On a generalization of P(3,n)
Agustin Moreno
利用偏序集表示论的技巧，给出了一个关于一个正整数划分为三个多边形数的个数公式。
By using techniques of poset representation theory, we present a formula for the number of partitions of a positive integer into three polygonal numbers.
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We compute all the equivariant Euler characteristics of the $Σ_n$-poset of partitions of the $n$ element set.
We present a method of computing the generating function $f_P(\x)$ of $P$-partitions of a poset $P$. The idea is to introduce two kinds of transformations on posets and compute $f_P(\x)$ by recursively applying these transformations. As an application, we consider the partially ordinal sum $P_n$ of $n$ copies of a given poset, which generalizes both the direct sum and the ordinal sum. We show that the sequence $\{f_{P_n}(\x)\}_{n\ge 1}$ satisfies a finite system of recurrence relations with respect to $n$. We illustrate the method by several examples, including a kind of 3-rowed posets and the…
A poset structure on quasifibonacci partitions
本文研究了正整数划分为不同拟Fibonacci数的问题。一个有向图和偏序集结构被构造在这样的划分的集合上。进一步讨论了这些偏序集之间的对称关系和递归关系。最后证明了Robbins关于拟Fibonacci幂级数系数的结果的一个强推广。
This paper studies topological properties of the lattices of non-crossing partitions of types A and B and of the poset of injective words. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This strengthens the well-known facts that these posets are Cohen-Macaulay. Our results rely on a new poset fiber theorem which turns out to be a useful tool to prove double (homotopy) Cohen-Macaulayness of a poset. Applications to complexes of injective words are also included.
We prove that the posets of connected components of intersections of toric and elliptic arrangements defined by root systems are EL-shellable and we compute their homotopy type. Our method rests on Bibby's description of such posets by means of "labeled partitions": after giving an EL-labeling and counting homology chains for general posets of labeled partitions, we obtain the stated results by considering the appropriate subposets.
Poset Partitions and the Combinatorics of the $\textbf{cd}$-Index
本文提出了一种新的欧拉 poset 类，称为 S-分割 poset，证明其具有非负的 cd 指数，并引入了 SE-分割 poset 的半欧拉版本。
Plane partitions and rowmotion on rectangular and trapezoidal posets
本文定义矩形与梯形偏序集标号间的双有理映射，热带化后给出平面分拆双射，证明其关于行移动等变，解决猜想并表明有限阶。
Posets of Hyper $(b,t)$-ary Partitions are Distributive Lattices
该研究推广了超二进制分拆偏序集为分配格的结果，证明超 $(b,t)$ 元分拆（部分为 $b$ 的幂，重数至多 $t\ge b$）的偏序集也是分配格，其并不可约元素同构于栅栏偏序集。
The $P$-partition generating function of a (naturally labeled) poset $P$ is a quasisymmetric function enumerating order-preserving maps from $P$ to $\mathbb{Z}^+$. Using the Hopf algebra of posets, we give necessary conditions for two posets to have the same generating function. In particular, we show that they must have the same number of antichains of each size, as well as the same shape (as defined by Greene). We also discuss which shapes guarantee uniqueness of the $P$-partition generating function and give a method of constructing pairs of non-isomorphic posets with the same generating fu…
We define semi-pointed partition posets, which are a generalisation of partition posets and show that they are Cohen-Macaulay. We then use multichains to compute the dimension and the character for the action of the symmetric groups on their homology. We finally study the associated incidence Hopf algebra, which is similar to the Fa{à} di Bruno Hopf algebra.
An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemerédi proved that any on-line algorithm could be forced to use $\binom{w+1}{2}$ chains to partition a poset of width $w$. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In a survey paper by Bosek et al., it is shown that Szemerédi's argument could be improved to obtain a lower bound almost twice as good. Variants of the problem were considered where the class is restricted to posets of bounde…
Homology of generalized partition posets
我们定义了一个偏序集的分区相关联的操作数。我们证明了操作数是Koszul当且仅当偏序集是Cohen-Macaulay。一方面，这种刻画允许我们计算偏序集的同调。这个同调由Koszul对偶运算符给出。另一方面，我们得到了新的方法来证明一个操作数是Koszul。
We adapt here the computation of characters on incidence Hopf algebras introduced by W. Schmitt in the 1990s to a family mixing bounded and unbounded posets. We then apply our results to the family of hypertree posets and partition posets. As a consequence, we obtain some enumerative formulas and a new proof for the computation of the Moebius numbers of the hypertree posets. Moreover, we compute the coproduct of the incidence Hopf algebra and recover a known formula for the number of hypertrees with fixed valency set and edge sizes set.
The minimum number of chains in a noncrossing partition of a poset
本文引入偏序集非交叉划分概念，并研究其中所含链的最小数目，给出了一般偏序集情形下该数量的刻画，从而反映偏序集的结构复杂性。
Block-transitive designs with a poset of imprimitive partitions
本文研究了具有特定偏序集的块设计，通过广义wreath产品理论，给出了块传递2-设计的条件，并提供了多个偏序集的显式例子。
Polyunsaturated Posets and Graphs and the Greene-Kleitman Theorem
有限偏序集的链划分给k个反链的并的大小带来了一个自然的上界。一个链划分是k-饱和的，如果达到这个界限。Greene和Kleitman证明了，对于每个k，每个有限偏序集都有一个同时k-和k+1-饱和的链划分。West证明了Greene-Kleitman定理在强意义上是最好可能的，对每个c \ge 4，展示了一个偏序集，它具有基数为c的最长链，并且对任何不同的、非连续的k，l < c，没有k-和l-饱和链划分。我们称这种偏序集为多不饱和偏序集。给出了具有给定高度、宽度和基数的多不饱和偏序集存在的充要条件。我们证明这些结果在更一般的情况下，图满足类似的格林-克莱特曼定理。最后，我们讨论了反链分拆的类似结果。
For each composition $\vec{c}$ we show that the order complex of the poset of pointed set partitions $Π^{\bullet}_{\vec{c}}$ is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition $\vec{c}$. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module $S^{B}$ where $B$ is a border strip associated to the composition. We also study the filter of pointed set partitions generated by a knapsack integer partition and show the analogous results on homotopy type and action on the top homol…
We consider the poset of vector partitions of $[n]$ into $s$ components, denoted $Π_{n,s}$, which was first defined by Stanley in 1978. In 1986, Sagan showed that this poset is CL-shellable, and hence has the homotopy type of a wedge of spheres of dimension $(n-2)$. We extend on this result to show that $Π_{n,s}$ is edge-lexicographic shellable. We then use this edge-labeling to find a recursive expression for the number of spheres, and show that when $s=1$ the number of spheres is equal to the number of complete non-ambiguous trees, first defined in 2014 by Aval, Boussicault, Bouvel and Silim…
We describe some open problems related to homology representations of subposets of the partition lattice, beginning with questions first raised in Stanley's work on group actions on posets.
Poset probability in two-row partition posets
针对两行划分λ=(λ1,λ2)，给出划分偏序集Pλ中偏序概率Prob(Pλ; α<β)的显式公式，并利用钩长公式和Jacobi-Trudi-Aitken行列式公式将其表达为有理式，同时计算当臂长趋于无穷且差有界时的极限概率。
Anderson established a connection between core partitions and order ideals of certain posets by mapping a partition to its $β$-set. In this paper, we give a characterization of the poset $P_{(s,s+1,s+2)}$ whose order ideals correspond to $(s,s+1,s+2)$-core partitions. Using this characterization, we obtain the number of $(s,s+1,s+2)$-core partitions, the maximum size and the average size of an $(s,s+1,s+2)$-core partition, confirming three conjectures posed by Amdeberhan.
For any finite poset $P$ we have the poset of isotone maps $\text{Hom}(P,\mathbb{N})$, also called $P^{op}$-partitions. To any poset ideal ${\mathcal J}$ in $\text{Hom}(P,\mathbb{N})$, finite or infinite, we associate monomial ideals: the letterplace ideal $L({\mathcal J},P)$ and the Alexander dual co-letterplace ideal $L(P,{\mathcal J})$, and study them. We derive a class of monomial ideals in $k[x_p, p \in P]$ called $P$-stable. When $P$ is a chain we establish a duality on strongly stable ideals. We study the case when ${\mathcal J}$ is a principal poset ideal. When $P$ is a chain we constr…
On the structure of the d-indivisible noncrossing partition posets
本文研究d-不可分非交叉划分偏序集，通过生成函数、Hopf代数对映体、EL-标号及d-停车树等工具，获得新计数结果并揭示其结构性质。
A note on the Möbius uncertainty principle for posets
本文研究了偏序集的莫比乌斯不确定原理的两个一般化，提出新的条件并推翻了之前的猜想。
The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of $G$. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the noncrossing partitions of $L_G$. Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that…
We examine the lattice of all order congruences of a finite poset from the viewpoint of combinatorial algebraic topology. We will prove that the order complex of the lattice of all nontrivial order congruences (or order-preserving partitions) of a finite $n$-element poset $P$ with $n\geq 3$ is homotopy equivalent to a wedge of spheres of dimension $n-3$. If $P$ is connected, then the number of spheres is equal to the number of linear extensions of $P$. In general, the number of spheres is equal to the number of cyclic extensions of $P$.
Consider the noncrossing set partitions of an $n$-element set which either do not contain the block $\{n-1,n\}$, or which do not contain the singleton block $\{n\}$ whenever $1$ and $n-1$ are in the same block. In this article we study the subposet of the noncrossing partition lattice induced by these elements, and show that it is a supersolvable lattice, and therefore lexicographically shellable. We give a combinatorial model for the NBB bases of this lattice and derive an explicit formula for the value of its Möbius function between least and greatest element. This work is motivated by a rec…
A sefl-dual poset on objects counted by the Catalan numbers
我们研究了132个按降序排列的避免$n$ -置换的偏序集$P$。我们表明，这个偏序集是“粗化”的研究偏序集$Q$的非交叉分区。换句话说，如果$x<y$在$Q$中，则$f(y)<f(x)$在$P$中，其中$f$是从非交叉分区集合到132-避免排列集合的规范双射。这使我们能够证明$P$的许多性质。
We introduce the notion of a \emph{Whitney dual} of a graded poset. Two posets are Whitney duals to each other if (the absolute value of) their Whitney numbers of the first and second kind are interchanged between the two posets. We define new types of edge and chain-edge labelings which we call \emph{Whitney labelings}. We prove that every graded poset with a Whitney labeling has a Whitney dual. Moreover, we show how to explicitly construct a Whitney dual using a technique involving quotient posets. As applications of our main theorem, we show that geometric lattices, the lattice of noncrossi…
We define combinatorially a partial order on the set partitions and show that it is equivalent to the Bruhat-Chevalley-Renner order on the upper triangular matrices. By considering subposets consisting of set partitions with a fixed number of blocks, we introduce and investigate "Stirling posets." As we show, the Stirling posets have a hierarchy and they glue together to give the whole set partition poset. Moreover, we show that they (Stirling posets) are graded and EL-shellable. We offer various reformulations of their length functions and determine the recurrences for their length generating…
Inverse limits of various posets
本文研究了各种偏序集的逆极限，探讨了它们的树和格结构，并扩展到带符号的限制增长函数及多种格结构。
Posets of annular non-crossing partitions of types B and D
研究了B型环形非交叉置换集$\sncb (p,q)$，并引入了相应的B型环形非交叉分拆集$\ncb (p,q)$，其中$p$和$q$是两个正整数.证明了$\sncb (p,q)$和$\ncb (p,q)$之间的自然双射是偏序集同构，其中$\sncb (p,q)$上的偏序是由超八面体群$B_{p+q}$导出的，而$\ncb (p,q)$上的偏序是由逆加细导出的.在$q=1$的情况下，我们证明了$\ncb (p,1)$是关于逆加细序的格。我们指出，一个类似的发展可以在D型中进行，其中人们得到$\sncd (p,q)$和$\ncd (p,q)$之间的规范同构。对于$q=1$，偏序集$\ncd (p,1)$与Athanasiadis和Reiner在2004年的一篇论文中构造的偏序集“$NC^{(D)} (p+1)$”重合，并且根据该论文的结果是一个格。
The gold partition conjecture and the Lexicographic sum of posets
本文证明了若有限偏序集Q满足Gold划分猜想，则其与任意有限偏序集P的字典序和也满足该猜想。
We use k-Schur functions to get the minimal boundary of the k-bounded partition poset. This permits to describe the central random walks on affine Grassmannian elements of type A and yields a polynomial expression for their drift. We also recover Rietsch's parametriza-tion of totally nonnegative unitriangular Toeplitz matrices without using quantum cohomology of flag varieties. All the homeomorphisms we define can moreover be made explicit by using the combinatorics of k-Schur functions and elementary computations based on Perron-Frobenius theorem.
The $(P, ω)$-partition generating function of a labeled poset $(P, ω)$ is a quasisymmetric function enumerating certain order-preserving maps from $P$ to $\mathbb{Z}^+$. We study the expansion of this generating function in the recently introduced type 1 quasisymmetric power sum basis $\{ψ_α\}$. Using this expansion, we show that connected, naturally labeled posets have irreducible $P$-partition generating functions. We also show that series-parallel posets are uniquely determined by their partition generating functions. We conclude by giving a combinatorial interpretation for the coefficients…
A generalization of Chebyshev polynomials and non rooted posets
本文给出了切比雪夫多项式的一个推广，并利用它描述了偏序默比乌斯函数{a1，... as，c| ai<c}，其中包含了对Björner，Sagan，Vatter猜想的肯定回答。[5，10]
We introduce the notions of Schröder shape and of Schröder tableau, which provide some kind of analogs of the classical notions of Young shape and Young tableau. We investigate some properties of the partial order given by containment of Schröder shapes. Then we propose an algorithm which is the natural analog of the well known RS correspondence for Young tableaux, and we characterize those permutations whose insertion tableaux have some special shapes. The last part of the article relates the notion of Schröder tableau with those of interval order and of weak containment (and strong avoidance…
To every labeled poset (P,ω), one can associate a quasisymmetric generating function for its (P,ω)-partitions. We ask: when do two labeled posets have the same generating function? Since the special case corresponding to skew Schur function equality is still open, a complete classification of equality among (P,ω) generating functions is likely too much to expect. Instead, we determine necessary conditions and separate sufficient conditions for two labeled posets to have equal generating functions. We conclude with a classification of all equalities for labeled posets with small numbers of line…
On $P$-partitions Extended by Two-Rowed Plane Partitions
本文引入 $φ_{z}$、$ψ_{z}$ 两个算子，建立两行平面分拆扩展偏序集的 $P$-分拆新计数公式，并推广到多种平面分拆与偏序集结构。
Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We present a unifying perspective on ASMs and other combinatorial objects by studying a certain tetrahedral poset and its subposets. We prove the order ideals of these subposets are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self-complementary plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux, Catalan objects, tournaments, and totally symmetric plane partitions. …
Interpreting the two variable Distance enumerator of the Shi hyperplane arrangement
给出了在$n$维中Shi超平面排列$\Sh_n$的距离计数器的双变量精化$D_{\Sh_n}(q,t)$的系数的解释.这个两个变量的细化是由Stanley \cite{stan-rota}为一般的$r$ -扩展的施超平面安排定义的。当$r=1$ .我们定义了数$(n+1)^{n-1}$的三个自然三维划分。第一个产生于停车功能的长度$n$，第二个从特殊偏序集上的$n$顶点定义的Athanasiadis和第三个从生成树上的$n+1$顶点。我们把这三种划分分别称为停车划分、树偏序集划分和生成树划分。我们发现，停车分区的部分之一是相同的边缘标记树的数量与标签集$\{1,2,...,n\}$上$n+1$未标记的顶点。我们证明了停车分区majorises树偏序集分区和猜想，生成树分区也majorises树偏序集分区。
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 …
In this paper, we investigate the notion of partition of a finite partially ordered set (poset, for short). We will define three different notions of partition of a poset, namely, monotone, regular, and open partition. For each of these notions we will find three equivalent definitions, that will be shown to be equivalent. We start by defining partitions of a poset in terms of fibres of some surjection having the poset as domain. We then obtain combinatorial characterisations of such notions in terms of blocks, without reference to surjection. Finally, we give a further, equivalent definition …
To an extended permutohedron we associate the weighted integer points enumerator, whose principal specialization is the $f$-polynomial. In the case of poset cones it refines Gessel's $\mathsf{P}$-partitions enumerator. We show that this enumerator is a quasisymmetric function obtained by universal morphism from the Hopf algebra of posets.
A self-dual poset on objects counted by the Catalan numbers and a type-B analogue
我们引入了两个偏序集，$P^A_n$和$P^B_n$，与类型A和类型B的非交叉分划格具有相同的基数。$P^A_n$和$P^B_n$的基集是对称和超八面体群的子集，由避免某些模式的排列组成。顺序关系由下降集的（严格）包含给出。在每种情况下，通过一个显式的保序双射，我们证明了限制置换的偏序集是非交叉划分上加细序的扩展。这些置换偏序集的几个结构性质如下，包括自对偶和强Sperner性质。我们还讨论了偏序集$Q^A_n$和$Q^B_n$同样与非交叉分区，定义的对称和超八面体群的适当的模式避免子集的过剩集。
The immersion poset on partitions
本文定义分拆上的浸入偏序集以刻画 $GL_N(\mathbb{C})$ 不可约表示的浸入关系，研究表的单射、钩形与两列分拆、标准浸入极大元，并证明相关幂和函数的 Schur 正性。
Hooks are prominent in representation theory (of symmetric groups) and they play a role in number theory (via cranks associated to Ramanujan's congruences). A partition of a positive integer $n$ has a Young diagram representation. To each cell in the diagram there is an associated statistic called hook length, and if a number $t$ is absent from the diagram then the partition is called a $t$-core. A partition is an $(s,t)$-core if it is both an $s$- and a $t$-core. Since the work of Anderson on $(s,t)$-cores, the topic has received a growing attention. This paper expands the discussion to multi…
Peterson and Proctor obtained a formula which expresses the multivariate generating function for $P$-partitions on a $d$-complete poset $P$ as a product in terms of hooks in $P$. In this paper, we give a skew generalization of Peterson--Proctor's hook formula, i.e., a formula for the generating function for $(P \setminus F)$-partitions for a $d$-complete poset $P$ and its order filter $F$, by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant $K$-theory of Kac--Moody partial flag varieties. This generalization provides…
A generalization of RSK to $d$-complete posets
本文通过定义 Robinson-Schensted-Knuth 双射在 d-完全偏序集上的推广，给出了 d-完全偏序集钩长公式的一个初等纯组合新证明。
The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. The chain polynomials of the partition lattices and their standard type $B$ analogues are shown to have only real roots. The real-rootedness of the chain polynomial is conjectured for all geometric lattices and is shown to be preserved by the pyramid and the prism operations on Cohen--Macaulay posets. As a result, new families of convex polytopes whose face lattices have real-rooted chain polynomials are presented. An application to the face enumeration of the second barycentri…
Colored posets and colored quasisymmetric functions
着色拟对称函数与经典拟对称函数一样，也是一个具有自然峰子代数的霍普夫代数.我们展示了这些代数是如何作为着色偏序集代数的图像出现的。为了实现这种方法，我们引入了$P$ -分区和富集的$P$ -分区的有色类似物。我们还框架我们的结果在阿吉亚尔，Bergeron和Sottile的组合霍普夫代数和它的有色模拟理论。
First-Fit is a greedy algorithm for partitioning the elements of a poset into chains. Let $\textrm{FF}(w,Q)$ be the maximum number of chains that First-Fit uses on a $Q$-free poset of width $w$. A result due to Bosek, Krawczyk, and Matecki states that $\textrm{FF}(w,Q)$ is finite when $Q$ has width at most $2$. We describe a family of posets $\mathcal{Q}$ and show that the following dichotomy holds: if $Q\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \le 2^{c(\log w)^2}$ for some constant $c$ depending only on $Q$, and if $Q\not\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \ge 2^w - 1$.
Let $X$ be a finite set, $Z \subseteq X$ and $y \notin X$. Marcel Erné showed in 1981, that the number of posets on $X$ containing $Z$ as an antichain equals the number of posets $R$ on $X \cup \{ y \}$ in which the points of $Z \cup \{ y \}$ are exactly the maximal points of $R$. We prove the following generalization: For every poset $Q$ with carrier $Z$, the number of posets on $X$ containing $Q$ as an induced sub-poset equals the number of posets $R$ on $X \cup \{ y \}$ which contain $Q^d + A_y$ as an induced sub-poset and in which the maximal points of $Q^d + A_y$ are exactly the maximal p…
Three families of posets depending on a nonnegative integer parameter $m$ are introduced. The underlying sets of these posets are enumerated by the $m$-Fuss Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. The three families of posets are related: they fit into a chain for the order extension relation and they share some properties. Two associative algebras are constructed as quotients of generalizations of the Malvenuto-Reutenauer algebra. Their products describe intervals of our analogues of Stanley lattices and…
Lie-operads and operadic modules from poset cohomology
本文建立了一种一般形式体系，在具有自然递归结构的偏序集族上同调上产生算畴与算畴模结构，该框架超越了装饰划分与算畴Koszul对偶性，并包含元交换李算畴和Kontsevich树算畴等实例。
Generalizations of Eulerian partially ordered sets, flag numbers, and the Mobius function
一个偏序集是r-厚的，如果每个非空开区间至少包含r个元素.本文研究了分次r-厚偏序集的标志向量，证明了包含它们的最小凸锥同构于所有分次偏序集的标志向量的锥。它还定义了Mobius函数的k-模拟和k-欧拉偏序集，它们是2k-厚的。给出了k-欧拉偏序集的几个刻画。证明了k-欧拉偏序集的旗向量的广义Dehn-Sommerville方程。证明了一个新的不等式对于秩8的欧拉偏序集是有效的和尖锐的。
In 1980, Edelman defined a poset on objects called the noncrossing 2-partitions. They are closely related with noncrossing partitions and parking functions. To some extent, his definition is a precursor of the parking space theory, in the framework of finite reflection groups. We present some enumerative and topological properties of this poset. In particular, we get a formula counting certain chains, that encompasses formulas for Whitney numbers (of both kinds). We prove shellability of the poset, and compute its homology as a representation of the symmetric group. We moreover link it with tw…
Exploring Parking Functions: Poset and Polytope Perspectives
本文针对经典组合对象停车函数，提出从非交叉划分偏序集和多胞体两种视角研究其结构与性质，为停车函数研究提供新的分析角度。
In a previous paper (arXiv:1608.02262), we used computer-assisted methods to find explicit expressions for the moments of the size of a uniform random (n,n+1)-core partition with distinct parts. In particular, we conjectured that the distribution is asymptotically normal. However, our analysis hinged on a characterization of (n,n+1)-core partitions given by Straub, which is not readily generalized to other families of simultaneous core partitions. In another paper (arXiv:1611.05775) with Doron Zeilberger, we made use of the characterization in terms of posets to analyze (2n+1,2n+3)-core partit…
A New Approach to Order Polynomials of Labeled Posets and Their Generalizations
本文首先利用$(P, ω)$的$ω$ -图的邻接矩阵给出了有限标号偏序集$(P, ω)$的序多项式$Ω(\Pw; t)$和欧拉多项式$e(\Pw; λ)$的计算公式。然后，我们导出了$Ω(\Pw; t)$和$e(\Pw; λ)$的各种递推公式，并讨论了这些公式在Bernoulli数和伯努利多项式中的一些应用.最后，我们给出了一个在向量空间上使用单个线性算子的递归算法。该算法提供了一种构造一类新的标记偏序集不变量$(\Pw)$的统一方法，包括阶多项式$Ω(\Pw; t)$和不变量$\tilde e(\Pw; λ) =\frac {e(\Pw; λ)}{(1-λ)^{|P|+1}}$ .著名的拟对称函数不变量的标记偏序集和进一步推广我们的建设进行了讨论。
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