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1.
We determine the order dimension of the strong Bruhat order on finite Coxeter groups of types A, B and H. The order dimension is determined using a generalization of a theorem of Dilworth: dim (P)=width(Irr(P)), whenever P satisfies a simple order-theoretic condition called here the dissective property (or clivage). The result for dissective posets follows from an upper bound and lower bound on the dimension of any finite poset. The dissective property is related, via MacNeille completion, to the distributive property of lattices. We show a similar connection between quotients of the strong Bruhat order with respect to parabolic subgroups and lattice quotients. 相似文献
2.
John R. Stembridge 《Journal of Algebraic Combinatorics》2002,15(3):291-301
Given a finite Weyl group W with root system , assign the weight to each covering pair in the Bruhat order related by the reflection corresponding to . Extending this multiplicatively to chains, we prove that the sum of the weights of all maximal chains in the Bruhat order has an explicit product formula, and prove a similar result for a weighted sum over maximal chains in the Bruhat ordering of any parabolic quotient of W. Several variations and open problems are discussed. 相似文献
3.
John R. Stembridge 《Journal of Algebraic Combinatorics》2007,25(2):141-148
We give a short, self-contained derivation of the Möbius function for the Bruhat orderings of Coxeter groups and their parabolic quotients. 相似文献
4.
The Bruhat Order on the Involutions of the Symmetric Group 总被引:1,自引:0,他引:1
Federico Incitti 《Journal of Algebraic Combinatorics》2004,20(3):243-261
In this paper we study the partially ordered set of the involutions of the symmetric group S
n with the order induced by the Bruhat order of S
n. We prove that this is a graded poset, with rank function given by the average of the number of inversions and the number of excedances, and that it is lexicographically shellable, hence Cohen-Macaulay, and Eulerian. 相似文献
5.
Axel Hultman 《Journal of Combinatorial Theory, Series A》2009,116(3):564-580
We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w∈Sn is at most the number of elements below w in the Bruhat order, and (B) that equality holds if and only if w avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups.A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns. 相似文献
6.
Françcois Hennecart 《The Ramanujan Journal》2005,9(1-2):123-130
The restricted order of an asymptotic basis A is the least integer h, if it exists, such that every sufficiently large integer is the sum of h or fewer distinct elements of A. We show that any asymptotic basis of order 2 has a restricted order at most equal to 4. We also provide an example of an additive basis of order 2 whose restricted order is 4.To Jean-Louis Nicolas2000 Mathematics Subject Classification: Primary—11B13 相似文献
7.
We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold
G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas
for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters,
and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold
and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin
polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees.
A.P. was supported in part by National Science Foundation grant DMS-0201494 and by Alfred P. Sloan Foundation research fellowship.
R.S. was supported in part by National Science Foundation grant DMS-9988459. 相似文献
8.
The notion of a free triangle representation of a partially ordered set was first introduced by Josh Laison [4] as a generalization of the ideas of interval and trapezoid representations. A free triangle representation assigns a triangle to each element of a partially ordered set, with all triangles having one vertex on each of two parallel baselines and a third ‘free’ vertex between the two baselines. In a previous paper [1] we presented an example of an infinite non-unit free triangle order. In this paper we use some of the same ideas to construct an example of a finite, albeit more complicated, non-unit free triangle order.The majority of the content in this paper (theorems, proofs, etc.) was prepared before Ken's untimely death in March of 2005. 相似文献
9.
本文主要讨论满足交换性的两个元素乘积的阶的问题。设G是一个群,a,b∈G,ab=ba,a的阶为n1,b的阶为n2。一般情况下ab的阶不等于n1与n2的最小公倍数。例如当a=b,n1=n2=4时,ab=a2,ο(ab)=2≠[n1,n2]。本文获得了ab的阶等于n1与n2的最小公倍数的几个充分条件。 相似文献
10.
Bridget Eileen Tenner 《Journal of Combinatorial Theory, Series A》2007,114(5):888-905
The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. The permutations with boolean principal order ideals are characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. Moreover, the permutations whose principal order ideals have a form related to boolean posets are also completely described. It is determined when the set of permutations avoiding a particular set of patterns is an order ideal, and the rank generating functions of these ideals are computed. Finally, the Bruhat order in types B and D is studied, and the elements with boolean principal order ideals are characterized and enumerated by length. 相似文献
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12.
本文研究一类含非定线性项的二阶Hamilton系统多周期解问题.在位势函数满足超二次齐次条件下,利用临界点理论中对称型越山定理,证明了系统存在无穷多个给定周期的周期解. 相似文献
13.
本文通过对传统高阶马尔可夫链模型的状态空间进行阶数重构,导出一个在重构状态空间上的降阶马尔可夫模型。理论分析证明,降阶马尔可夫链模型不但可以描述传统高阶马尔可夫链模型的全部性态,更能表达较传统模型细微的随机结构。然后,应用降阶模型对我国股票指数的动态变化进行实证分析,讨论了阶数的选取和高阶马尔可夫性检验,分析了股票市场内在波动结构。最后,对股指序列作出短期与长期的预测分析。 相似文献
14.
We show that the principal order ideal of 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.
This article is largely based on results from the second author’s M.Sc. thesis [15]. 相似文献
15.
We introduce two numerical invariants of orders that measure how close a poset is to having the fixed point property. We give general properties of those invariants and link them to known results on the fixed point property. 相似文献
16.
文中把二次曲线的几何性质的研究转化成条件极值问题,但又不关心问题的解,而是利用Lagrange乘数来研究二次曲线的几何性质,找到了用Lagrange 乘数判别二次曲线形状的方法,给出了用Lagrange乘数计算二次曲线的对称轴和轴长的公式. 相似文献
17.
Gábor Elek 《K-Theory》1998,13(1):1-22
We prove that, for any exact category M, any element of K1(M)can be described in terms of a pair of admissible monomorphisms A X, B Y and an isomorphism :A X/A Y B Y/B X. 相似文献
18.
廖靖宇 《数学的实践与认识》2011,41(20)
受关于最大次序统计量数学期望恒等式的启发,给出了最小次序统计量数学期望的恒等式,并利用概率方法进行了证明.两个恒等式是对概率论知识的补充和推广 相似文献
19.
In this paper we introduce two sequences of operator functions and their dual functions:f_k(t)=((tlog t)~k-(t-1)~k)/(log~k 1t)(k=1,2,...),gk(t)=((t-1)~k-log~k t)/(log~k 1t)(k=1,2,...)and f_k~*=(t~klog~(k 1)t)/((tlog t)~k-(t-1)~k)(k=1,2,...),g_k~*(t)=(t~klog~(k 1)t)/((t-1)~k-log~k t)(k=1,2,...)definedon(0, ∞). We find that they are all operator monotone functions with respect to the strictly chaotic order and some ordinary orders among positive invertible operators.Indeed,we extend the results of the operator monotone function(tlog t -t 1)/(log~2t)which is widely used in the theory of heat transfer of the heat engineering and fluid mechanics[1]. 相似文献
20.