On the Numbers of Orbits of Permutations under an Operator Related to Eulerian Numbers

In two previous papers an operator on permutations was introduced and its applications to Eulerian numbers were discussed by means of periods and orbits under the operator. In this paper, observing particular subsequences of permutations, an explicit formula for the number of orbits is given for each period. Several identities concerning the number of orbits and its related numbers are also derived.AMS Subject Classification: 05A05, 05A10.     

避免$2$字母符号模式的具有偶数个符号的带符号排列

Let Dn be the set of all signed permutations on [n] = {1,... ,n} with even signs, and let ：Dn（T） be the set of all signed permutations in Dn which avoids a set T of signed patterns. In this paper, we find all the cardinalities of the sets Dn（T） where T B2. Some of the cardinalities encountered involve inverse binomial coefficients, binomial coefficients, Catalan numbers, and Fibonacci numbers.     

Combinatorial Statistics on Alternating Permutations

We consider two combinatorial statistics on permutations. One is the genus. The other, , is defined for alternating permutations, as the sum of the number of descents in the subwords formed by the peaks and the valleys. We investigate the distribution of on genus zero permutations and Baxter permutations. Our q-enumerative results relate the statistic to lattice path enumeration, the rank generating function and characteristic polynomial of noncrossing partition lattices, and polytopes obtained as face-figures of the associahedron.     

Alternating Eulerian polynomials and left peak polynomials

We first present grammatical interpretations for the alternating Eulerian polynomials of types A and B. As applications, we then derive several properties of the type B alternating Eulerian polynomials, including recurrence relations, generating function and unimodality. And then, we establish an interesting connection between alternating Eulerian polynomials of type B and left peak polynomials, which implies that the type B alternating Eulerian polynomials have gamma-vectors that alternate in sign.     

图的符号星控制数与因子

利用图的因子研究图的符号星控制数,简化了文献中已有的结果.     

图的k符号边控制数

设G=(V,E)是一个图,一个函数f:E→{-1,+1},如果对于G中至少k条边e有sum from e'∈N[e]f(e')≥1成立,则称f为图G的一个k符号边控制函数.一个图的k符号边控制数定义为γ_(ks)^/(G)=min{∑_(e∈E(G))f(e)|f为图G的一个k符号边控制函数}.主要给出了一个图G的k符号边控制数γ_(ks)/(G)=min{∑_(e∈E(G))f(e)|f为图G的一个k符号边控制函数}.主要给出了一个图G的k符号边控制数γ_(ks)^/(G)的若干新下限,并确定了路和圈的k符号边控制数.     

关于图的符号星控制数

设G=(V,E)是一个图,u∈V,则E(u)表示u点所关联的边集.一个函数f:E→{-1,1}如果满足■f(e)≥1对任意v∈V成立,则称f为图G的一个符号星控制函数,图G的符号星控制数定义为γ'_(ss)(G)=min{■f(e):f为图G的一个符号星控制函数}.给出了几类特殊图的符号星控制数,主要包含完全图,正则偶图和完全二部图.     

两个包含欧拉数的恒等式的等价性

基于n维多项式空间中基之间的线性变换，证明了两个包含欧拉数的恒等式是等价．     

关于图的团符号控制数

引入了图的团符号控制的概念,给出了n阶图G的团符号控制数γks(G)的若干下限,确定了几类特殊图的团符号控制数,并提出了若干未解决的问题和猜想.     

Betti Numbers of Toric Varieties and Eulerian Polynomials

It is well known that the Eulerian polynomials, which count permutations in S _n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the S _n -orbit for a generic weight in the weight lattice of S _n . Therefore, the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra.

In this article we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a nongeneric weight, namely, a weight fixed by only the simple reflections J = {s _n , s _n?1, s _n?2,…, s _n?k+2, s _n?k+1} for some k with respect to the A _n root lattice. Furthermore, they give rise to certain rationally smooth toric varieties X(J) that come naturally from the theory of algebraic monoids. Using effectively the theory of reductive algebraic monoids and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincaré polynomial of X(J) in terms of the Eulerian polynomials.     

两个包含欧拉数的恒等式的等价性(英文)

基于n维多项式空间中基之间的线性变换,证明了两个包含欧拉数的恒等式是等价.     

图的好符号星控制数

引入了图的好符号星控制的概念,求出了欧拉图、完全二部图、完全图和轮图的好符号星控制数,并改进了图的符号星控制数的两个上界.     

图的三种符号星控制数

引入了图的符号星k限定控制的概念,从而求出了星图和轮图的符号星k控制数.还刻画了满足γ′_(ss)(G)=1/2(2r+s)的图,基中γ′_(ss)(G)表示图G的符号星控制数.最后对图的符号星部分控制的已有结果作了改进.     

图的逆符号边全控制数

设γ_(st)(G)是图G的逆符号边全控制数,p(n,k)是广义Petersen图.得到了γ_(st)(G)的两个上界,并且确定了γ_(st)(p(n,k)).     

关于图的符号边全控制数

引入了图的符号边全控制的概念,给出了一个连通图G的符号边全控制数γs′t(G)的下限,确定所有n阶树T的最小符号边全控制数,并刻划了满足γs′t(G)=E(G)的所有连通图G,最后还提出了一个关于γs′t(G)上界的猜想.     

关于图的符号边全控制数

Let G = （V,E） be a graph.A function f ： E → {-1,1} is said to be a signed edge total dominating function （SETDF） of G if e ∈N（e） f（e ） ≥ 1 holds for every edge e ∈ E（G）.The signed edge total domination number γ st （G） of G is defined as γ st （G） = min{ e∈E（G） f（e）｜f is an SETDF of G}.In this paper we obtain some new lower bounds of γ st （G）.     

On a Family of Conjectures of Joel Lewis on Alternating Permutations

We prove generalized versions of some conjectures of Joel Lewis on the number of alternating permutations avoiding certain patterns. Our main tool is the perhaps surprising observation that a classic bijection on pattern avoiding permutations often preserves the alternating property.     

On Classifying Subsets of Natural Numbers by Their Computable Permutations

We introduce some alternative definitions of the concept of computable automorphism of a set of natural numbers. We study their relationships and investigate whether some classes of sets having isomorphic groups of automorphisms coincide with other classes of sets usual in computability. Finally, we show that the classification of sets by these groups of automorphisms is nontrivial.     

李善兰对Stirling数和Euler数的研究

我国清代有影响的数学家李善兰(1811—1882)在他的名著《垛积比类》(1867)中提出并研究了两类系数表(依原著顺序为第二表和第七表),获得了重要成果。它们实际就是现代组合数学中的两种重要计数函数。     

两类特殊图的逆符号边控制数

设G=(V,E)是一个图,对于图G的一个函数f:E→{-1,1},如果对任意e∈E(G),均有Σe′∈N[e]f(e′)≤1,则称f为图G的一个逆符号边控制函数.图G的逆符号边控制数γ′s(G)=max{Σe∈E(G)f(e)|f为图G的一个逆符号边控制函数}.在逆符号边控制数定义基础上,得到了所有轮图和扇图的逆符号边控制数.