Binomial Coefficients Math 217 Probability and Statistics
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Combinatorial Proof of an Abel-Type Identity∗
Combinatorial Proof of an Abel-type Identity∗ P. Mark Kayll Department of Mathematical Sciences, University of Montana Missoula MT 59812-0864, USA [email protected] David Perkins Department of Mathematics and Computer Science Houghton College, Houghton NY 14744, USA [email protected] Identity (1) below resulted from our investigation in [21] of chip-firing games on complete graphs Kn, for n ≥ 1; see, e.g., [2] for antecedents. The left side expresses the sum of the probabilities of a game experiencing firing sequences of each possible length ℓ = 0, 1,...,n. This note gives a combinatorial proof that these probabilities sum to unity. We first manipulate n n − 1 n ℓℓ−1(n + 1 − ℓ)n−1−ℓ + = 1 (1) n + 1 µℓ¶ n(n + 1)n−1 Xℓ=1 into a form amenable to combinatorial proof. Multiplying by n(n+1)n and n n+1 using the relation ℓ = ℓ (n + 1 − ℓ)/(n + 1), we transform (1) to the equivalent form ¡ ¢ ¡ ¢ n n + 1 ℓℓ−2(n + 1 − ℓ)n−1−ℓℓ(n + 1 − ℓ) = 2n(n + 1)n−1. (2) µ ℓ ¶ Xℓ=1 To see that (2) holds, first observe that the right side enumerates the pairs (T,~e ), where T is a spanning tree of Kn+1 for which one edge e (of 2000 MSC: Primary 05A19; Secondary 05C30, 60C05. ∗Preprint to appear in J. Combin. Math. Combin. Comput. 1 its n edges) has been distinguished and oriented (in one of two possible directions). The left side also enumerates these pairs. Given (T,~e ), notice that deleting the oriented edge ~e from T leaves behind a spanning forest of Kn+1 with two components L, R (that we may consider ordered from left to right). -
Multipermutations 1.1. Generating Functions
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector JOURNALOF COMBINATORIALTHEORY, Series A 67, 44-71 (1994) The r- Multipermutations SEUNGKYUNG PARK Department of Mathematics, Yonsei University, Seoul 120-749, Korea Communicated by Gian-Carlo Rota Received October 1, 1992 In this paper we study permutations of the multiset {lr, 2r,...,nr}, which generalizes Gesse| and Stanley's work (J. Combin. Theory Ser. A 24 (1978), 24-33) on certain permutations on the multiset {12, 22..... n2}. Various formulas counting the permutations by inversions, descents, left-right minima, and major index are derived. © 1994 AcademicPress, Inc. 1. THE r-MULTIPERMUTATIONS 1.1. Generating Functions A multiset is defined as an ordered pair (S, f) where S is a set and f is a function from S to the set of nonnegative integers. If S = {ml ..... mr}, we write {m f(mD, .... m f(mr)r } for (S, f). Intuitively, a multiset is a set with possibly repeated elements. Let us begin by considering permutations of multisets (S, f), which are words in which each letter belongs to the set S and for each m e S the total number of appearances of m in the word is f(m). Thus 1223231 is a permutation of the multiset {12, 2 3, 32}. In this paper we consider a special set of permutations of the multiset [n](r) = {1 r, ..., nr}, which is defined as follows. DEFINITION 1.1. An r-multipermutation of the set {ml ..... ran} is a permutation al ""am of the multiset {m~ .... -
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 . -
Math 958–Topics in Discrete Mathematics Spring Semester 2018 TR 09:30–10:45 in Avery Hall (AVH) 351 1 Instructor
Math 958{Topics in Discrete Mathematics Spring Semester 2018 TR 09:30{10:45 in Avery Hall (AVH) 351 1 Instructor Dr. Tri Lai Assistant Professor Department of Mathematics University of Nebraska - Lincoln Lincoln, NE 68588-0130, USA Email: [email protected] Website: http://www.math.unl.edu/$\sim$tlai3/ Office: Avery Hall 339 Office Hours: By Appointment 2 Prerequisites Math 450. 3 Contacting me The best way to contact with me is by email, [email protected]. Please put \[MATH 958]" at the beginning of the title and make sure to include your whole name in your email. Using your official UNL email to contact me is strongly recommended. My office is Avery Hall 339. My office hours are by appoinment. 4 Course Description Bijective Combinatorics is a branch of combinatorics focusing on bijections between mathematical objects. The fact is that there many totally different objects in mathematics that are actually equinumerous, in this case, we always want to seek for a simple bijection between them. For example, the number of ways to divide that a convex (n + 2)-gon into triangles by non-intersecting diagonals is equal to the number of full binary trees with n + 1 leaves. Both objects are counted 1 2n by the well known Catalan number Cn = n+1 n . Do you know that there are more than two hundred (!!) mathematical objects are counted by the Catalan number? In this course, we will go over a number of such \Catalan objects" and investigate the beautiful bijections between them. One more example is the well-known theorem by Euler about integer partitions stating that the number of ways to write a positive integer n as the sum of distinct positive integers is equal to the number of ways to write n as the sum of odd positive integers. -
Basic Combinatorics
Basic Combinatorics Carl G. Wagner Department of Mathematics The University of Tennessee Knoxville, TN 37996-1300 Contents List of Figures iv List of Tables v 1 The Fibonacci Numbers From a Combinatorial Perspective 1 1.1 A Simple Counting Problem . 1 1.2 A Closed Form Expression for f(n) . 2 1.3 The Method of Generating Functions . 3 1.4 Approximation of f(n) . 4 2 Functions, Sequences, Words, and Distributions 5 2.1 Multisets and sets . 5 2.2 Functions . 6 2.3 Sequences and words . 7 2.4 Distributions . 7 2.5 The cardinality of a set . 8 2.6 The addition and multiplication rules . 9 2.7 Useful counting strategies . 11 2.8 The pigeonhole principle . 13 2.9 Functions with empty domain and/or codomain . 14 3 Subsets with Prescribed Cardinality 17 3.1 The power set of a set . 17 3.2 Binomial coefficients . 17 4 Sequences of Two Sorts of Things with Prescribed Frequency 23 4.1 A special sequence counting problem . 23 4.2 The binomial theorem . 24 4.3 Counting lattice paths in the plane . 26 5 Sequences of Integers with Prescribed Sum 28 5.1 Urn problems with indistinguishable balls . 28 5.2 The family of all compositions of n . 30 5.3 Upper bounds on the terms of sequences with prescribed sum . 31 i CONTENTS 6 Sequences of k Sorts of Things with Prescribed Frequency 33 6.1 Trinomial Coefficients . 33 6.2 The trinomial theorem . 35 6.3 Multinomial coefficients and the multinomial theorem . 37 7 Combinatorics and Probability 39 7.1 The Multinomial Distribution . -
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. -
Discrete Math for Computer Science Students
Discrete Math for Computer Science Students Cliff Stein Ken Bogart Scot Drysdale Dept. of Industrial Engineering Dept. of Mathematics Dept. of Computer Science and Operations Research Dartmouth College Dartmouth College Columbia University ii c Kenneth P. Bogart, Scot Drysdale, and Cliff Stein, 2004 Contents 1 Counting 1 1.1 Basic Counting . 1 The Sum Principle . 1 Abstraction . 2 Summing Consecutive Integers . 3 The Product Principle . 3 Two element subsets . 5 Important Concepts, Formulas, and Theorems . 6 Problems . 7 1.2 Counting Lists, Permutations, and Subsets. 9 Using the Sum and Product Principles . 9 Lists and functions . 10 The Bijection Principle . 12 k-element permutations of a set . 13 Counting subsets of a set . 13 Important Concepts, Formulas, and Theorems . 15 Problems . 16 1.3 Binomial Coefficients . 19 Pascal’s Triangle . 19 A proof using the Sum Principle . 20 The Binomial Theorem . 22 Labeling and trinomial coefficients . 23 Important Concepts, Formulas, and Theorems . 24 Problems . 25 1.4 Equivalence Relations and Counting (Optional) . 27 The Symmetry Principle . 27 iii iv CONTENTS Equivalence Relations . 28 The Quotient Principle . 29 Equivalence class counting . 30 Multisets . 31 The bookcase arrangement problem. 32 The number of k-element multisets of an n-element set. 33 Using the quotient principle to explain a quotient . 34 Important Concepts, Formulas, and Theorems . 34 Problems . 35 2 Cryptography and Number Theory 39 2.1 Cryptography and Modular Arithmetic . 39 Introduction to Cryptography . 39 Private Key Cryptography . 40 Public-key Cryptosystems . 42 Arithmetic modulo n .................................. 43 Cryptography using multiplication mod n ...................... 47 Important Concepts, Formulas, and Theorems . 48 Problems . 49 2.2 Inverses and GCDs . -
Combinatorial Proofs of Fibonomial Identities 1
COMBINATORIAL PROOFS OF FIBONOMIAL IDENTITIES ARTHUR T. BENJAMIN AND ELIZABETH REILAND Abstract. Fibonomial coefficients are defined like binomial coefficients, with integers re- placed by their respective Fibonacci numbers. For example, 10 = F10F9F8 . Remarkably, 3 F F3F2F1 n is always an integer. In 2010, Bruce Sagan and Carla Savage derived two very nice k F combinatorial interpretations of Fibonomial coefficients in terms of tilings created by lattice paths. We believe that these interpretations should lead to combinatorial proofs of Fibonomial identities. We provide a list of simple looking identities that are still in need of combinatorial proof. 1. Introduction What do you get when you cross Fibonacci numbers with binomial coefficients? Fibono- mial coefficients, of course! Fibonomial coefficients are defined like binomial coefficients, with integers replaced by their respective Fibonacci numbers. Specifically, for n ≥ k ≥ 1, n F F ··· F = n n−1 n−k+1 k F F1F2 ··· Fk For example, 10 = F10F9F8 = 55·34·21 = 19;635. Fibonomial coefficients resemble binomial 3 F F3F2F1 1·1·2 n coefficients in many ways. Analogous to the Pascal Triangle boundary conditions 1 = n and n n n n n = 1, we have 1 F = Fn and n F = 1. We also define 0 F = 1. n n−1 n−1 Since Fn = Fk+(n−k) = Fk+1Fn−k + FkFn−k−1, Pascal's recurrence k = k + k−1 has the following analog. Identity 1.1. For n ≥ 2, n n − 1 n − 1 = Fk+1 + Fn−k−1 : k F k F k − 1 F n As an immediate corollary, it follows that for all n ≥ k ≥ 1, k F is an integer. -
A Combinatorial Proof of the Log- Concavity of the Numbers Of
Journal of Combinatorial Theory, Series A 90, 293303 (2000) doi:10.1006Âjcta.1999.3040, available online at http:ÂÂwww.idealibrary.com on A Combinatorial Proof of the Log-Concavity of the Numbers of Permutations with k Runs MiklosBona1 University of Florida, Gainesville, Florida 32611 E-mail: bonaÄmath.ufl.edu and Richard Ehrenborg2 School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 E-mail: jrgeÄmath.ias.edu Communicated by the Managing Editors Received February 26, 1999 We combinatorially prove that the number R(n, k) of permutations of length n having k runs is a log-concave sequence in k, for all n. We also give a new com- binatorial proof for the log-concavity of the Eulerian numbers. 2000 Academic Press 1. INTRODUCTION Let p= p1 p2 }}}pn be a permutation of the set [1, 2, ..., n] written in the one-line notation. We say that p get changes direction at position i, if either pi&1<pi>pi+ j ,orpi&1>pi>pi+1, in other words, when pi is either a peak or a valley. We say that p has k runs if there are k&1 indices i so that p changes direction at these positions. So, for example, p=3561247 has 3 runs as p changes direction when i=3 and when i=4. A geometric way to represent a permutation and its runs by a diagram is shown in Fig. 1. The runs are the line segments (or edges) between two consecutive entries where p changes direction. So a permutation has k runs if it can be represented by k line segments so that the segments go ``up'' and ``down'' 1 The paper was written while the author's stay at IAS was supported by Trustee Ladislaus von Hoffmann, the Arcana Foundation.