Polynomial Statistics, Necklace Polynomials, and the Arithmetic Dynamical Mordell-Lang Conjecture By
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Fast Algorithms to Generate Necklaces, Unlabeled Necklaces, and Irreducible Polynomials Over GF(2)
Journal of Algorithms 37, 267᎐282Ž. 2000 doi:10.1006rjagm.2000.1108, available online at http:rrwww.idealibrary.com on Fast Algorithms to Generate Necklaces, Unlabeled Necklaces, and Irreducible Polynomials over GFŽ. 2 1 Kevin Cattell HP Research Labs, Santa Rosa, California E-mail: kevin᎐ [email protected] Frank Ruskey, Joe Sawada, and Micaela Serra Department of Computer Science, Uni¨ersity of Victoria, Victoria, British Columbia, Canada E-mail: [email protected]; [email protected], [email protected] and C. Robert Miers Department of Mathematics and Statistics, Uni¨ersity of Victoria, Victoria, British Columbia, Canada E-mail: [email protected] Received August 22, 1998 Many applications call for exhaustive lists of strings subject to various con- straints, such as inequivalence under group actions. A k-ary necklace is an equivalence class of k-ary strings under rotationŽ. the cyclic group . A k-ary unlabeled necklace is an equivalence class of k-ary strings under rotation and permutation of alphabet symbols. We present new, fast, simple, recursive algo- rithms for generatingŽ. i.e., listing all necklaces and binary unlabeled necklaces. These algorithms have optimal running times in the sense that their running times are proportional to the number of necklaces produced. The algorithm for generat- ing necklaces can be used as the basis for efficiently generating many other equivalence classes of strings under rotation and has been applied to generating bracelets, fixed density necklaces, and chord diagrams. As another application, we describe the implementation of a fast algorithm for listing all degree n irreducible and primitive polynomials over GFŽ. -
APPLICATIONS of M¨OBIUS INVERSION on PARTIALLY ORDERED SETS Contents 1. Introduction 1 2. Basic Definitions for Partially Order
APPLICATIONS OF MOBIUS¨ INVERSION ON PARTIALLY ORDERED SETS DAVID WU Abstract. In this paper, we discuss how the theory of partially ordered sets plays an important role in combinatorics. First, we outline definitions for partially ordered sets. Next, we introduce incidence algebra and M´obiusin- version. Then, we present three applications of partially ordered set theory: the Inclusion-Exclusion Principle, Euler's Totient function (or phi function), and P´olya's Enumeration Theorem. For P´olya's Enumeration Theorem, we assume prior knowledge of basic group theory. Contents 1. Introduction 1 2. Basic Definitions for Partially Ordered Sets (Posets) 2 3. Incidence Algebra 3 4. Applications of M¨obiusinversion 5 4.1. Inclusion-Exclusion Principle 6 4.2. Classical M¨obiusFunction 7 5. P´olya's Enumeration Theorem 9 5.1. A Galois connection between permutations and partitions 10 5.2. Generating Functions 11 5.3. Proof of P´olya's Enumeration Theorem 13 5.4. Examples with Necklaces 16 Acknowledgments 18 References 18 1. Introduction Many problems in combinatorics deal with counting a particular combinatorial object. A common strategy is to construct an algebraic object to do this counting (ex. generating functions). In this paper, the combinatorial object of interest is the partially ordered set, and the algebraic object we use is incidence algebra. In particular, we are most interested in the Kronker delta, zeta, and M¨obiusfunction in incidence algebra. These functions give rise to a technique called the M¨obius inversion formula, a counting tool with far-reaching applications. In section 2, we begin by introducing preliminary definitions for partially ordered sets. -
Basic Counting
CHAPTER 1 Basic Counting Introduction Before beginning, we must confront some matters of notation. Two words that we shall often use are set and list. Both words refer to collections of objects. There is no standard notation for lists. Some of those in use are apple banana pear peach a list of four items ... apple, banana, pear, peach commas added for clarity ... and (apple, banana, pear, peach) parentheses added. The notation for sets is standard: the items are separated by commas and surround by curly brackets as in {apple, banana, pear, peach}. The curly bracket notation for sets is so well established that you can normally assume it means a set—but beware, Mathematica R uses curly brackets for lists. What is the difference between a set and a list? Quite a bit, and nothing. “Set” means a collection of distinct objects in which the order doesn’t matter. Thus {apple, peach, pear} and {peach, apple, pear} are the same sets, and the set {apple, peach, apple} is the same as the set {apple, peach}. In other words, repeated elements are treated as if they occurred only once. Thus two sets are the same if and only if each element that is in one set is in both. In a list, order is important and repeated objects are usually allowed. Thus (apple, peach) (peach, apple) and (apple, peach, apple) are three different lists. Two lists are the same if and only if they have exactly the same items in exactly the same positions. Thus, sets and lists are different. On the other hand, people talk about things like “unordered lists,” “sets with repetition,” and so on. -
Number of Necklaces Number of Bracelets Necklace Example
From Wikipedia, the free encyclopedia In combinatorics, a k-ary necklace of length n is an equivalence class of n-character strings over an alphabet of size k, taking all rotations as equivalent. It represents a structure with n circularly connected beads of up to k different colors. A k-ary bracelet, also referred to as a turnover (or free) necklace, is a necklace such that strings may also be equivalent under reflection. That is, given two strings, if each is the reverse of the other then they belong to the same equivalence class. For this reason, a necklace might also be called a fixed necklace to distinguish it from a turnover necklace. Technically, one may classify a necklace as an orbit of the action of the cyclic group on n-character strings, and a bracelet as an orbit of the dihedral group's action. This enables application of Pólya enumeration theorem for enumeration of necklaces and bracelets. 1 Equivalence classes Possible patterns of bracelets of length n 1.1 Number of necklaces corresponding to the k-th integer partition 1.2 Number of bracelets 2 Examples (set partitions up to rotation and reflection) 2.1 Necklace example 2.2 Bracelet example 3 Aperiodic necklaces 4 Products of necklaces 5 See also 6 References 7 External links The 3 bracelets with 3 red and 3 green beads. The one in the middle is chiral, so there are 4 necklaces. Compare box(6,9) in the triangle. Number of necklaces There are different k-ary necklaces of length n, where φ is Euler's [1] totient function. -
Witt Vectors and the Algebra of Necklaces*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN MATHEMATICS s&95-125 (1983) Witt Vectors and the Algebra of Necklaces* N. METROPOLIS Los Alamos National Laboratory, Los Alamos, New Mexico 87545 AND GIAN-CARLO ROTA+ Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Contents. 1. Introduction. 2. Words and primitive words. 3. Main results. 4. The Necklace ring. 5. Necklaces and placements. 6. Unital rational functions. 7. Carryless Witt vectors. 8. The Witt ring. 1. INTRODUCTION Given a set of a colors, a necklace is the result of placing n colored beads around a circle. A necklace which is asymmetric under rotation is said to be primitive. The number M(a, n) of primitive necklaces is computed by a formula that goes back to Col. Moreau of the French Army, to wit, where ,U is the classical Mobius function; M(a, n) is called a necklace polynomial. We shall be concerned with some remarkable identities satisfied by the polynomials M(a, n), which apparently have not been previously noticed. Specifically, we prove that M(a/?, n) can be expressed as a quadratic polynomial in M(a, i) and M&j), and generalized to more variables; that M(a’, n) can be expressed linearly and integrally in terms of M(a, i). The proofs given below of these identities are entirely combinatorial (Theorems l-6). * This work was performed under the auspices of the U.S. Department of Energy. -
Circular and Necklace Permutations
Circular and Necklace Permutations Hiroshi KAJIMOTO and Mai OSABE Department of Mathematical Science, Faculty of Education Nagasaki University, Nagasaki 852-8521, Japan e-mail: [email protected] (Received October 31, 2006) Abstract We enumerate circular and necklace permutations which have beads of several colors, and count the related combinatorial numbers. 0 Introduction Circular and necklace permutations are standard subjects of high school com- binatorics. On the other hand they serve a good introduction to more ad- vanced subject of combinatorics, that is the P´olya theory of counting under group action. In fact these enumerations were historically a precursor of the theory [2, 4, p.558]. In this article we give the enumeration of circular and necklace permuations according to the given number of beads for each colors, and of the related combinatorial numbers. A circular permutation is defined to be an equivalence class of sequences under the cyclic group Cn, i.e., two sequences a1a2 : : : an and b1b2 : : : bn are equivalent if one is a cyclic shift of the other. A necklace (or necklace permu- tation) is defined to be an equivalence class of sequences under the dihedral group Dn, i.e., two sequences a1a2 : : : an and b1b2 : : : bn are equivalent if one is a cyclic shift of the other or if circular permutaions of them are mutually reflective with respect to some diameter. We make full use of the counting theory and look back the classical ex- amples, and show how to use P´olya's cycle index efficiently. We use combi- natorial notation, especially relative to partitions, as in [4].