William Y. C. Chen, Deheng Xu
By a re-examination of MacMahon's original proof of his celebrated theorem on the distribution of the major indices over permutations, we give a reformulation of his argument in terms of the structure of labeled partitions. In this framework, we are able to establish a decomposition theorem for labeled partitions that leads to a simple bijective proof of Wachs' formula on the q-derangement numbers.
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The absolute values of the perfect matching derangement graph's eigenvalues almost follow the lexicographic order of partitions
本研究通过推导K_{2n}的完美匹配错位排列图ℳ_{2n}特征值的新递推公式，证明了其特征值绝对值存在与普通错位排列图类似的、近似遵循分划字典序的性质，拓展了错位排列图特征值序的相关结论。
Extensions of a set partition obtained by imposing bounds on the size of the parts and the coloring of some of the elements are examined. Combinatorial properties and the generating functions of some counting sequences associated with these partitions are established. Connections with Riordan arrays are presented.
We consider the Cayley graph on the symmetric group Sn generated by derangements. It is well known that the eigenvalues of this grpah are indexed by partitions of n. We investigate how these eigenvalues are determined by the shape of their corresponding partitions. In particular, we show that the sign of an eigenvalue is the parity of the number of cells below the first row of the corresponding Ferrers diagram. We also provide some lower and upper bounds for the absolute values of these eigenvalues.
Jack Derangements
本文通过引入主下 hook 积这一 Jack 类比工具，给出了 Jack 特征在无单元素分拆上求和的组合表达式，并由此解决了置换错排图和完美匹配错排图特征值的公开问题。
On the proportion of derangements in affine classical groups
研究仿射经典群中错位元素及其p次幂阶错位元素的比例公式，通过整数划分生成函数和多项式恒等式推导出精确表达式。
In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations in the sense that the colors are endowed with a cyclic structure. We use labeled partitions with colored permutations to derive the generating function of the $\mathrm{fmaj}_k$ indices of colored permutations. The second result is a combinatorial treatment of a relation on the q-derangement numbers with respect to colored permutations which leads to the formula of Chow for signed permutations and the formula of Faliharimalala and Zeng [10] on colored permutations. The third result is an…
The number of conjugate classes of derangements of order $n$ is the same as the number $h(n)$ of the restricted partitions with every portion greater than $1$. It is also equal to the number of isotopy classes of $2\times n$ Latin rectangles. Sometimes the exact value is necessary, while sometimes we need the approximation value. In this paper, a recursion formula of $h(n)$ will be obtained, also will some elementary approximation formulae with high accuracy for $h(n)$ be presented. Although we may obtain the value of $h(n)$ in some computer algebra system, it is still meaningful to find an ef…
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.
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…
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…
A labeled set partition is a partition of a set of integers whose arcs are labeled by nonzero elements of an abelian group $A$. Inspired by the action of the linear characters of the unitriangular group on its supercharacters, we define a group action of $A^n$ on the set of $A$-labeled partitions of an $(n+1)$-set. By investigating the orbit decomposition of various families of set partitions under this action, we derive new combinatorial proofs of Coker's identity for the Narayana polynomial and its type B analogue, and establish a number of other related identities. In return, we also prove …
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…
The Stirling number of the second kind $n\brace k$ counts the number of ways to partition a set of $n$ labeled balls into $k$ non-empty unlabeled cells. As an extension of this, we consider $b_1+b_2+\ldots+b_n$ balls with $b_1$ balls labeled $1$, $b_2$ balls labeled $2$, $\ldots$, $b_n$ balls labeled $n$ and $c_1+c_2+\ldots+c_k$ cells with $c_1$ cells labeled $1$, $c_2$ cells labeled $2$, $\ldots$, $c_k$ cells labeled $k$ and then we called the number of ways to partition the set of these balls into non-empty cells of these types as the \textit{mixed partition numbers}. As an application, we g…
We present a bijective algorithm with which an arbitrary permutation decomposes canonically into elementary blocks which we call families, which are sets with a specified number of ascents and descents. We show that families, arranged in an arbitrary order in a sequence, are in bijection with permutations. The permutation decomposes canonically, by inserting parentheses, into a tree having as nodes a class of permutations which we call primitive. Primitive permutations can be assembled from very simple data. The data for the trees into which a permutation decomposes can be written in a form si…
In this brief note a straightforward combinatorial proof for an identity directly connecting rooted forests and unordered set partitions is provided. Furthermore, references that put this type of identity in the context of forest volumes and multinomial identities are given.
We introduce and study the lattice of generalized partitions, called weighted partitions. This lattice possesses similar properties of the lattice of partitions. By use of the pictorial representation of a weighted partition, the total number is given by the successive Stirling transforms of the Stirling number of the second kind. We construct an explicit $EL$-labeling on the lattice, which implies this lattice is $EL$-shellable and hence shellable. We compute the Möbius function and the characteristic polynomial by use of a pictorial representation of a maximal decreasing chain. Further, a ma…
Let P be a polygon whose vertices have been colored (labeled) cyclically with the numbers 1,2,...,c. Motivated by conjectures of Propp, we are led to consider partitions of P into k-gons which are proper in the sense that each k-gon contains all c colors on its vertices. Counting the number of proper partitions involves a generalization of the k-Catalan numbers. We also show that in certain cases, any proper partition can be obtained from another by a sequence of moves called flips.
On the second largest eigenvalue of certain graphs in the perfect matching association scheme
本文研究了完美匹配关联方案中特定划分标记的矩阵的第二大特征值，提出并验证了关于其特征空间的猜想。
Labeling and folding multi-labeled trees
本文推广Erdős和Székely的标记算法至多标记树，利用多重集排序刻画其划分，并证明可标记系统发育网络等价于折叠过程稳定的网络，且给出相应双射。
On Ward Numbers and Increasing Schröder Trees
本文证明Ward数也计数递增Schröder树，构造了其与全划分树的保型双射，并通过加权推广和函数方程求解建立了与带标签有根树及Lagrange反演变体的新联系。
New Identities in the Character Table of Symmetric Groups involving Riordan Numbers
针对Amdeberhan提出的对称群特征标表带符号列和等式，证明其在特殊情况下成立，得到不可约特征标度数和的新等式及Riordan数的新组合解释，是首个施加分拆奇偶性条件的相关定理。
Regular Schur labeled skew shape posets and their 0-Hecke modules
该研究在Stanley的$P$-分拆猜想假设下，刻画了正则Schur标记斜形状偏序集的性质，分类其线性扩张集合与相关0-Hecke模，并构造了模关于Schur基的特异滤过。
An ordered set-partition (or preferential arrangement) of n labeled elements represents a single ``hierarchy''; these are enumerated by the ordered Bell numbers. In this note we determine the number of ``hierarchical orderings'' or ``societies'', where the n elements are first partitioned into m <= n subsets and a hierarchy is specified for each subset. We also consider the unlabeled case, where the ordered Bell numbers are replaced by the composition numbers. If there is only a single hierarchy, we show that the average rank of an element is asymptotic to n/(4 log 2) in the labeled case and t…
Stirling complexes
本文定义并研究资源分配问题产生的Stirling复形，证明其同伦等价于同维球面楔和，并给出球面计数公式，且同伦类型与树结构无关。
Arithmetic Properties of Mixed Stirling Numbers of the second kind
本文研究第二类混合斯特林数，建立其递推关系与指数生成函数，分析其模素数$p$及$p^2$的性质，推广经典Touchard同余式，揭示其具有区别于经典集合分拆的独特数论特征。
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