M. Barnabei, F. Bonetti, M. Silimbani
We define an analogue of signed Eulerian numbers $f_{n,k}$ for involutions of the symmetric group and derive some combinatorial properties of this sequence. In particular, we exhibit both an explicit formula and a recurrence for $f_{n,k}$ arising from the properties of its generating function.
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The Eulerian distribution on the involutions of the hyperoctahedral group is unimodal
本文将对称群对合上欧拉分布的单峰性推广到超八面体群（带色置换群）的对合上，并用带符号拟对称函数计算其生成函数。
We present an extensive study of the Eulerian distribution on the set of self evacuated involutions, namely, involutions corresponding to standard Young tableaux that are fixed under the Sch$\ddot{\textrm{u}}$tzenberger map. We find some combinatorial properties for the generating polynomial of such distribution, together with an explicit formula for its coefficients. Afterwards, we carry out an analogous study for the subset of self evacuated involutions without fixed points.
This paper is devoted to a detailed study of nowhere-zero flows on signed eulerian graphs. We generalise the well-known fact about the existence of nowhere-zero $2$-flows in eulerian graphs by proving that every signed eulerian graph that admits an integer nowhere-zero flow has a nowhere-zero $4$-flow. We also characterise signed eulerian graphs with flow number $2$, $3$, and $4$, as well as those that do not have an integer nowhere-zero flow. Finally, we discuss the existence of nowhere-zero $A$-flows on signed eulerian graphs for an arbitrary abelian group~$A$.
A signed circuit cover of a signed graph is a natural analog of a circuit cover of a graph, and is equivalent to a covering of its corresponding signed-graphic matroid with circuits. It was conjectured that a signed graph whose signed-graphic matroid has no coloops has a 6-cover. In this paper, we prove that the conjecture holds for signed Eulerian graphs.
In order to study signed Eulerian numbers, we introduce permutations of a particular type, called parity-alternate permutations, because they take even and odd entries alternately. The objective of this paper is twofold. The first is to derive several properties of those permutations, by subdividing them into even and odd ones. The second is to discuss relationships between those and signed Eulerian numbers. Divisibility properties by prime powers are also deduced for signed Eulerian numbers and several related numbers.
The numbers of even and odd permutations with a given ascent number are investigated using an operator that was previously introduced by the author. Their difference is called a signed Eulerian number. By means of the operator the recurrence relation for signed Eulerian numbers is deduced, which was obtained by an analytic method. Our approach is straightforward and enables us to deduce other properties including divisibility by prime powers.
In this paper, we study Eulerian polynomials for permutations and signed permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$. Properties of these polynomials, including recurrence relations and unimodality are discussed. In particular, we give a unified proof of the fact that these polynomials are unimodal with modes in the middle.
Phil Hanlon proved that the coefficients of the chromatic polynomial of a graph G are equal (up to sign) to the dimensions of the summands in a Hodge-type decomposition of the top homology of the coloring complex for G. We prove a type B analogue of this result for chromatic polynomials of signed graphs using hyperoctahedral Eulerian idempotents.
Let $\mathcal I_n$ and $\mathcal J_n$ denote the set of involutions and fixed-point free involutions of $\{1, \dots, n\}$, respectively, and let $\text{des}(π)$ denote the number of descents of the permutation $π$. We prove a conjecture of Guo and Zeng which states that $I_n(t) := \sum_{π\in \mathcal I_n} t^{\text{des}(π)}$ is $γ$-positive for $n \ge 1$ and $J_{2n}(t) := \sum_{π\in \mathcal J_{2n}} t^{\text{des}(π)}$ is $γ$-positive for $n \ge 9$. We also prove that the number of $(3412, 3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to the number of separable …
We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S_5, S_6, and S_7 for both permutations and involutions.
Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point free involutions) of {1,...,n} with k descents. Motivated by Brenti's conjecture which states that the sequence I_{n,0}, I_{n,1},..., I_{n,n-1} is log-concave, we prove that the two sequences I_{n,k} and J_{2n,k} are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers a_{n,k} such that $$ \sum_{k=0}^{n-1}I_{n,k}t^k=\sum_{k=0}^{\lfloor (n-1)/2\rfloor}a_{n,k}t^{k}(1+t)^{n-2k-1}. $$ This statement is stronger than the unimodality of I_{n,k} but is also interesting in its own right.
A signed circuit is a minimal signed graph (with respect to inclusion) that admits a nowhere-zero flow. We show that each flow-admissible signed graph on $m$ edges can be covered by signed circuits of total length at most $(3+2/3)\cdot m$, improving a recent result of Cheng et al. [manuscript, 2015]. To obtain this improvement we prove several results on signed circuit covers of trees of Eulerian graphs, which are connected signed graphs such that removing all bridges results in a collection of Eulerian graphs.
A symmetry of $(t,q)$-Eulerian numbers of type $B$ is combinatorially proved by defining an involution preserving many important statistics on the set of permutation tableaux of type $B$. This involution also proves a symmetry of the generating polynomial $\hat{D}_{n, k}(p,q,r)$ of number of crossings and alignments, and hence $q$-Eulerian numbers of type $A$ defined by L. Williams. By considering a restriction of our bijection, we were led to define a new statistic on the permutations of type $D$ and $(t,q)$-Eulerian numbers of type $D$, which is proved to have a nice symmetry as well. We con…
Signed circuit $6$-covers of signed $K_4$-minor-free graphs
本文针对符号图的符号回路覆盖问题，证明了Fan猜想对无K₄子式的符号图成立，即每个可覆盖的无K₄子式符号图都具有符号回路6-覆盖，并说明6-覆盖是最优可能的。
An involution in a Coxeter group has an associated set of involution words, a variation on reduced words. These words are saturated chains in a partial order first considered by Richardson and Springer in their study of symmetric varieties. In the symmetric group, involution words can be enumerated in terms of tableaux using appropriate analogues of the symmetric functions introduced by Stanley to accomplish the same task for reduced words. We adapt this approach to the group of signed permutations. We show that involution words for the longest element in the Coxeter group $C_n$ are in bijecti…
A relationship between signed Eulerian polynomials and the classical Eulerian polynomials on $\mathfrak{S}_n$ was given by Désarménien and Foata in 1992, and a refined version, called signed Euler-Mahonian identity, together with a bijective proof were proposed by Wachs in the same year. By generalizing this bijection, in this paper we extend the above results to the Coxeter groups of types $B_n$, $D_n$, and the complex reflection group $G(r,1,n)$, where the `sign' is taken to be any one-dimensional character. Some obtained identities can be further restricted on some particular set of permuta…
We prove several identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials, extending results of Stembridge, Petersen, and Brändén. Additionally, we find $q$-exponential generating functions for $q$-analogues of these descent statistic polynomials that also keep track of the inversion number or inverse major index. We also present identities relating several of these descent statistic polynomials to refinements of type B Eulerian polynomials and flag descent polynomials by the number of negative letters of a signed permutation. Our m…
We show that the principal order ideal below an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin paths is demonstrated.
Hamilton cycles for involutions of classical types
本文统一构造经典 Weyl 群对合在对合连接集下的 Hamilton 圈，并通过修改 B 型 Gray 码算法给出 D 型 Hamming 距离为二的最优构造。
The Eulerian numbers can D.I.E
本文探讨了 Eulerian 数的两个交替符号公式，并利用 D.I.E. 方法进行证明，展示了其在数学中的广泛应用。
This paper is concerned with multivariate refinements of the gamma-positivity of Eulerian polynomials by using the succession and fixed point statistics. Properties of the enumerative polynomials for permutations, signed permutations and derangements, including generating functions and gamma-positivity are studied, which generalize and unify earlier results of Athanasiadis, Brenti, Chow, Petersen, Roselle, Stembridge, Shin and Zeng. In particular, we derive a formula expressing the joint distribution of excedance number and negative number statistics over the type B derangements in terms of th…
The W-set of an element of a weak order poset is useful in the cohomological study of the closures of spherical subgroups in generalized flag varieties. We explicitly describe in a purely combinatorial manner the W-sets of the weak order posets of three different sets of involutions in the symmetric group, namely, the set of all involutions, the set of all fixed point free involutions, and the set of all involutions with signed fixed points (or "clans"). These distinguished sets of involutions parameterize Borel orbits in the classical symmetric spaces associated to the general linear group. I…
The well-known Worpitzky identity provides a connection between two bases of $\mathbb{Q}[x]$: The standard basis $(x+1)^n$ and the binomial basis ${{x+n-i} \choose {n}}$, where the Eulerian numbers for the Coxeter group of type $A$ (the symmetric group) serve as the entries of the transformation matrix. Brenti has generalized this identity to the Coxeter groups of types $B$ and $D$ (signed and even-signed permutations groups, respectively) using generating function techniques. Motivated by Foata-Schützenberger and Rawlings' proof for the Worpitzky identity in the symmetric group, we provide co…
A Lovász Theta Parameter and Theta Body for Signed Graphs
该研究为带符号图的平衡着色提出Lovász型半定参数与凸角层级结构，证明平衡完美性弱于关联双覆盖图的完美性，相关构造与广义Theta数等概念相关。
Permutation tableaux are combinatorial objects related with permutations and various statistics on them. They appeared in connection with total positivity in Grassmannians, and stationary probabilities in a PASEP model. In particular they gave rise to an interesting q-analog of Eulerian numbers. The purpose of this article is to study some combinatorial properties of type B permutation tableaux, defined by Lam and Williams, and links with signed permutation statistics. We show that many of the tools used for permutation tableaux generalize in this case, including: the Matrix Ansatz (a method o…
We provide a combinatorial approach to the largest power of $p$ in the number of permutations $π$ with $π^p=1$, for a fixed prime number $p$. With this approach, we find the largest power of $2$ in the number of involutions, in the signed sum of involutions and in the numbers of even or odd involutions.
Thin divisible designs graphs: an interplay between fixed-point free involutions of $(v,k,λ)$-graphs and symmetric weighing matrices
本文研究了薄可除设计图中加权矩阵、(v,k,λ)-图及符号图之间的相互作用，提出了两种新的递归构造方法用于正则对称Hadamard矩阵和强正则图，并发现了symplictic图中固定点自由置换的性质。
We examine Borel subgroup orbits in the classical symmetric space of type CI, which are parametrized by skew symmetric (n, n)-clans. We describe bijections between such clans, certain weighted lattice paths, and pattern-avoiding signed involutions, and we give a cell decomposition of the symmetric space in terms of collections of clans called sects. The largest sect with a conjectural closure order is isomorphic (as a poset) to the Bruhat order on partial involutions.
Atoms for signed permutations
本文研究带符号排列群中对合的原子组合结构，给出任意带符号对合原子集的简洁描述与分次偏序结构，并据此显式确定 B、C 型中 Brion 上同调公式的各项。
On a trace formula for counting Eulerian cycles
该研究将计数欧拉回路的经典问题与同调谱图理论结合，提出显式迹公式，得到以亏格为参数的固定参数可处理算法，并开发对称性归约减少计算量。
Triangular Recurrences, Generalized Eulerian Numbers, and Related Number Triangles
本文研究GKP型递推的组合数三角，分析其变换群作用与EGF表示，推导新定义的广义欧拉数与Hsu-Shiue广义斯特林数的关系及相关恒等式，包括组合意义下的闭式计算。
Weighted Ehrhart series and a type-$\mathsf{B}$ analogue of a formula of MacMahon
本文通过多面体的$h^*$-多项式给出了带符号多重集排列上主指标与下降统计量联合分布的生成多项式公式，并建立了A、B型广义欧拉数与多面体之间的联系，进而将结果推广到着色排列。
Permutation Wordle
本文提出排列Wordle游戏，通过最优策略分析排列恢复问题，并将解决数目与Eulerian数联系起来。
We define a new family of generalized Stirling permutations that can be interpreted in terms of ordered trees and forests. We prove that the number of generalized Stirling permutations with a fixed number of ascents is given by a natural three-parameter generalization of the well-known Eulerian numbers. We give the generating function for this new class of numbers and, in the simplest cases, we find closed formulas for them and the corresponding row polynomials. By using a non-trivial involution our generalized Eulerian numbers can be mapped onto a family of generalized Ward numbers, forming a…
$q$-enumeration of type B and D Eulerian polynomials based on parity of descents
本文研究了基于降数奇偶性的B和D型欧拉多项式$q$-枚举问题，通过引入四种统计量和B型翻转数，推导了相关生成函数及递推关系，同时给出了D型Coxeter群的五变量多项式及其指数生成函数。
We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions are, depending on the number of fixed points, (1) the Tracy-Widom distributions for the largest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy-Widom distributions. We also consider the second rows of the correspond…
Generalized Euler numbers and ordered set partitions
本研究针对广义欧拉数研究以代数方法为主的现状，提出将其表示为有序分划带符号和的组合模型，用符号反转对合、偏序集Möbius反演证明了递推、整性、同余式等新旧结果。
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