A Logarithmic Connection for Circular Permutation Enumeration

A general theorem is obtained for the enumeration of permutations equivalent under cyclic rotation. This result gives the generating function as the logarithm of a determinant which arises in the enumeration of a related linear permutation enumeration. Applications of this theorem are given to a number of classical enumerative problems.     

Pattern Avoidance in Ordered Set Partitions

In this paper we consider the enumeration of ordered set partitions avoiding a permutation pattern of length 2 or 3. We provide an exact enumeration for avoiding the permutation 12. We also give exact enumeration for ordered partitions with 3 blocks and ordered partitions with n?1 blocks avoiding a permutation of length 3. We use enumeration schemes to recursively enumerate 123-avoiding ordered partitions with any block sizes. Finally, we give some asymptotic results for the growth rates of the number of ordered set partitions avoiding a single pattern; including a Stanley-Wilf type result that exhibits existence of such growth rates.     

Enumerating 3-manifolds with lengths up to 9 by a canonical order

On previous works, we enumerated the prime links with lengths up to 10 and the prime link exteriors with lengths up to 9. In this paper, we make an enumeration of the first 133 closed 3-manifolds which are the 3-manifolds with lengths up to 9 by using the enumeration of the prime link exteriors.     

投资项目评价选择模型的遗传算法

针对一类常见的投资项目评价选择模型提出了一种快速有效的遗传算法。考虑到模型的特殊形式，算法采用了特别的编码方式、遗传算子和适应度确定方法，并与穷举法和隐枚举法进行了对比试验，结果表明该方法能快速有效地给出解决方案。     

Decomposed enumeration of extreme points in the linear programming problem

In this paper we consider the problem of enumeration of extreme points in the linear programming problem when the matrix is of block-angular type. It is shown how decomposition methods can be used. Finally application of decomposed enumeration to the problem of computing equilibrium prices in a capital market network is given as an example.     

关于K圈标号图的计数

本文研究含K个圈的标号图的计数问题,得出了有n个标定顶点且有K个交于一点的圈的连通图的计数公式,并得到了双圈连通标号图的计数公式,从而解决了K-2时连通图的计数问题。     

Development of Counting Skills: Role of Spontaneous Focusing on Numerosity and Subitizing-Based Enumeration

Children differ in how much they spontaneously pay attention to quantitative aspects of their natural environment. We studied how this spontaneous tendency to focus on numerosity (SFON) is related to subitizing-based enumeration and verbal and object counting skills. In this exploratory study, children were tested individually at the age of 4-5 years on these skills. Results showed 2 primary relationships in children's number skills development. Performance in a number sequence production task, which is closely related to ordinal number sequence without reference to cardinality, is directly associated with SFON. Second, the association of SFON and object counting skills, which require relating cardinal and ordinal aspects of number, is mediated by subitizing-based enumeration. This suggests that there are multiple pathways to enumeration skills during development.     

Quasi-minimal enumeration degrees and minimal Turing degrees

We show that there exists a set A such that A has quasi-minimal enumeration degree, and there are uncountably many sets B such that A is enumeration reducible to B and B has minimal Turing degree. Answering a related question raised by Solon, we also show that there exists a nontotal enumeration degree which is not e-hyperimmune.During the preparation of this paper, Slaman was partially supported by the HCM European Program no. ERBCHRXCT930415 (while he was visiting the University of Leeds), by NSF Grant DMS-9500878 and was a CNR Visiting Professor at the University of Siena.The preparation of this paper was partially supported by the HCM European Program no. ERBCHRXCT930415 and by MURST 60%.     

Enumeration problems for classes of self-similar graphs

We describe a general construction principle for a class of self-similar graphs. For various enumeration problems, we show that this construction leads to polynomial systems of recurrences and provide methods to solve these recurrences asymptotically. This is shown for different examples involving classical self-similar graphs such as the Sierpiński graphs. The enumeration problems we investigate include counting independent subsets, matchings and connected subsets.     

Development of Counting Skills: Role of Spontaneous Focusing on Numerosity and Subitizing-Based Enumeration

Children differ in how much they spontaneously pay attention to quantitative aspects of their natural environment. We studied how this spontaneous tendency to focus on numerosity (SFON) is related to subitizing-based enumeration and verbal and object counting skills. In this exploratory study, children were tested individually at the age of 4–5 years on these skills. Results showed 2 primary relationships in children's number skills development. Performance in a number sequence production task, which is closely related to ordinal number sequence without reference to cardinality, is directly associated with SFON. Second, the association of SFON and object counting skills, which require relating cardinal and ordinal aspects of number, is mediated by subitizing-based enumeration. This suggests that there are multiple pathways to enumeration skills during development.     

An operator approach to some graph enumeration problems

An operator approach to some graph enumeration problems is developed together with the formal procedures related to the enumeration. Both annihilation and creation operators are defined for vertices, edges and Euler characteristics of a graph. An application to forest enumeration leads to compact expressions exhibiting the duality between the operators.     

State matrix recursion method and monomer–dimer problem

The exact enumeration of pure dimer coverings on the square lattice was obtained by Kasteleyn, Temperley and Fisher in 1961. In this paper, we consider the monomer–dimer covering problem (allowing multiple monomers) which is an outstanding unsolved problem in lattice statistics. We have developed the state matrix recursion method that allows us to compute the number of monomer–dimer coverings and to know the partition function with monomer and dimer activities. This method proceeds with a recurrence relation of so-called state matrices of large size. The enumeration problem of pure dimer coverings and dimer coverings with single boundary monomer is revisited in partition function forms. We also provide the number of dimer coverings with multiple vacant sites. The related Hosoya index and the asymptotic behavior of its growth rate are considered. Lastly, we apply this method to the enumeration study of domino tilings of Aztec diamonds and more generalized regions, so-called Aztec octagons and multi-deficient Aztec octagons.     

Primal—Dual Methods for Vertex and Facet Enumeration

Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (resp. vertex) to the vertex (resp. halfspace) representation is called vertex enumeration (resp. facet enumeration ). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (resp. facet) enumeration problem is the facet (resp. vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primal—dual algorithms use a solution to the easy direction as an oracle to help solve the seemingly hard direction. Received July 31, 1997, and in revised form March 8, 1998.     

函数图及函数有向图计数式的组合证明

本文给出了函数有向图计数式的一个简短证明     

A note on the enumeration degrees of 1-generic sets

We show that every nonzero \({\Delta^{0}_{2}}\) enumeration degree bounds the enumeration degree of a 1-generic set. We also point out that the enumeration degrees of 1-generic sets, below the first jump, are not downwards closed, thus answering a question of Cooper.     

An application of Fourier transforms on finite Abelian groups to an enumeration arising from the Josephus problem

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We analyze an enumeration associated with the Josephus problem by applying a Fourier transform to a multivariate generating function. This yields a formula for the enumeration that reduces to a simple expression under a condition we call local prime abundance. Under this widely held condition, we prove (Corollary 3.4) that the proportion of Josephus permutations in the symmetric group Sn that map t to k (independent of the choice of t and k) is 1/n. Local prime abundance is intimately connected with a well-known result of S.S. Pillai, which we exploit for the purpose of determining when it holds and when it fails to hold. We pursue the first case where it fails, reducing an intractable DFT computation of the enumeration to a tractable one. A resulting computation shows that the enumeration is nontrivial for this case.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=DnZi-Znuk-A.     

一些组合地图新算法的实现

本文主要讨论组合地图列举问题.刘的一部专著中提出了一个判定两个地图是否同构的算法.该算法的时间复杂度为O(m2),其中m为下图的规模.在此基础上,本文给出一个用于地图列举以及进而计算任意连通下图的地图亏格分布的通用算法.本文所得结果比之前文献中所给结果更优.     

Some set partition statistics in non-crossing partitions and generating functions

In this paper we shall give the generating functions for the enumeration of non-crossing partitions according to some set partition statistics explicitly, which are based on whether a block is singleton or not and is inner or outer. Using weighted Motzkin paths, we find the continued fraction form of the generating functions. There are bijections between non-crossing partitions, Dyck paths and non-nesting partitions, hence we can find applications in the enumeration of Dyck paths and non-nesting partitions. We shall also study the integral representation of the enumerating polynomials for our statistics. As an application of integral representation, we shall give some remarks on the enumeration of inner singletons in non-crossing partitions, which is equivalent to one of udu's at high level in Dyck paths investigated in [Y. Sun, The statistic “number of udu's” in Dyck paths, Discrete Math. 284 (2004) 177-186].     

Neural correlates of counting large numerosity

The mastery of counting numerosities larger than those correctly estimated by infants or non-human species is an important foundation for the development of higher level calculation skills. The cognitive processes involved in counting are related to spatial attention, language, and number processing. However, the respective involvement of language- and/or visuo-spatial-based brain systems during counting is still under debate. In the present functional magnetic resonance imaging study, we asked 27 right-handed participants to perform an enumeration task on visual arrays of bars that varied in numerosity. Each enumeration condition was contrasted to a color-detection condition that was numerically and spatially matched to the counting condition. The results showed a behavioral discontinuity in response time slopes between large (6–10) and small (1–5) numerosities during enumeration, suggesting that during large enumeration, participants engaged counting processes. Comparing brain regional activity during the enumeration of large numerosity to the enumeration of smaller numerosity, we found increased activation in the bilateral fronto-parietal attentional network, the inferior parietal gyri/intraparietal sulci, and the left ventral premotor and left inferior temporal areas. These results indicated that in adults who master enumeration, counting more than five items requires the strong involvement of spatial attention and eye movements, as well as numerical magnitude processes. Counting large numerosity also recruited verbal working memory areas, subtending a subvocal articulatory code and a visual representation of numbers.     

On Dedekind’s problem for complete simple games

We state an integer linear programming formulation for the unique characterization of complete simple games, i.e. a special subclass of monotone Boolean functions. In order to apply the parametric Barvinok algorithm to obtain enumeration formulas for these discrete objects we provide a tailored decomposition of the integer programming formulation into a finite list of suitably chosen sub-cases. As for the original enumeration problem of Dedekind on Boolean functions we have to introduce some parameters to be able to derive exact formulas for small parameters. Recently, Freixas et al. have proven an enumeration formula for complete simple games with two types of voters. We will provide a shorter proof and a new enumeration formula for complete simple games with two minimal winning vectors.