Bridget Eileen Tenner
The interval poset of a permutation catalogues the intervals that appear in its one-line notation, according to set inclusion. We study this poset, describing its structural, characterizing, and enumerative properties.
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A king-non-attacking permutation is a permutation $π\in S_n$ such that $|π(i)-π(i-1)|\neq 1$ for each $i \in \{2,\dots,n\}$. We investigate the structure of the poset of these permutations under the containment relation, and also provide some results on its Möbius function.
For a permutation $ω\in S_{n}$ Denoncourt constructed a poset $M_ω$ which is the set of join-irreducibles of the Lehmer codes of the permutations in $[e, ω]$ in the inversion order on $S_{n}$. In this paper we show that $M_ω$ is a $B_{2}$-free poset if and only if $ω$ is a 3412-3421-avoiding permutation.
We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P$ that avoid the pattern $π$ is denoted $Av_P(π)$. We extend a proof of Simion and Schmidt to show that $Av_P(132) \leq Av_P(123)$ for any poset $P$, and we exactly classify the posets for which equality holds.
Geometric view of interval poset permutations
本文提出区间偏序的新几何视角，建立大小为$n$的置换的区间偏序集合与$n+1$边凸多边形特定剖分集的一一对应，研究区间偏序子集并揭示其与多边形剖分的关联。
We introduce Lipschitz functions on a finite partially ordered set $P$ and study the associated Lipschitz polytope $L(P)$. The geometry of $L(P)$ can be described in terms of descent-compatible permutations and permutation statistics that generalize descents and big ascents. For ranked posets, Lipschitz polytopes are centrally-symmetric and Gorenstein, which implies symmetry and unimodality of the statistics. Finally, we define $(P,k)$-hypersimplices as generalizations of classical hypersimplices and give combinatorial interpretations of their volumes and $h^*$-vectors.
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。
A structure $\cal S$ is quasi-projective if for every structure $\cal T$, for every homomorphism $f : {\cal S} \rightarrow {\cal T}$ and every epimorphism $j: {\cal S}\rightarrow {\cal T}$ there is an endomorphism $ϕ$ of $\cal S$ such that $ϕ\circ j=f$. In this paper, we characterise the quasi-projective posets and lattices of arbitrary cardinalities, finite permutations, graphs and digraphs of arbitrary cardinalities with loops and without loops, finite hypergraphs, and finite point-line geometries.
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…
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…
The combined work of Bousquet-Mélou, Claesson, Dukes, Jelínek, Kitaev, Kubitzke and Parviainen has resulted in non-trivial bijections among ascent sequences, (2+2)-free posets, upper-triangular integer matrices, and pattern-avoiding permutations. To probe the finer behavior of these bijections, we study two types of restrictions on ascent sequences. These restrictions are motivated by our results that their images under the bijections are natural and combinatorially significant. In addition, for one restriction, we are able to determine the effect of poset duality on the corresponding ascent s…
We define a family of combinatorial objects, which we call Baxter posets. We prove that Baxter posets are counted by the Baxter numbers by showing that they are the adjacency posets of diagonal rectangulations. Given a diagonal rectangulation, we describe the cover relations in the associated Baxter poset. Given a Baxter poset, we describe a method for obtaining the associated Baxter permutation and the associated twisted Baxter permutation.
A permutation $σ=[σ_1,\dots,σ_n] \in S_n$ is called a {\em cylindrical king permutation} if $ |σ_{i+1}-σ_{i}|>1$ for each $1\leq i \leq n-1$ and $|σ_1-σ_n|>1$. The name comes from the the way one can see these permutations as describing locations of $n$ kings on a chessboard of order $n\times n$ in such a way that (each row and each column contains exactly one king and) no two kings are attacking each other, with the additional condition that a king can move off a certain row and reappear at the beginning of that row. In a recent paper, we dealt with the more general set of 'king permutations'…
On naturally labelled posets and permutations avoiding 12-34
本文建立了{3, 2+2}-free自然标号偏序集与避免12-34排列间的双射，通过级数分析探究其渐近增长，并揭示了3-free自然标号偏序集与其他组合对象及区间序层级结构的联系。
We introduce vincular pattern posets, then we consider in particular the quasi-consecutive pattern poset, which is defined by declaring $σ\leq τ$ whenever the permutation $τ$ contains an occurrence of the permutation $σ$ in which all the entries are adjacent in $τ$ except at most the first and the second. We investigate the Möbius function of the quasi-consecutive pattern poset and we completely determine it for those intervals $[σ,τ]$ such that $σ$ occurs precisely once in $τ$.
On Cohen-Macaulay posets of dimension two and permutation graphs
本文刻画了二维 Cohen-Macaulay 偏序集为可壳化且强连通的偏序集，并利用余可比图与置换图的关系，给出了 Cohen-Macaulay 置换图的特征描述。
We introduce a notion of pattern occurrence that generalizes both classical permutation patterns as well as poset containment. Many questions about pattern statistics and avoidance generalize naturally to this setting, and we focus on functional complexity problems -- particularly those that arise by constraining the order dimensions of the pattern and text posets. We show that counting the number of induced, injective occurrences among dimension 2 posets is #P-hard; enumerating the linear extensions that occur in realizers of dimension 2 posets can be done in polynomial time, while for uncons…
Canon Permutation Posets
本文从Stanley的$(P,\omega)$-分拆视角研究典范置换，推广Elizalde的定义，证明下降多项式的乘积结构与回文性，并引入弱下降多项式与两栖构造。
We examine the poset $P$ of 132-avoiding $n$-permutations ordered by descents. We show that this poset is the "coarsening" of the well-studied poset $Q$ of noncrossing partitions . In other words, if $x<y$ in $Q$, then $f(y)<f(x)$ in $P$, where $f$ is the canonical bijection from the set of noncrossing partitions onto that of 132-avoiding permutations. This enables us to prove many properties of $P$.
A poset is {\it $(\3+\1)$-free} if it contains no induced subposet isomorphic to the disjoint union of a 3-element chain and a 1-element chain. These posets are of interest because of their connection with interval orders and their appearance in the $(\3+\1)$-free Conjecture of Stanley and Stembridge. The dimension 2 posets $P$ are exactly the ones which have an associated permutation $π$ where $i\prec j$ in $P$ if and only if $i<j$ as integers and $i$ comes before $j$ in the one-line notation of $π$. So we say that a permutation $π$ is {\it $(\3+\1)$-free} or {\it $(\3+\1)$-avoiding} if its p…
The confluent binomial posets with the atomic function A(n) = max (1,2^(n-2)) are classified. In particular, it is shown that in general there are non-isomorphic intervals of the same length.
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.
A nice factorization is given for the characteristic polynomials of intervals in some posets of leaf-labeled forests of rooted binary trees.
An occurrence of a consecutive permutation pattern $p$ in a permutation $π$ is a segment of consecutive letters of $π$ whose values appear in the same order of size as the letters in $p$. The set of all permutations forms a poset with respect to such pattern containment. We compute the Möbius function of intervals in this poset, providing what may be called a complete solution to the problem. For most intervals our results give an immediate answer to the question. In the remaining cases, we give a polynomial time algorithm to compute the Möbius function. In particular, we show that the Möbius …
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, …
We show that the order complex of intervals of a poset, ordered by inclusion, is a Tchebyshev triangulation of the order complex of the original poset. Besides studying the properties of this transformation, we show that the dual of the type $B$ permutohedron is combinatorially equivalent to the suspension of the order complex of the poset of intervals of a Boolean algebra (with the minimum and maximum elements removed).
(2+2)-free posets, ascent sequences and pattern avoiding permutations
本文通过建立四类组合对象（(2+2)-自由偏序集、对合、新型模式避免排列、上升序列）之间的双射，保持统计量并确定生成函数，最终解决Pudwell关于带横杠模式排列的猜想。
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…
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.
Motivation coming from the study of affine Weyl groups, a structure of ranked poset is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.
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…
Interval hypergraphic polytopes (or deformed associahedra), Tamari interval posets, and weeping willows
研究区间超图的多面体（Loday结合体的变形），刻画其顶点偏序集为Tamari区间偏序集，并给出简单多面体的区间超图特征及其顶点偏序集（垂柳）。
Counting interval sizes in the poset of monotone Boolean functions
研究单调布尔函数偏序集 $D_n$ 中区间大小计数的计算问题，提出一种资源感知算法，成功实现了 $D_7$ 中区间大小的计算。
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.
We introduce two partially ordered sets, $P^A_n$ and $P^B_n$, of the same cardinalities as the type-A and type-B noncrossing partition lattices. The ground sets of $P^A_n$ and $P^B_n$ are subsets of the symmetric and the hyperoctahedral groups, consisting of permutations which avoid certain patterns. The order relation is given by (strict) containment of the descent sets. In each case, by means of an explicit order-preserving bijection, we show that the poset of restricted permutations is an extension of the refinement order on noncrossing partitions. Several structural properties of these per…
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.
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 …
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.
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…
Efficient counting of permutation patterns via double posets
本文将 corner trees 刻画为 twin-tree double posets，并把 Even-Zohar/Leng 算法推广到一类 tree double posets，使更多排列模式可在 O(n^{5/3}) 时间内计数。
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。
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.
Recently it has been shown that all non-trivial closed permutation groups containing the automorphism group of the random poset are generated by two types of permutations: the first type are permutations turning the order upside down, and the second type are permutations induced by so-called rotations. In this paper we introduce rotations for finite posets, which can be seen as the poset counterpart of Seidel-switch for finite graphs. We analyze some of their combinatorial properties, and investigate in particular the question of when two finite posets are rotation-equivalent. We moreover give…
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…
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 $P$.
Lehmer Codes and the Reverse-Complement Mapping from (32-1)-Avoiding Permutations to (3-21)-Avoiding Permutations
该研究以避免特定vincular模式的排列为对象，借助莱默码将其转化为加权偏序集，证明其极大元由斐波那契数计数，且经典逆补映射在极大元间构成双射，揭示了偏序集结构的对称性。
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…
Pattern Avoidance in Task-Precedence Posets
本文将经典模式避免推广到三层任务前序偏序集（菱形），用Sage提出并证明枚举猜想，发现避免132与广义Dyck路径的双射，并给出下降数生成函数；应用场景为机器人仓库分拣中避免重包裹压轻包裹。
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…
In this paper, we are interested in the combinatorial analysis of the whole genome duplication - random loss model of genome rearrangement initiated in a paper of Chaudhuri, Chen, Mihaescu, and Rao in SODA 2006 and continued by Bouvel and Rossin in 2007. In this model, genomes composed of n genes are modeled by permutations of the set of integers [1..n], that can evolve through duplication-loss steps. It was previously shown that the class of permutations obtained in this model after a given number p of steps is a class of pattern-avoiding permutations of finite basis. The excluded patterns we…
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。
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…
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。
We study several aspects of the Möbius function, $μ[σ,π]$, on the poset of permutations under the pattern containment order. First, we consider cases where the lower bound of the poset is indecomposable. We show that $μ[σ,π]$ can be computed by considering just the indecomposable permutations contained in the upper bound. We apply this to the case where the upper bound is an increasing oscillation, and give a method for computing the value of the Möbius function that only involves evaluating simple inequalities. We then consider conditions on an interval which guarantee that the value of the M…
Let $π\in \mathfrak{S}_m$ and $σ\in \mathfrak{S}_n$ be permutations. An occurrence of $π$ in $σ$ as a consecutive pattern is a subsequence $σ_i σ_{i+1} \cdots σ_{i+m-1}$ of $σ$ with the same order relations as $π$. We say that patterns $π, τ\in \mathfrak{S}_m$ are strongly c-Wilf equivalent if for all $n$ and $k$, the number of permutations in $\mathfrak{S}_n$ with exactly $k$ occurrences of $π$ as a consecutive pattern is the same as for $τ$. In 2018, Dwyer and Elizalde conjectured (generalizing a conjecture of Elizalde from 2012) that if $π, τ\in \mathfrak{S}_m$ are strongly c-Wilf equivalen…
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.
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。
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。
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。
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…
Let $\mathfrak{S}_n$ and $\mathfrak{B}_n$ denote the respective sets of ordinary and bigrassmannian (BG) permutations of order $n$, and let $(\mathfrak{S}_n,\leq)$ denote the Bruhat ordering permutation poset. We study the restricted poset $(\mathfrak{B}_n,\leq)$, first providing a simple criterion for comparability. This criterion is used to show that that the poset is connected, to enumerate the saturated chains between elements, and to enumerate the number of maximal elements below $r$ fixed elements. It also quickly produces formulas for $β(ω)$ ($α(ω)$ respectively), the number of BG permu…
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