Anders Björner, Jonathan David Farley
It is shown that the numbers $c_i$ of chains of length $i$ in the proper part $L\setminus\{0,1\}$ of a distributive lattice $L$ of length $\ell +2$ satisfy the inequalities $$c_0<...<c_{\lfloor{\ell /2}\rfloor} \quad{and}\quad c_{\lfloor{3 \ell /4}\rfloor}>...>c_{\ell}.$$ This proves 75 % of the inequalities implied by the Neggers unimodality conjecture.
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Rank Polynomials of Fence Posets are Unimodal
通过引入圆形栅栏偏序集并证明其秩多项式的对称性与单峰性，本文证实了 Morier-Genoud 等人关于栅栏偏序集秩多项式单峰性的猜想，并将 rowmotion 的同均值结果推广至圆形情况。
Stanley (1986) introduced the order polytope and chain polytope of a partially ordered set and showed that they are related by a piecewise-linear homeomorphism. In this paper we view order and chain polytopes as instances of distributive and anti-blocking polytopes, respectively. Both these classes of polytopes are defined in terms of the componentwise partial order on $\mathbb R^n$. We generalize Stanley's PL-homeomorphism to a large class of distributive polyhedra using infinite walks in marked networks.
We prove the unimodality of the Ehrhart $δ$-polynomial of the chain polytope of the zig-zag poset, which was conjectured by Kirillov. First, based on a result due to Stanley, we show that this polynomial coincides with the $W$-polynomial for the zig-zag poset with some natural labeling. Then, its unimodality immediately follows from a result of Gasharov, which states that the $W$-polynomials of naturally labeled graded posets of rank $1$ or $2$ are unimodal.
Young's lattice $L(m,n)$ consists of partitions having $m$ parts of size at most $n$, ordered by inclusion of the corresponding Ferrers diagrams. K. O'Hara gave the first constructive proof of the unimodality of the Gaussian polynomials by expressing the underlying ranked set of $L(m,n)$ as a disjoint union of products of centered rank-unimodal subsets. We construct a finer decomposition which is compatible with the partial order on Young's lattice, at the cost of replacing the cartesian product with a more general poset extension. As a corollary, we obtain an explicit chain decomposition whic…
Associated with any composition beta=(a,b,...) is a corresponding fence poset F(beta) whose covering relations are x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... The distributive lattice L(beta) of all lower order ideals of F(beta) is important in the theory of cluster algebras. In addition, its rank generating function r(q;beta) is used to define q-analogues of rational numbers. Oguz and Ravichandran recently showed that its coefficients satisfy an interlacing condition, proving a conjecture of McConville, Smyth and Sagan, which in turn implies a previous conjecture o…
A simple but elegant result of Rival states that every sublattice $L$ of a finite distributive lattice $\mathcal{P}$ can be constructed from $\mathcal{P}$ by removing a particular family $\mathcal{I}_L$ of its irreducible intervals. Applying this in the case that $\mathcal{P}$ is a product of a finite set $\mathcal{C}$ of chains, we get a one-to-one correspondence $L \mapsto \mathcal{D}_\mathcal{P}(L)$ between the sublattices of $\mathcal{P}$ and the preorders spanned by a canonical sublattice $\mathcal{C}^\infty$ of $\mathcal{P}$. We then show that $L$ is a tight sublattice of the product of …
A distributive lattice structure ${\mathbf M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a distributive lattice. It is natural to ask which lattices are MDLs. We show that if a plane bipartite graph $G$ is elementary, then ${\mathbf M}(G)$ is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice $\mathbf{L}$ is an MDL if and only if each factor in any cartesian product decomposition of $\mathbf{L}$ …
Chain algebras of finite distributive lattices
本文定义并研究有限分配格极大链给出的环面代数，证明其正规且 Cohen–Macaulay，并刻画平面格情形的 Hibi 环结构、二次 Gröbner 基与 $h$-向量，以及高维格理想的生成元次数下界。
We characterize the finite distributive lattices which admit a complete valuation, that is bijective over a set of consecutive natural numbers, with the additional conditions of completeness (Definition 2.3). We prove that such lattices are downset lattices of finite posets of dimension at most two, and determine a realizer through a recursive relation between weights on the poset associated to valuation. The relation shows that the weights count chains in the complementary poset. Conversely, we prove that a valuation defined on a poset of dimension at most two, through the weight function whi…
The Computational Complexity of the Ungar Games on Distributive Lattices
本文解决了分配格上Ungar游戏结果判定的计算复杂性问题，证明其在$l(P) \le 3$的二分图Hasse图偏序集上为PSPACE完全，$l(P) \le 2$时为NP完全，$l(P) \le 1$时属于LOGSPACE。
In prior work, the authors, along with M. McClard, R. A. Proctor, and N. J. Wildberger, studied certain distributive lattice models for the `Weyl bialternants' (aka `Weyl characters') associated with the rank two root systems/Weyl groups. These distributive lattices were uniformly described as lattices of order ideals taken from certain grid-like posets, although the arguments connecting the lattices to Weyl bialternants were case-by-case depending on the type of the rank two root system. Using this connection with Weyl bialternants, these lattices were shown to be rank symmetric and rank unim…
Covering Distributive Lattices by Intervals
该研究将有限分配格的凸子集视为区间并集，证明当子集边界为原子集与coatoms集时存在最小区间覆盖，并给出偏序集的更一般结论及相关推测的支撑定理。
We propose an original approach to the problem of rankunimodality for Dyck lattices. It is based on a well known recursive construction of Dyck paths originally developed in the context of the ECO methodology, which provides a partition of Dyck lattices into saturated chains. Even if we are not able to prove that Dyck lattices are rank-unimodal, we describe a family of polynomials (which constitutes a polynomial analog of ballot numbers) and a succession rule which appear to be useful in addressing such a problem. At the end of the paper, we also propose and begin a systematic investigation of…
In this paper we are interested in functionals defined on completely distributive lattices and which are invariant under mappings preserving {arbitrary} joins and meets. We prove that the class of nondecreasing invariant functionals coincides with the class of Sugeno integrals associated with $\{0,1\}$-valued capacities, the so-called term functionals, thus extending previous results both to the infinitary case as well as to the realm of completely distributive lattices. Furthermore, we show that, in the case of functionals over complete chains, the nondecreasing condition is redundant. Charac…
It will be shown that the toric ring of the chain polytope of a finite partially ordered set is an algebra with straightening laws on a finite distributive lattice. Thus in particular every chain polytope possesses a regular unimodular triangulation arising from a flag complex.
Toric varieties associated with distributive lattices arise as a fibre of a flat degeneration of a Schubert variety in a minuscule. The singular locus of these varieties has been studied by various authors. In this article we prove that the number of diamonds incident on a lattice point $\a$ in a product of chain lattices is more than or equal to the codimension of the lattice. Using this we also show that the lattice varieties associated with product of chain lattices is smooth.
We provide a direct proof that a finite graded lattice with a maximal chain of left modular elements is supersolvable. This result was first established via a detour through EL-labellings in [McNamara-Thomas] by combining results of McNamara and Liu. As part of our proof, we show that the maximum graded quotient of the free product of a chain and a single-element lattice is finite and distributive.
Sperner theorems for unrelated copies of some partially ordered sets in a powerset lattice and minimum generating sets of powers of distributive lattices
该研究确定了含0、1元的有限偏序集在幂集格中的最大无关副本数Sp(U,n)并给出部分偏序集的估计，证明其与分配格幂的最小生成集函数互为左伴随，可计算多种分配格幂的最小生成集，为密码认证协议提供了适用的格结构。
A visual approach to symmetric chain decompositions of finite Young lattices
本文通过将有限Young格的序结构等价于膨胀单纯形格点的自然序及类型$A_n$根系统的权图，提出一种视觉方法，并完全图示描述了$L(3, n)$的对称链分解。
In this paper, we introduce the graph $G(S)$ of a bounded semilattice $S$, which is a generalization of the intersection graph of the substructures of an algebraic structure. We prove some general theorems about these graphs; as an example, we show that if $S$ is a product of three or more chains, then $G(S)$ is Eulerian if and only if either the length of every chain is even or all the chains are of length one. We also show that if $G(S)$ contains a cycle, then $girth(G(S)) = 3$. Finally, we show that if $(S,+,\cdot,0,1)$ is a dually atomic bounded distributive lattice whose set of dual atoms…
Two classes of posets with real-rooted chain polynomials
本文证明两类偏序集具有实根链多项式，其中一类为Cohen-Macaulay单纯偏序集的秩选择子偏序集，另一类为与有限Coxeter群相关的非交叉划分格，并展示了特定排列的下降计数器的实根性质。
Arc permutations and unimodal permutations were introduced in the study of triangulations and characters. This paper studies combinatorial properties and structures on these permutations. First, both sets are characterized by pattern avoidance. It is also shown that arc permutations carry a natural affine Weyl group action, and that the number of geodesics between a distinguished pair of antipodes in the associated Schreier graph, as well as the number of maximal chains in the weak order on unimodal permutations, are both equal to twice the number of standard Young tableaux of shifted staircas…
We are interested in representations and characterizations of lattice polynomial functions f:L^n -> L, where L is a given bounded distributive lattice. In companion papers [arXiv 0901.4888, arXiv 0808.2619], we investigated certain representations and provided various characterizations of these functions both as solutions of certain functional equations and in terms of necessary and sufficient conditions. In the present paper, we investigate these representations and characterizations in the special case when L is a chain, i.e., a totally ordered lattice. More precisely, we discuss representat…
Lattice Conditional Independence models are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be described via a transitive directed acyclic graph (TDAG) in which, as is normal for causal models, the conditional independence is in terms of conditioning on ancestors in the graph. We demonstrate that a parallel stream of research in algebra, the theory of Hibi ideals, not only maps directly to the LCI models but gives a vehicle to generalise the theo…
Ungarian Markov Chains
本文定义有限格上的 Ungarian Markov 链，分析其由顶到底的期望步数，并利用修剪格的脊与最后通过渗流给出经典 Cambrian 格和 $\nu$-Tamari 格的渐近上界。
We prove that the number of 01-fillings of a given stack polyomino (a polyomino with justified rows whose lengths form a unimodal sequence) with at most one 1 per column which do not contain a fixed-size northeast chain and a fixed-size southeast chain, depends only on the set of row lengths of the polyomino. The proof is via a bijection between fillings of stack polyominoes which differ only in the position of one row and uses the Hecke insertion algorithm by Buch, Kresch, Shimozono, Tamvakis, and Yong and the jeu de taquin for increasing tableaux of Thomas and Yong. Moreover, our bijection g…
The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of $n$. This paper proves the equivalence of the two definitions for all $m\geq 1$ and all $n\leq 4$. We also give a bijective proof of the joint symmetry property $C^{(m)}_n(q,t)=C^{(m)}_n(t,q)$ for all $m\geq 1$ and all $n\leq 4$. The proof is based on a general approach for proving joint symmetry that dissects a collection of objects into chains, and then passes…
Antichain polynomials of products of chains and minuscule posets
本文研究链与极小偏序集乘积的反链多项式，给出计数公式、回文性条件，证明零点性质并构造双射，解决猜想并给出反例。
Chow polynomials of rank-uniform labeled posets
研究秩均匀标记偏序集的Chow多项式实根性质，揭示其与matroid代数几何结构的关联。
Young's partition lattice $L(m,n)$ consists of unordered partitions having $m$ parts where each part is at most $n$. Using methods from complex algebraic geometry, R. Stanley proved that $L(m,n)$ is rank-symmetric, unimodal, and strongly Sperner. Moreover, he conjectured that $L(m,n)$ has a stronger property called symmetric chain decomposition. Despite many efforts, this conjecture has only been proved for $\min(m,n)\leq 4$. In this paper, we decompose $L(m,n)$ into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, …
Rowmotion Markov Chains
本文将确定性 rowmotion 推广为带概率参数的切换与半分配格马尔可夫链，刻画不可约性、平稳分布及混合时间，并证明布尔格情形具有 cutoff 现象。
Generalized snake posets, order polytopes, and lattice-point enumeration
本文研究了广义蛇偏序集的顺序聚类的Ehrhart理论，推导了其$h^*$-多项式的显式公式和递推关系，并证明了其$h^*$-向量的界性。
In 1978, motivated by E. Hückel's work in quantum chemistry, I. Gutman introduced the concept of the energy of a finite simple graph $G$ as the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. At the time of writing, the MathSciNet search for "Title=(graph energy) AND Review Text=(eigenvalue)" returns 351 publications, most of which going after Gutman's definition. A congruence $α$ of a finite algebra $A$ turns $A$ into a simple graph: we connect $x\neq y\in A$ by an edge iff $(x,y)\inα$; we let En$(α)$ be the energy of this graph. We introduce the congruence energ…
Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo
将戴德金数问题转化为布尔格理想格的层比重建问题，利用可逆固定层马尔可夫链估计层比，从而估计戴德金数M(n)，并发现n=9时中心层附近存在双肩特征。
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