII. Grundsatz in paragraph 1 in M. Pasch, Vorlesungen ueber neuere Geometrie. Fu…

Obstructions to Shellability

We consider a simplicial complex generaliztion of a result of Billera and Meyers that every nonshellable poset contains the smallest nonshellable poset as an induced subposet. We prove that every nonshellable $2$-dimensional simplicial complex contains a nonshellable induced subcomplex with less than $8$ vertices. We also establish CL-shellability of interval orders and as a consequence obtain a formula for the Betti numbers of any interval order.

贪婪 Tamari 偏序集中的区间

Intervals in the greedy Tamari posets

本文研究 m-Dyck 路径上贪婪 Tamari 序的区间计数，通过递归方法和“催化”参数得到简单公式，并证明该公式同时计数 (m+1)-星座平面地图，还恢复了普通 m-Tamari 格的已知区间计数结果。

欧拉偏序集与$Z$-多项式

Eulerian posets and $Z$-polynomials

该研究解决了Proudfoot提出的问题，证明欧拉偏序集的$Z$-多项式与对应区间偏序集的环面$h$-多项式一致，还得到周多项式与维罗纳变换的关联结果。

(-1)-临界偏序集的形态

Morphologie des posets (-1)-critiques

本文针对有向图的不可分解性与临界顶点概念，定义了(-1)-临界有向图，并对作为传递有向图的偏序集，完成了(-1)-临界偏序集的特征刻画工作。

The poset of posets

Let X be a finite set. This paper describes some topological and combinatorial properties of the poset Ω_X of order relations on X. In particular, the homotopy type of all the intervals in Ω_X is precisely determined, and the Möbius function of Ω_X is computed.

与排列双射的网络

Networks bijective to permutations

本文研究由源点、汇点和中性点组成的与排列双射的网络，通过多米诺或Rothe图构造边集，引入森林对象证明其与逆排列网络双射，并证明网络偏序集为有限分次格且具有$EL$-标号，进而证明超可解性并计算Möbius函数。

色函数、区间序与递增森林

Chromatic functions, interval orders and increasing forests

本研究将色拟对称函数从单位区间序推广到一般区间序，证明其为递增生成森林相关拟对称函数的加权和，并推广相关恒等式与猜想。

The Mobius Function of the Permutation Pattern Poset

A permutation τcontains another permutation σas a pattern if τhas a subsequence whose elements are in the same order with respect to size as the elements in σ. This defines a partial order on the set of all permutations, and gives a graded poset P. We give a large class of pairs of permutations whose intervals in P have Mobius function 0. Also, we give a solution to the problem when σoccurs precisely once in τ, and σand τsatisfy certain further conditions, in which case the Mobius function is shown to be either -1, 0 or 1. We conjecture that for intervals [σ,τ] consisting of permutations avoid…

Barcode Posets: Combinatorial Properties and Connections

A barcode is a finite multiset of intervals on the real line, $B = \{ (b_i, d_i)\}_{i=1}^n$. Barcodes are important objects in topological data analysis, where they serve as summaries of the persistent homology groups of a filtration. The combinatorial properties of barcodes have also been studied, mainly in the context of interval orders and interval graphs. In this paper, we define a new family of maps from the space of barcodes with $n$ bars to the permutation sets of various multisets, known as multipermutations. These multipermutations provide new combinatorial invariants on the space of …