Multipermutations 1.1. Generating Functions
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Principle of Inclusion-Exclusion
Inclusion / Exclusion — §3.1 67 Principle of Inclusion-Exclusion Example. Suppose that in this class, 14 students play soccer and 11 students play basketball. How many students play a sport? Solution. Let S be the set of students who play soccer and B be the set of students who play basketball. Then, |S ∪ B| = |S| + |B| . Inclusion / Exclusion — §3.1 68 Principle of Inclusion-Exclusion When A = A1 ∪···∪Ak ⊂U (U for universe) and the sets Ai are pairwise disjoint,wehave|A| = |A1| + ···+ |Ak |. When A = A1 ∪···∪Ak ⊂U and the Ai are not pairwise disjoint, we must apply the principle of inclusion-exclusion to determine |A|: |A1 ∪ A2| = |A1| + |A2|−|A1 ∩ A2| |A1 ∪ A2 ∪ A3| = |A1| + |A2| + |A3|−|A1 ∩ A2|−|A1 ∩ A3| −|A2 ∩ A3| + |A1 ∩ A2 ∩ A3| |A1 ∪···∪Am| = |Ai |− |Ai ∩ Aj | + Ai ∩ Aj ∩ Ak ··· ! ! ! " " " " It may be more convenient to apply inclusion/exclusion where the Ai are forbidden subsets of U,inwhichcase . Inclusion / Exclusion — §3.1 69 mmm...PIE The key to using the principle of inclusion-exclusion is determining the right choice of Ai .TheAi and their intersections should be easy to count and easy to characterize. Notation: π = p1p2 ···pn is the one-line notation for a permutation of [n] whose first element is p1,secondelementisp2,etc. Example. How many permutations p = p1p2 ···pn are there in which at least one of p1 and p2 are even? Solution. Let U be the set of n-permutations. Let A1 be the set of permutations where p1 is even. Let A2 be the set of permutations where p2 is even. -
MTH 102A - Part 1 Linear Algebra 2019-20-II Semester
MTH 102A - Part 1 Linear Algebra 2019-20-II Semester Arbind Kumar Lal1 February 3, 2020 1Indian Institute of Technology Kanpur 1 2 • Matrix A = behaves like the scalar 3 when multiplied with 2 1 1 1 2 1 1 x = . That is, = 3 . 1 2 1 1 1 • Physically: The Linear function f (x) = Ax magnifies the nonzero 1 2 vector ∈ C three (3) times. 1 1 1 • Similarly, A = −1 . So, behaves by changing the −1 −1 1 direction of the vector −1 1 2 2 • Take A = . Do I have a nonzero x ∈ C which gets 1 3 magnified by A? • So I am looking for x 6= 0 and α s.t. Ax = αx. Using x 6= 0, we have Ax = αx if and only if [αI − A]x = 0 if and only if det[αI − A] = 0. √ α − 1 −2 2 • det[αI − A] = det = α − 4α + 1. So α = 2 ± 3. −1 α − 3 √ √ 1 + 3 −2 • Take α = 2 + 3. To find x, solve √ x = 0: −1 3 − 1 √ 3 − 1 using GJE, for instance. We get x = . Moreover 1 √ √ √ 1 2 3 − 1 3 + 1 √ 3 − 1 Ax = = √ = (2 + 3) . 1 3 1 2 + 3 1 n • We call λ ∈ C an eigenvalue of An×n if there exists x ∈ C , x 6= 0 s.t. Ax = λx. We call x an eigenvector of A for the eigenvalue λ. We call (λ, x) an eigenpair. • If (λ, x) is an eigenpair of A, then so is (λ, cx), for each c 6= 0, c ∈ C. -
Inclusion‒Exclusion Principle
Inclusionexclusion principle 1 Inclusion–exclusion principle In combinatorics, the inclusion–exclusion principle (also known as the sieve principle) is an equation relating the sizes of two sets and their union. It states that if A and B are two (finite) sets, then The meaning of the statement is that the number of elements in the union of the two sets is the sum of the elements in each set, respectively, minus the number of elements that are in both. Similarly, for three sets A, B and C, This can be seen by counting how many times each region in the figure to the right is included in the right hand side. More generally, for finite sets A , ..., A , one has the identity 1 n This can be compactly written as The name comes from the idea that the principle is based on over-generous inclusion, followed by compensating exclusion. When n > 2 the exclusion of the pairwise intersections is (possibly) too severe, and the correct formula is as shown with alternating signs. This formula is attributed to Abraham de Moivre; it is sometimes also named for Daniel da Silva, Joseph Sylvester or Henri Poincaré. Inclusion–exclusion illustrated for three sets For the case of three sets A, B, C the inclusion–exclusion principle is illustrated in the graphic on the right. Proof Let A denote the union of the sets A , ..., A . To prove the 1 n Each term of the inclusion-exclusion formula inclusion–exclusion principle in general, we first have to verify the gradually corrects the count until finally each identity portion of the Venn Diagram is counted exactly once. -
40 Chapter 6 the BINOMIAL THEOREM in Chapter 1 We Defined
Chapter 6 THE BINOMIAL THEOREM n In Chapter 1 we defined as the coefficient of an-rbr in the expansion of (a + b)n, and r tabulated these coefficients in the arrangement of the Pascal Triangle: n Coefficients of (a + b)n 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 ... n n We then observed that this array is bordered with 1's; that is, ' 1 and ' 1 for n = 0 n 0, 1, 2, ... We also noted that each number inside the border of 1's is the sum of the two closest numbers on the previous line. This property may be expressed in the form n n n%1 (1) % ' . r&1 r r This formula provides an efficient method of generating successive lines of the Pascal Triangle, but the method is not the best one if we want only the value of a single binomial coefficient for a 100 large n, such as . We therefore seek a more direct approach. 3 It is clear that the binomial coefficients in a diagonal adjacent to a diagonal of 1's are the 40 n numbers 1, 2, 3, ... ; that is, ' n. Now let us consider the ratios of binomial coefficients 1 to the previous ones on the same row. For n = 4, these ratios are: (2) 4/1, 6/4 = 3/2, 4/6 = 2/3, 1/4. For n = 5, they are (3) 5/1, 10/5 = 2, 10/10 = 1, 5/10 = 1/2, 1/5. -
Q-Combinatorics
q-combinatorics February 7, 2019 1 q-binomial coefficients n n Definition. For natural numbers n; k,the q-binomial coefficient k q (or just k if there no ambi- guity on q) is defined as the following rational function of the variable q: (qn − 1)(qn−1 − 1) ··· (q − 1) : (1.1) (qk − 1)(qk−1 − 1) ··· (q − 1) · (qn−k − 1)(qn−k−1 − 1) ··· (q − 1) For n = 0, k = 0, or n = k, we interpret the corresponding products 1, as the value of the empty product. If we exclude certain roots of unity from the domain of q, (1.1) is equal to (qn − 1)(qn−1 − 1) ··· (qn−k+1 − 1) (1.2) (qk − 1)(qk−1 − 1) ··· (q − 1) · 1 (qn−1 + ::: + q + 1) ··· (q2 + q + 1)(q + 1) · 1 = : (qk−1 + ::: + q + 1) ··· (q2 + q + 1)(q + 1) · 1(qn−k−1 + ::: + q + 1) ··· (q2 + q + 1)(q + 1) · 1 As the notation starts to be overwhelming, we introduce to q-analogue of the positive integer n, n−1 denoted by [n] or [n]q, as q + ::: + q + 1; and the q-analogue of the factorial of the positive integer n, denoted by [n]! or [n]q!, as [n]q! = [1]q · [2]q ··· [n]q. With this additional notation, we also can say n [n] ! = q ; (1.3) k q [k]q![n − k]q! when we avoid certain roots of unity with q. It is easy too see from (1.1) that n n = = 1 0 q n q and that for every 0 ≤ k ≤ n the symmetry rule n n = k q n − k q holds. -
The Binomial Theorem and Combinatorial Proofs Shagnik Das
The Binomial Theorem and Combinatorial Proofs Shagnik Das Introduction As we live our lives, we are faced with innumerable decisions on a daily basis1. Can I get away with wearing these clothes again before I have to wash them? Which frozen pizza should I have for dinner tonight? What excuse should I use for skipping the gym today? While it may at times seem tiring to be faced with all these choices on a regular basis, it is this exercise of free will that separates us from the machines we live with.2 This multitude of choices, then, provides some measure of our own vitality | the more options we have, the more alive we truly are. What, then, could be more important than being able to count how many options one actually has?4 As we have already seen during our first week of lectures, one of the main protagonists in these counting problems is the binomial coefficient, whose definition is given below. n Definition 1. Given non-negative integers n and k, the binomial coefficient k denotes the number of subsets of size k of a set of n elements. Equivalently, it is the number of unordered choices of k distinct elements from a set of n elements. However, the binomial coefficient leads a double life. Not only does it have the above definition, but also the formula below, which we proved in lecture. Proposition 2. For non-negative integers n and k, n nk n(n − 1)(n − 2) ::: (n − k + 1) n! = = = : k k! k! k!(n − k)! In this note, we will prove several more facts about this most fascinating of creatures. -
Chapter 3.3, 4.1, 4.3. Binomial Coefficient Identities
Chapter 3.3, 4.1, 4.3. Binomial Coefficient Identities Prof. Tesler Math 184A Winter 2019 Prof. Tesler Binomial Coefficient Identities Math 184A / Winter 2019 1 / 36 Table of binomial coefficients n k k = 0 k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 n = 0 1 0 0 0 0 0 0 n = 1 1 1 0 0 0 0 0 n = 2 1 2 1 0 0 0 0 n = 3 1 3 3 1 0 0 0 n = 4 1 4 6 4 1 0 0 n = 5 1 5 10 10 5 1 0 n = 6 1 6 15 20 15 6 1 n n! Compute a table of binomial coefficients using = . k k! (n - k)! We’ll look at several patterns. First, the nonzero entries of each row are symmetric; e.g., row n = 4 is 4 4 4 4 4 0 , 1 , 2 , 3 , 4 = 1, 4, 6, 4, 1 , n n which reads the same in reverse. Conjecture: k = n-k . Prof. Tesler Binomial Coefficient Identities Math 184A / Winter 2019 2 / 36 A binomial coefficient identity Theorem For nonegative integers k 6 n, n n n n = including = = 1 k n - k 0 n First proof: Expand using factorials: n n! n n! = = k k! (n - k)! n - k (n - k)! k! These are equal. Prof. Tesler Binomial Coefficient Identities Math 184A / Winter 2019 3 / 36 Theorem For nonegative integers k 6 n, n n n n = including = = 1 k n - k 0 n Second proof: A bijective proof. We’ll give a bijection between two sets, one counted by the left n n side, k , and the other by the right side, n-k . -
Basic Operations for Fuzzy Multisets
Basic operations for fuzzy multisets A. Riesgoa, P. Alonsoa, I. D´ıazb, S. Montesc aDepartment of Mathematics, University of Oviedo, Spain bDepartment of Informatics, University of Oviedo, Spain cDepartment of Statistics and O.R., University of Oviedo, Spain Abstract In their original and ordinary formulation, fuzzy sets associate each element in a reference set with one number, the membership value, in the real unit interval [0; 1]. Among the various existing generalisations of the concept, we find fuzzy multisets. In this case, membership values are multisets in [0; 1] rather than single values. Mathematically, they can be also seen as a generalisation of the hesitant fuzzy sets, but in this general environment, the information about repetition is not lost, so that, the opinions given by the experts are better managed. Thus, we focus our study on fuzzy multisets and their basic operations: complement, union and intersection. Moreover, we show how the hesitant operations can be worked out from an extension of the fuzzy multiset operations and investigate the important role that the concepts of order and sorting sequences play in the basic difference between these two related approaches. Keywords: Fuzzy multiset; Hesitant fuzzy set; Complement; Aggregate union; Aggregate intersection. 1. Introduction The concept of a fuzzy set was originally introduced by Lotfi A. Zadeh as an extension of classical set theory [15]. Fuzzy sets aim to account for real-life situations where there is either limited knowledge or some sort of implicit ambiguity about whether an element should be considered a member of a set. This extension from the classical (or crisp) sets to the fuzzy ones is accomplished by replacing the Boolean characteristic function of a set, which maps each element of the universal set to either 0 (non-member) or 1 (member), with a more sophisticated membership function that maps elements into the real interval [0; 1]. -
Finding Direct Partition Bijections by Two-Directional Rewriting Techniques
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 285 (2004) 151–166 www.elsevier.com/locate/disc Finding direct partition bijections by two-directional rewriting techniques Max Kanovich Department of Computer and Information Science, University of Pennsylvania, 3330 Walnut Street, Philadelphia, PA 19104, USA Received 7 May 2003; received in revised form 27 October 2003; accepted 21 January 2004 Abstract One basic activity in combinatorics is to establish combinatorial identities by so-called ‘bijective proofs,’ which consists in constructing explicit bijections between two types of the combinatorial objects under consideration. We show how such bijective proofs can be established in a systematic way from the ‘lattice properties’ of partition ideals, and how the desired bijections are computed by means of multiset rewriting, for a variety of combinatorial problems involving partitions. In particular, we fully characterizes all equinumerous partition ideals with ‘disjointly supported’ complements. This geometrical characterization is proved to automatically provide the desired bijection between partition ideals but in terms of the minimal elements of the order ÿlters, their complements. As a corollary, a new transparent proof, the ‘bijective’ one, is given for all equinumerous classes of the partition ideals of order 1 from the classical book “The Theory of Partitions” by G.Andrews. Establishing the required bijections involves two-directional reductions technique novel in the sense that forward and backward application of rewrite rules heads, respectively, for two di?erent normal forms (representing the two combinatorial types). It is well-known that non-overlapping multiset rules are con@uent. -
Symmetric Relations and Cardinality-Bounded Multisets in Database Systems
Symmetric Relations and Cardinality-Bounded Multisets in Database Systems Kenneth A. Ross Julia Stoyanovich Columbia University¤ [email protected], [email protected] Abstract 1 Introduction A relation R is symmetric in its ¯rst two attributes if R(x ; x ; : : : ; x ) holds if and only if R(x ; x ; : : : ; x ) In a binary symmetric relationship, A is re- 1 2 n 2 1 n holds. We call R(x ; x ; : : : ; x ) the symmetric com- lated to B if and only if B is related to A. 2 1 n plement of R(x ; x ; : : : ; x ). Symmetric relations Symmetric relationships between k participat- 1 2 n come up naturally in several contexts when the real- ing entities can be represented as multisets world relationship being modeled is itself symmetric. of cardinality k. Cardinality-bounded mul- tisets are natural in several real-world appli- Example 1.1 In a law-enforcement database record- cations. Conventional representations in re- ing meetings between pairs of individuals under inves- lational databases su®er from several consis- tigation, the \meets" relationship is symmetric. 2 tency and performance problems. We argue that the database system itself should pro- Example 1.2 Consider a database of web pages. The vide native support for cardinality-bounded relationship \X is linked to Y " (by either a forward or multisets. We provide techniques to be im- backward link) between pairs of web pages is symmet- plemented by the database engine that avoid ric. This relationship is neither reflexive nor antire- the drawbacks, and allow a schema designer to flexive, i.e., \X is linked to X" is neither universally simply declare a table to be symmetric in cer- true nor universally false. -
An Overview of the Applications of Multisets1
Novi Sad J. Math. Vol. 37, No. 2, 2007, 73-92 AN OVERVIEW OF THE APPLICATIONS OF MULTISETS1 D. Singh2, A. M. Ibrahim2, T. Yohanna3 and J. N. Singh4 Abstract. This paper presents a systemization of representation of multisets and basic operations under multisets, and an overview of the applications of multisets in mathematics, computer science and related areas. AMS Mathematics Subject Classi¯cation (2000): 03B65, 03B70, 03B80, 68T50, 91F20 Key words and phrases: Multisets, power multisets, dressed epsilon sym- bol, multiset ordering, Dershowtiz-Mana ordering 1. Introduction A multiset (mset, for short) is an unordered collection of objects (called the elements) in which, unlike a standard (Cantorian) set, elements are allowed to repeat. In other words, an mset is a set to which elements may belong more than once, and hence it is a non-Cantorian set. In this paper, we endeavour to present an overview of basics of multiset and applications. The term multiset, as Knuth ([46], p. 36) notes, was ¯rst suggested by N.G.de Bruijn in a private communication to him. Owing to its aptness, it has replaced a variety of terms, viz. list, heap, bunch, bag, sample, weighted set, occurrence set, and ¯reset (¯nitely repeated element set) used in di®erent contexts but conveying synonimity with mset. As mentioned earlier, elements are allowed to repeat in an mset (¯nitely in most of the known application areas, albeit in a theoretical development in¯nite multiplicities of elements are also dealt with (see [11], [37], [51], [72] and [30], in particular). The number of copies (([17], P. -
8.6 the BINOMIAL THEOREM We Remake Nature by the Act of Discovery, in the Poem Or in the Theorem
pg476 [V] G2 5-36058 / HCG / Cannon & Elich cr 11-30-95 MP1 476 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers (b) Based on your results for (a), guess the minimum number of handshakes. See Exercise 19, page 469. number of moves required if you start with an arbi- Show trary number of n disks. (Hint: Tohelpseeapat- n~n 2 1! tern,add1tothenumberofmovesforn51, 2, 3, f ~n! 5 for every positive integer n. 4.) 2 (c) A legend claims that monks in a remote monastery 56. Suppose n is an odd positive integer not divisible by 3. areworkingtomoveasetof64disks,andthatthe Show that n 2 2 1 is divisible by 24. (Hint: Consider the world will end when they complete their sacred three consecutive integers n 2 1, n, n 1 1. Explain task. Moving one disk per second without error, 24 why the product ~n 2 1!~n 1 1! must be divisible by 3 hours a day, 365 days a year, how long would it take andby8.) to move all 64 disks? 57. Finding Patterns If an 5 Ï24n 1 1, (a) write out 55. Number of Handshakes Suppose there are n people the first five terms of the sequence $an%. (b) What odd at a party and that each person shakes hands with every integers occur in $an%? (c) Explain why $an% contains all other person exactly once. Let f~n! denote the total primes greater than 3. (Hint: UseExercise56.) 8.6 THE BINOMIAL THEOREM We remake nature by the act of discovery, in the poem or in the theorem.