Named numbres. Ngày 25 tháng 11 năm () Named numbres Ngày 25 tháng 11 năm / 7
|
|
|
- Dominick Skinner
- 5 years ago
- Views:
Transcription
1 Named numbres Ngày 25 tháng 11 năm 2011 () Named numbres Ngày 25 tháng 11 năm / 7
2 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. () Named numbres Ngày 25 tháng 11 năm / 7
3 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. 2 1, 3, 5, 7, 9... the odd integers. () Named numbres Ngày 25 tháng 11 năm / 7
4 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. 2 1, 3, 5, 7, 9... the odd integers. 3 2, 3, 5, 7, 11, 13,... () Named numbres Ngày 25 tháng 11 năm / 7
5 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. 2 1, 3, 5, 7, 9... the odd integers. 3 2, 3, 5, 7, 11, 13,... () Named numbres Ngày 25 tháng 11 năm / 7
6 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. 2 1, 3, 5, 7, 9... the odd integers. 3 2, 3, 5, 7, 11, 13,... the prime numbers. 4 1, 1, 2, 3, 5, 8, 13,... () Named numbres Ngày 25 tháng 11 năm / 7
7 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. 2 1, 3, 5, 7, 9... the odd integers. 3 2, 3, 5, 7, 11, 13,... the prime numbers. 4 1, 1, 2, 3, 5, 8, 13,... () Named numbres Ngày 25 tháng 11 năm / 7
8 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. 2 1, 3, 5, 7, 9... the odd integers. 3 2, 3, 5, 7, 11, 13,... the prime numbers. 4 1, 1, 2, 3, 5, 8, 13,... Fibonacci numbers. ( n ) 5 () Named numbres Ngày 25 tháng 11 năm / 7
9 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. 2 1, 3, 5, 7, 9... the odd integers. 3 2, 3, 5, 7, 11, 13,... the prime numbers. 4 1, 1, 2, 3, 5, 8, 13,... Fibonacci numbers. ( n ) 5 () Named numbres Ngày 25 tháng 11 năm / 7
10 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. 2 1, 3, 5, 7, 9... the odd integers. 3 2, 3, 5, 7, 11, 13,... the prime numbers. 4 1, 1, 2, 3, 5, 8, 13,... Fibonacci numbers. ( 5 n ) binomials. ( 6 1 2n ) n+1 n () Named numbres Ngày 25 tháng 11 năm / 7
11 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. 2 1, 3, 5, 7, 9... the odd integers. 3 2, 3, 5, 7, 11, 13,... the prime numbers. 4 1, 1, 2, 3, 5, 8, 13,... Fibonacci numbers. ( 5 n ) binomials. ( 6 1 2n ) n+1 n () Named numbres Ngày 25 tháng 11 năm / 7
12 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. 2 1, 3, 5, 7, 9... the odd integers. 3 2, 3, 5, 7, 11, 13,... the prime numbers. 4 1, 1, 2, 3, 5, 8, 13,... Fibonacci numbers. ( 5 n ) binomials. ( 6 1 2n ) n+1 n the Catalan numbers. () Named numbres Ngày 25 tháng 11 năm / 7
13 Fibonacci, Catalan, Stirling, Euler, Bernoulli Many sequences are famous. 1 1, 2, 3, 4,... the integers. 2 1, 3, 5, 7, 9... the odd integers. 3 2, 3, 5, 7, 11, 13,... the prime numbers. 4 1, 1, 2, 3, 5, 8, 13,... Fibonacci numbers. ( 5 n ) binomials. ( 6 1 2n ) n+1 n the Catalan numbers. The reason they are named is because they appear in many forms in mathematics and other sciences. () Named numbres Ngày 25 tháng 11 năm / 7
14 Stirling Numbers Stirling Numbers are named after the Scottish mathematician James Stirling who introduced them in the 18 th century. There are two inds of Stirling numbers (with various notation): [ ] n Stirling numbers of the first ind: = c(n, ) { n Stirling numbers of the second ind: } = S(n, ) Both numbers describe combinatorial counting that lead to a triangular recurrence relation similar to the binomial coefficients. () Named numbres Ngày 25 tháng 11 năm / 7
15 Stirling numbers of the second ind: Question In how many ways can you partition the set {a, b, c, d, e} into two non-empty subsets? () Named numbres Ngày 25 tháng 11 năm / 7
16 Stirling numbers of the second ind: Question In how many ways can you partition the set {a, b, c, d, e} into two non-empty subsets? Answer We can just list the subsets: {{a}, {, b, c, d, e}}, {{b}, {a, c, d, e}}..., {{a, b}, {c, d, e}},... {{d, e}, {, a, b, c}}. For a total of: () Named numbres Ngày 25 tháng 11 năm / 7
17 Stirling numbers of the second ind: Question In how many ways can you partition the set {a, b, c, d, e} into two non-empty subsets? Answer We can just list the subsets: {{a}, {, b, c, d, e}}, {{b}, {a, c, d, e}}..., {{a, b}, {c, d, e}},... {{d, e}, {, a, b, c}}. For a total of: () Named numbres Ngày 25 tháng 11 năm / 7
18 Stirling numbers of the second ind: Question In how many ways can you partition the set {a, b, c, d, e} into two non-empty subsets? Answer We can just list the subsets: {{a}, {, b, c, d, e}}, {{b}, {a, c, d, e}}..., {{a, b}, {c, d, e}},... {{d, e}, {, a, b, c}}. For a total of: = 15 different partitions. Definition { n } The Stirling number of the second ind is the number of ways to partition an n-set into non-empty subsets. () Named numbres Ngày 25 tháng 11 năm / 7
19 From the example we can derive a recurrence relation for these numbers: () Named numbres Ngày 25 tháng 11 năm / 7
20 From the example we can derive a recurrence relation for these numbers: Let B be an n set. We can divide the patitions into two sets A: partitions that include a fixed singleton {x 0 } and B: the rest. () Named numbres Ngày 25 tháng 11 năm / 7
21 From the example we can derive a recurrence relation for these numbers: Let B be an n set. We can divide the patitions into two sets A: partitions that include a fixed singleton {x 0 } and B: the rest. Clearly, there are S(n 1, 1) distinct partitions in A. () Named numbres Ngày 25 tháng 11 năm / 7
22 From the example we can derive a recurrence relation for these numbers: Let B be an n set. We can divide the patitions into two sets A: partitions that include a fixed singleton {x 0 } and B: the rest. Clearly, there are S(n 1, 1) distinct partitions in A. As for the set B for every partition B \ {x 0 } into subset we can add x 0 to any one of the parttions yielding: { } { } { } n n 1 n 1 = + 1 () Named numbres Ngày 25 tháng 11 năm / 7
23 From the example we can derive a recurrence relation for these numbers: Let B be an n set. We can divide the patitions into two sets A: partitions that include a fixed singleton {x 0 } and B: the rest. Clearly, there are S(n 1, 1) distinct partitions in A. As for the set B for every partition B \ {x 0 } into subset we can add x 0 to any one of the parttions yielding: { } { } { } n n 1 n 1 = + 1 This triangular relation is very similar to Pascal s identity for binomials. () Named numbres Ngày 25 tháng 11 năm / 7
24 An application The polynomials of degree n form a vector space over the field R, and so do the polynomials {1, x, x(x 1), x(x 1)(x 2)...}. A common notation for x(x 1)... (x j + 1) is x j This means that the polynomials x can be expressed as linear combination of these polynomials: x = a i x i i=0 () Named numbres Ngày 25 tháng 11 năm / 7
25 An application The polynomials of degree n form a vector space over the field R, and so do the polynomials {1, x, x(x 1), x(x 1)(x 2)...}. A common notation for x(x 1)... (x j + 1) is x j This means that the polynomials x can be expressed as linear combination of these polynomials: What are the coefficients a i? x = a i x i i=0 () Named numbres Ngày 25 tháng 11 năm / 7
26 Claim: x n = n =0 { n } x Chứng minh. () Named numbres Ngày 25 tháng 11 năm / 7
27 Claim: x n = n =0 { n } x Chứng minh. 1 a. x x = x +1 + x. () Named numbres Ngày 25 tháng 11 năm / 7
28 Claim: x n = n =0 { n } x Chứng minh. 1 a. x x = x +1 + x. 2 n 1 { x x n 1 n 1 = x =0 n 1 { n 1 =0 } x = n 1 =0 } { x n Follows from the triangular relation. { n 1 } x = } (x +1 + x ) = n =0 { n } x () Named numbres Ngày 25 tháng 11 năm / 7
29 The Stirling numbers of the first ind are defined by a closely related relation: It counts in how many ways you can arrange n objects into disjoint cycles. So for example, the partitions {[1, 3][2, 5, 4]} and {[1, 3][2, 4, 5]} are distinct but {[3, 1], [5, 4, 2]} is the same as the first partition. () Named numbres Ngày 25 tháng 11 năm / 7
30 The Stirling numbers of the first ind are defined by a closely related relation: It counts in how many ways you can arrange n objects into disjoint cycles. So for example, the partitions {[1, 3][2, 5, 4]} and {[1, 3][2, 4, 5]} are distinct but {[3, 1], [5, 4, 2]} is the same as the first partition. We leave it to you to show that: 1 [ n ] [ n 1 = (n 1) ] [ n ] () Named numbres Ngày 25 tháng 11 năm / 7
31 The Stirling numbers of the first ind are defined by a closely related relation: It counts in how many ways you can arrange n objects into disjoint cycles. So for example, the partitions {[1, 3][2, 5, 4]} and {[1, 3][2, 4, 5]} are distinct but {[3, 1], [5, 4, 2]} is the same as the first partition. We leave it to you to show that: 1 [ n ] [ n 1 = (n 1) 2 Let x n = x(x + 1)... (x + n 1) ] [ n ] () Named numbres Ngày 25 tháng 11 năm / 7
32 The Stirling numbers of the first ind are defined by a closely related relation: It counts in how many ways you can arrange n objects into disjoint cycles. So for example, the partitions {[1, 3][2, 5, 4]} and {[1, 3][2, 4, 5]} are distinct but {[3, 1], [5, 4, 2]} is the same as the first partition. We leave it to you to show that: 1 [ n ] [ n 1 = (n 1) 2 Let x n = x(x + 1)... (x + n 1) 3 x n = n x =0 ] [ n ] () Named numbres Ngày 25 tháng 11 năm / 7
33 The Stirling numbers of the first ind are defined by a closely related relation: It counts in how many ways you can arrange n objects into disjoint cycles. So for example, the partitions {[1, 3][2, 5, 4]} and {[1, 3][2, 4, 5]} are distinct but {[3, 1], [5, 4, 2]} is the same as the first partition. We leave it to you to show that: 1 [ n ] [ n 1 = (n 1) 2 Let x n = x(x + 1)... (x + n 1) 3 x n = n 4 x =0 n =0 [ n ] = n! ] [ n ] () Named numbres Ngày 25 tháng 11 năm / 7
CS2800 Fall 2013 October 23, 2013
Discrete Structures Stirling Numbers CS2800 Fall 203 October 23, 203 The text mentions Stirling numbers briefly but does not go into them in any depth. However, they are fascinating numbers with a lot
are the q-versions of n, n! and . The falling factorial is (x) k = x(x 1)(x 2)... (x k + 1).
Lecture A jacques@ucsd.edu Notation: N, R, Z, F, C naturals, reals, integers, a field, complex numbers. p(n), S n,, b(n), s n, partition numbers, Stirling of the second ind, Bell numbers, Stirling of the
Binomial Coefficient Identities/Complements
Binomial Coefficient Identities/Complements CSE21 Fall 2017, Day 4 Oct 6, 2017 https://sites.google.com/a/eng.ucsd.edu/cse21-fall-2017-miles-jones/ permutation P(n,r) = n(n-1) (n-2) (n-r+1) = Terminology
On Arithmetic Properties of Bell Numbers, Delannoy Numbers and Schröder Numbers
A tal given at the Institute of Mathematics, Academia Sinica (Taiwan (Taipei; July 6, 2011 On Arithmetic Properties of Bell Numbers, Delannoy Numbers and Schröder Numbers Zhi-Wei Sun Nanjing University
Fibonacci numbers. Chapter The Fibonacci sequence. The Fibonacci numbers F n are defined recursively by
Chapter Fibonacci numbers The Fibonacci sequence The Fibonacci numbers F n are defined recursively by F n+ = F n + F n, F 0 = 0, F = The first few Fibonacci numbers are n 0 5 6 7 8 9 0 F n 0 5 8 55 89
A combinatorial bijection on k-noncrossing partitions
A combinatorial bijection on k-noncrossing partitions KAIST Korea Advanced Institute of Science and Technology 2018 JMM Special Session in honor of Dennis Stanton January 10, 09:30 09:50 I would like to
What you learned in Math 28. Rosa C. Orellana
What you learned in Math 28 Rosa C. Orellana Chapter 1 - Basic Counting Techniques Sum Principle If we have a partition of a finite set S, then the size of S is the sum of the sizes of the blocks of the
Generating Functions
Semester 1, 2004 Generating functions Another means of organising enumeration. Two examples we have seen already. Example 1. Binomial coefficients. Let X = {1, 2,..., n} c k = # k-element subsets of X
MATH 802: ENUMERATIVE COMBINATORICS ASSIGNMENT 2
MATH 80: ENUMERATIVE COMBINATORICS ASSIGNMENT KANNAPPAN SAMPATH Facts Recall that, the Stirling number S(, n of the second ind is defined as the number of partitions of a [] into n non-empty blocs. We
Counting. Mukulika Ghosh. Fall Based on slides by Dr. Hyunyoung Lee
Counting Mukulika Ghosh Fall 2018 Based on slides by Dr. Hyunyoung Lee Counting Counting The art of counting is known as enumerative combinatorics. One tries to count the number of elements in a set (or,
Sets. A set is a collection of objects without repeats. The size or cardinality of a set S is denoted S and is the number of elements in the set.
Sets A set is a collection of objects without repeats. The size or cardinality of a set S is denoted S and is the number of elements in the set. If A and B are sets, then the set of ordered pairs each
Combinatorics of inverse semigroups. 1 Inverse semigroups and orders
Combinatorics of inverse semigroups This is an exposition of some results explained to me by Abdullahi Umar, together with some further speculations of my own and some with Ni Rusuc. 1 Inverse semigroups
The candidates are advised that they must always show their working, otherwise they will not be awarded full marks for their answers.
MID SWEDEN UNIVERSITY TFM Examinations 2006 MAAB16 Discrete Mathematics B Duration: 5 hours Date: 7 June 2006 There are EIGHT questions on this paper and you should answer as many as you can in the time
Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting
Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 39 Chapter Summary The Basics
MATH 310 Course Objectives
MATH 310 Course Objectives Upon successful completion of MATH 310, the student should be able to: Apply the addition, subtraction, multiplication, and division principles to solve counting problems. Apply
and the compositional inverse when it exists is A.
Lecture B jacques@ucsd.edu Notation: R denotes a ring, N denotes the set of sequences of natural numbers with finite support, is a generic element of N, is the infinite zero sequence, n 0 R[[ X]] denotes
Warm-up Quantifiers and the harmonic series Sets Second warmup Induction Bijections. Writing more proofs. Misha Lavrov
Writing more proofs Misha Lavrov ARML Practice 3/16/2014 and 3/23/2014 Warm-up Using the quantifier notation on the reference sheet, and making any further definitions you need to, write the following:
Homework #2 Solutions Due: September 5, for all n N n 3 = n2 (n + 1) 2 4
Do the following exercises from the text: Chapter (Section 3):, 1, 17(a)-(b), 3 Prove that 1 3 + 3 + + n 3 n (n + 1) for all n N Proof The proof is by induction on n For n N, let S(n) be the statement
COMBINATORIAL COUNTING
COMBINATORIAL COUNTING Our main reference is [1, Section 3] 1 Basic counting: functions and subsets Theorem 11 (Arbitrary mapping Let N be an n-element set (it may also be empty and let M be an m-element
Bijective Proofs with Spotted Tilings
Brian Hopkins, Saint Peter s University, New Jersey, USA Visiting Scholar, Mahidol University International College Editor, The College Mathematics Journal MUIC Mathematics Seminar 2 November 2016 outline
A RESULT ON RAMANUJAN-LIKE CONGRUENCE PROPERTIES OF THE RESTRICTED PARTITION FUNCTION p(n, m) ACROSS BOTH VARIABLES
#A63 INTEGERS 1 (01) A RESULT ON RAMANUJAN-LIKE CONGRUENCE PROPERTIES OF THE RESTRICTED PARTITION FUNCTION p(n, m) ACROSS BOTH VARIABLES Brandt Kronholm Department of Mathematics, Whittier College, Whittier,
Discrete Mathematics, Spring 2004 Homework 4 Sample Solutions
Discrete Mathematics, Spring 2004 Homework 4 Sample Solutions 4.2 #77. Let s n,k denote the number of ways to seat n persons at k round tables, with at least one person at each table. (The numbers s n,k
Combinatorial Unification of Binomial-Like Arrays
University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2010 Combinatorial Unification of Binomial-Lie Arrays James Stephen Lindsay University
Counting Peaks and Valleys in a Partition of a Set
1 47 6 11 Journal of Integer Sequences Vol. 1 010 Article 10.6.8 Counting Peas and Valleys in a Partition of a Set Toufi Mansour Department of Mathematics University of Haifa 1905 Haifa Israel toufi@math.haifa.ac.il
Applications of Chromatic Polynomials Involving Stirling Numbers
Applications of Chromatic Polynomials Involving Stirling Numbers A. Mohr and T.D. Porter Department of Mathematics Southern Illinois University Carbondale, IL 6290 tporter@math.siu.edu June 23, 2008 The
Discrete Mathematics. Kishore Kothapalli
Discrete Mathematics Kishore Kothapalli 2 Chapter 4 Advanced Counting Techniques In the previous chapter we studied various techniques for counting and enumeration. However, there are several interesting
The solutions to the two examples above are the same.
One-to-one correspondences A function f : A B is one-to-one if f(x) = f(y) implies that x = y. A function f : A B is onto if for any element b in B there is an element a in A such that f(a) = b. A function
GENERALIZATION OF AN IDENTITY OF ANDREWS
GENERALIZATION OF AN IDENTITY OF ANDREWS arxiv:math/060480v1 [math.co 1 Apr 006 Eduardo H. M. Brietze Instituto de Matemática UFRGS Caixa Postal 15080 91509 900 Porto Alegre, RS, Brazil email: brietze@mat.ufrgs.br
1. Determine (with proof) the number of ordered triples (A 1, A 2, A 3 ) of sets which satisfy
UT Putnam Prep Problems, Oct 19 2016 I was very pleased that, between the whole gang of you, you solved almost every problem this week! Let me add a few comments here. 1. Determine (with proof) the number
Figurate Numbers: presentation of a book
Figurate Numbers: presentation of a book Elena DEZA and Michel DEZA Moscow State Pegagogical University, and Ecole Normale Superieure, Paris October 2011, Fields Institute Overview 1 Overview 2 Chapter
The Inclusion Exclusion Principle
The Inclusion Exclusion Principle 1 / 29 Outline Basic Instances of The Inclusion Exclusion Principle The General Inclusion Exclusion Principle Counting Derangements Counting Functions Stirling Numbers
Some thoughts concerning power sums
Some thoughts concerning power sums Dedicated to the memory of Zoltán Szvetits (929 20 Gábor Nyul Institute of Mathematics, University of Debrecen H 00 Debrecen POBox 2, Hungary e mail: gnyul@scienceunidebhu
Binomial coefficients and k-regular sequences
Binomial coefficients and k-regular sequences Eric Rowland Hofstra University New York Combinatorics Seminar CUNY Graduate Center, 2017 12 22 Eric Rowland Binomial coefficients and k-regular sequences
1 Basic Combinatorics
1 Basic Combinatorics 1.1 Sets and sequences Sets. A set is an unordered collection of distinct objects. The objects are called elements of the set. We use braces to denote a set, for example, the set
Lecture 4: Counting, Pigeonhole Principle, Permutations, Combinations Lecturer: Lale Özkahya
BBM 205 Discrete Mathematics Hacettepe University http://web.cs.hacettepe.edu.tr/ bbm205 Lecture 4: Counting, Pigeonhole Principle, Permutations, Combinations Lecturer: Lale Özkahya Resources: Kenneth
Pascal triangle variation and its properties
Pascal triangle variation and its properties Mathieu Parizeau-Hamel 18 february 2013 Abstract The goal of this work was to explore the possibilities harnessed in the the possible variations of the famous
TILING PROOFS OF SOME FORMULAS FOR THE PELL NUMBERS OF ODD INDEX
#A05 INTEGERS 9 (2009), 53-64 TILING PROOFS OF SOME FORMULAS FOR THE PELL NUMBERS OF ODD INDEX Mark Shattuck Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300 shattuck@math.utk.edu
{ 0! = 1 n! = n(n 1)!, n 1. n! =
Summations Question? What is the sum of the first 100 positive integers? Counting Question? In how many ways may the first three horses in a 10 horse race finish? Multiplication Principle: If an event
Generating Functions
8.30 lecture notes March, 0 Generating Functions Lecturer: Michel Goemans We are going to discuss enumeration problems, and how to solve them using a powerful tool: generating functions. What is an enumeration
CMSC Discrete Mathematics SOLUTIONS TO FIRST MIDTERM EXAM October 18, 2005 posted Nov 2, 2005
CMSC-37110 Discrete Mathematics SOLUTIONS TO FIRST MIDTERM EXAM October 18, 2005 posted Nov 2, 2005 Instructor: László Babai Ryerson 164 e-mail: laci@cs This exam contributes 20% to your course grade.
Dramatis Personae. Robin Whitty. Riemann s Hypothesis, Rewley House, 15 September Queen Mary University of London
Queen Mary University of London Riemann s Hypothesis, Rewley House, 15 September 2013 Symbols Z = {..., 3, 2, 1,0,1,2,3,4,...} the integers R = the real numbers integers, fractions, irrationals C = the
Yi Wang Department of Applied Mathematics, Dalian University of Technology, Dalian , China (Submitted June 2002)
SELF-INVERSE SEQUENCES RELATED TO A BINOMIAL INVERSE PAIR Yi Wang Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China (Submitted June 2002) 1 INTRODUCTION Pairs of
Discrete Mathematics and its Applications
Discrete Mathematics and its Applications Ngày 14 tháng 9 năm 2011 () Discrete Mathematicsand its Applications Ngày 14 tháng 9 năm 2011 1 / 6 Logic-2 (Applications) Applications of logic are abundent.
(n = 0, 1, 2,... ). (2)
Bull. Austral. Math. Soc. 84(2011), no. 1, 153 158. ON A CURIOUS PROPERTY OF BELL NUMBERS Zhi-Wei Sun and Don Zagier Abstract. In this paper we derive congruences expressing Bell numbers and derangement
Basic counting techniques. Periklis A. Papakonstantinou Rutgers Business School
Basic counting techniques Periklis A. Papakonstantinou Rutgers Business School i LECTURE NOTES IN Elementary counting methods Periklis A. Papakonstantinou MSIS, Rutgers Business School ALL RIGHTS RESERVED
McGill University Faculty of Science. Solutions to Practice Final Examination Math 240 Discrete Structures 1. Time: 3 hours Marked out of 60
McGill University Faculty of Science Solutions to Practice Final Examination Math 40 Discrete Structures Time: hours Marked out of 60 Question. [6] Prove that the statement (p q) (q r) (p r) is a contradiction
Indistinguishable objects in indistinguishable boxes
Counting integer partitions 2.4 61 Indistinguishable objects in indistinguishable boxes When placing k indistinguishable objects into n indistinguishable boxes, what matters? We are partitioning the integer
Rook theory and simplicial complexes
Rook theory and simplicial complexes Ira M. Gessel Department of Mathematics Brandeis University A Conference to Celebrate The Mathematics of Michelle Wachs University of Miami, Coral Gables, Florida January
Discrete mathematics , Fall Instructor: prof. János Pach
Discrete mathematics 016-017, Fall Instructor: prof. János Pach - covered material - Lecture 1. Counting problems To read: [Lov]: 1.. Sets, 1.3. Number of subsets, 1.5. Sequences, 1.6. Permutations, 1.7.
On Certain Sums of Stirling Numbers with Binomial Coefficients
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 18 (2015, Article 15.9.6 On Certain Sums of Stirling Numbers with Binomial Coefficients H. W. Gould Department of Mathematics West Virginia University
Some Catalan Musings p. 1. Some Catalan Musings. Richard P. Stanley
Some Catalan Musings p. 1 Some Catalan Musings Richard P. Stanley Some Catalan Musings p. 2 An OEIS entry A000108: 1,1,2,5,14,42,132,429,... Some Catalan Musings p. 2 An OEIS entry A000108: 1,1,2,5,14,42,132,429,...
A LINEAR BINOMIAL RECURRENCE AND THE BELL NUMBERS AND POLYNOMIALS
Applicable Analysis and Discrete Mathematics, 1 (27, 371 385. Available electronically at http://pefmath.etf.bg.ac.yu A LINEAR BINOMIAL RECURRENCE AND THE BELL NUMBERS AND POLYNOMIALS H. W. Gould, Jocelyn
Quiz 3 Reminder and Midterm Results
Quiz 3 Reminder and Midterm Results Reminder: Quiz 3 will be in the first 15 minutes of Monday s class. You can use any resources you have during the quiz. It covers all four sections of Unit 3. It has
Some Congruences for the Partial Bell Polynomials
3 47 6 3 Journal of Integer Seuences, Vol. 009), Article 09.4. Some Congruences for the Partial Bell Polynomials Miloud Mihoubi University of Science and Technology Houari Boumediene Faculty of Mathematics
Thesis submitted in partial fulfillment of the requirement for The award of the degree of. Masters of Science in Mathematics and Computing
SOME n-color COMPOSITION Thesis submitted in partial fulfillment of the requirement for The award of the degree of Masters of Science in Mathematics and Computing Submitted by Shelja Ratta Roll no- 301203014
Algorithmic Approach to Counting of Certain Types m-ary Partitions
Algorithmic Approach to Counting of Certain Types m-ary Partitions Valentin P. Bakoev Abstract Partitions of integers of the type m n as a sum of powers of m (the so called m-ary partitions) and their
Derangements with an additional restriction (and zigzags)
Derangements with an additional restriction (and zigzags) István Mező Nanjing University of Information Science and Technology 2017. 05. 27. This talk is about a class of generalized derangement numbers,
Investigating Geometric and Exponential Polynomials with Euler-Seidel Matrices
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 14 (2011), Article 11.4.6 Investigating Geometric and Exponential Polynomials with Euler-Seidel Matrices Ayhan Dil and Veli Kurt Department of Mathematics
MULTI-RESTRAINED STIRLING NUMBERS
MULTI-RESTRAINED STIRLING NUMBERS JI YOUNG CHOI DEPARTMENT OF MATHEMATICS SHIPPENSBURG UNIVERSITY SHIPPENSBURG, PA 17257, U.S.A. Abstract. Given positive integers n, k, and m, the (n, k)-th m- restrained
A MATRIX GENERALIZATION OF A THEOREM OF FINE
A MATRIX GENERALIZATION OF A THEOREM OF FINE ERIC ROWLAND To Jeff Shallit on his 60th birthday! Abstract. In 1947 Nathan Fine gave a beautiful product for the number of binomial coefficients ( n, for m
Taylor Expansions with Finite Radius of Convergence Work of Robert Jentzsch, 1917
Polynomial Families with Interesting Zeros Robert Boyer Drexel University April 30, 2010 Taylor Expansions with Finite Radius of Convergence Work of Robert Jentzsch, 1917 Partial Sums of Geometric Series
Solutions to Exercises Chapter 5: The Principle of Inclusion and Exclusion
Solutions to Exercises Chapter 5: The Principle of Inclusion and Exclusion An opinion poll reports that the percentage of voters who would be satisfied with each of three candidates A, B, C for President
2030 LECTURES. R. Craigen. Inclusion/Exclusion and Relations
2030 LECTURES R. Craigen Inclusion/Exclusion and Relations The Principle of Inclusion-Exclusion 7 ROS enumerates the union of disjoint sets. What if sets overlap? Some 17 out of 30 students in a class
Math Circle: Recursion and Induction
Math Circle: Recursion and Induction Prof. Wickerhauser 1 Recursion What can we compute, using only simple formulas and rules that everyone can understand? 1. Let us use N to denote the set of counting
Summing Series: Solutions
Sequences and Series Misha Lavrov Summing Series: Solutions Western PA ARML Practice December, 0 Switching the order of summation Prove useful identity 9 x x x x x x Riemann s zeta function ζ is defined
Sequences that satisfy a(n a(n)) = 0
Sequences that satisfy a(n a(n)) = 0 Nate Kube Frank Ruskey October 13, 2005 Abstract We explore the properties of some sequences for which a(n a(n)) = 0. Under the natural restriction that a(n) < n the
Random partition of a set 2/28/01
Random partition of a set 220 A partition of the set i is a collection of disjoint nonempty sets whose union equals i. e.g. The & partitions of the set i ß#ß$ are ß #ß $ ß # ß $ ß $ ß # #ß$ ß ß # ß $ We
Algorithms: Background
Algorithms: Background Amotz Bar-Noy CUNY Amotz Bar-Noy (CUNY) Algorithms: Background 1 / 66 What is a Proof? Definition I: The cogency of evidence that compels acceptance by the mind of a truth or a fact.
Review problems solutions
Review problems solutions Math 3152 December 15, 2017 1. Use the binomial theorem to prove that, for all n 1 and k satisfying 0 k n, ( )( ) { n i ( 1) i k 1 if k n; i k 0 otherwise. ik Solution: Using
d-regular SET PARTITIONS AND ROOK PLACEMENTS
Séminaire Lotharingien de Combinatoire 62 (2009), Article B62a d-egula SET PATITIONS AND OOK PLACEMENTS ANISSE KASAOUI Université de Lyon; Université Claude Bernard Lyon 1 Institut Camille Jordan CNS UM
Basic Discrete Mathematics, Spring 2015
Basic Discrete Mathematics, Spring 2015 Department of Mathematics, University of Notre Dame Instructor: David Galvin Abstract This document includes homeworks, quizzes and exams from the Spring 2015 incarnation
Ex. Here's another one. We want to prove that the sum of the cubes of the first n natural numbers is. n = n 2 (n+1) 2 /4.
Lecture One type of mathematical proof that goes everywhere is mathematical induction (tb 147). Induction is essentially used to show something is true for all iterations, i, of a sequence, where i N.
Chapter 5: Integer Compositions and Partitions and Set Partitions
Chapter 5: Integer Compositions and Partitions and Set Partitions Prof. Tesler Math 184A Winter 2017 Prof. Tesler Ch. 5: Compositions and Partitions Math 184A / Winter 2017 1 / 32 5.1. Compositions A strict
Basic Proof Examples
Basic Proof Examples Lisa Oberbroeckling Loyola University Maryland Fall 2015 Note. In this document, we use the symbol as the negation symbol. Thus p means not p. There are four basic proof techniques
MATH39001 Generating functions. 1 Ordinary power series generating functions
MATH3900 Generating functions The reference for this part of the course is generatingfunctionology by Herbert Wilf. The 2nd edition is downloadable free from http://www.math.upenn. edu/~wilf/downldgf.html,
Catalan numbers Wonders of Science CESCI, Madurai, August
Catalan numbers Wonders of Science CESCI, Madurai, August 25 2009 V.S. Sunder Institute of Mathematical Sciences Chennai, India sunder@imsc.res.in August 25, 2009 Enumerative combinatorics Enumerative
Parity Theorems for Statistics on Domino Arrangements
Parity Theorems for Statistics on Domino Arrangements Mar A Shattuc Mathematics Department University of Tennessee Knoxville, TN 37996-1300 shattuc@mathutedu Carl G Wagner Mathematics Department University
Convex Domino Towers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 20 (2017, Article 17.3.1 Convex Domino Towers Tricia Muldoon Brown Department of Mathematics Armstrong State University 11935 Abercorn Street Savannah,
REVIEW QUESTIONS. Chapter 1: Foundations: Sets, Logic, and Algorithms
REVIEW QUESTIONS Chapter 1: Foundations: Sets, Logic, and Algorithms 1. Why can t a Venn diagram be used to prove a statement about sets? 2. Suppose S is a set with n elements. Explain why the power set
The Inclusion Exclusion Principle and Its More General Version
The Inclusion Exclusion Principle and Its More General Version Stewart Weiss June 28, 2009 1 Introduction The Inclusion-Exclusion Principle is typically seen in the context of combinatorics or probability
THE LOG-BEHAVIOR OF THE SEQUENCE FOR THE PARTIAL SUM OF A LOG-CONVEX SEQUENCE. 1. Introduction
SARAJEVO JOURNAL OF MATHEMATICS Vol.13 (26), No.2, (2017), 163 178 DOI: 10.5644/SJM.13.2.04 THE LOG-BEHAVIOR OF THE SEQUENCE FOR THE PARTIAL SUM OF A LOG-CONVEX SEQUENCE FENG-ZHEN ZHAO Abstract. In this
Chapter 5: Integer Compositions and Partitions and Set Partitions
Chapter 5: Integer Compositions and Partitions and Set Partitions Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 5: Compositions and Partitions Math 184A / Fall 2017 1 / 46 5.1. Compositions A strict
Example 1. The sample space of an experiment where we flip a pair of coins is denoted by:
Chapter 8 Probability 8. Preliminaries Definition (Sample Space). A Sample Space, Ω, is the set of all possible outcomes of an experiment. Such a sample space is considered discrete if Ω has finite cardinality.
Counting Palindromic Binary Strings Without r-runs of Ones
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 16 (013), Article 13.8.7 Counting Palindromic Binary Strings Without r-runs of Ones M. A. Nyblom School of Mathematics and Geospatial Science RMIT University
5. Sequences & Recursion
5. Sequences & Recursion Terence Sim 1 / 42 A mathematician, like a painter or poet, is a maker of patterns. Reading Sections 5.1 5.4, 5.6 5.8 of Epp. Section 2.10 of Campbell. Godfrey Harold Hardy, 1877
Approaches differ: Catalan numbers
ISSN: 2455-4227 Impact Factor: RJIF 5.12 www.allsciencejournal.com Volume 2; Issue 6; November 2017; Page No. 82-89 Approaches differ: Catalan numbers 1 Mihir B Trivedi, 2 Dr. Pradeep J Jha 1 Research
Construction of latin squares of prime order
Construction of latin squares of prime order Theorem. If p is prime, then there exist p 1 MOLS of order p. Construction: The elements in the latin square will be the elements of Z p, the integers modulo
3.9 My Irrational and Imaginary Friends A Solidify Understanding Task
3.9 My Irrational and Imaginary Friends A Solidify Understanding Task Part 1: Irrational numbers Find the perimeter of each of the following figures. Express your answer as simply as possible. 2013 www.flickr.com/photos/lel4nd
GENERATING FUNCTIONS OF CENTRAL VALUES IN GENERALIZED PASCAL TRIANGLES. CLAUDIA SMITH and VERNER E. HOGGATT, JR.
GENERATING FUNCTIONS OF CENTRAL VALUES CLAUDIA SMITH and VERNER E. HOGGATT, JR. San Jose State University, San Jose, CA 95. INTRODUCTION In this paper we shall examine the generating functions of the central
Polynomial analogues of Ramanujan congruences for Han s hooklength formula
Polynomial analogues of Ramanujan congruences for Han s hooklength formula William J. Keith CELC, University of Lisbon Email: william.keith@gmail.com Detailed arxiv preprint: 1109.1236 Context Partition
Advanced Counting Techniques. Chapter 8
Advanced Counting Techniques Chapter 8 Chapter Summary Applications of Recurrence Relations Solving Linear Recurrence Relations Homogeneous Recurrence Relations Nonhomogeneous Recurrence Relations Divide-and-Conquer
Unit 5: Sequences, Series, and Patterns
Unit 5: Sequences, Series, and Patterns Section 1: Sequences and Series 1. Sequence: an ordered list of numerical terms 2. Finite Sequence: has a first term (a beginning) and a last term (an end) 3. Infinite
Universal Juggling Cycles
Universal Juggling Cycles Fan Chung y Ron Graham z Abstract During the past several decades, it has become popular among jugglers (and juggling mathematicians) to represent certain periodic juggling patterns
Notes on Paths, Trees and Lagrange Inversion
Notes on Paths, Trees and Lagrange Inversion Today we are going to start with a problem that may seem somewhat unmotivated, and solve it in two ways. From there, we will proceed to discuss applications
2 Generating Functions
2 Generating Functions In this part of the course, we re going to introduce algebraic methods for counting and proving combinatorial identities. This is often greatly advantageous over the method of finding
Exercises for Chapter 1
Solution Manual for A First Course in Abstract Algebra, with Applications Third Edition by Joseph J. Rotman Exercises for Chapter. True or false with reasons. (i There is a largest integer in every nonempty
Diploma Programme. Mathematics HL guide. First examinations 2014
Diploma Programme First examinations 2014 Syllabus 17 Topic 1 Core: Algebra The aim of this topic is to introduce students to some basic algebraic concepts and applications. 1.1 Arithmetic sequences and
Math 3121, A Summary of Sections 0,1,2,4,5,6,7,8,9
Math 3121, A Summary of Sections 0,1,2,4,5,6,7,8,9 Section 0. Sets and Relations Subset of a set, B A, B A (Definition 0.1). Cartesian product of sets A B ( Defintion 0.4). Relation (Defintion 0.7). Function,
D I S C R E T E M AT H E M AT I C S H O M E W O R K
D E PA R T M E N T O F C O M P U T E R S C I E N C E S C O L L E G E O F E N G I N E E R I N G F L O R I D A T E C H D I S C R E T E M AT H E M AT I C S H O M E W O R K W I L L I A M S H O A F F S P R
Applications. More Counting Problems. Complexity of Algorithms
Recurrences Applications More Counting Problems Complexity of Algorithms Part I Recurrences and Binomial Coefficients Paths in a Triangle P(0, 0) P(1, 0) P(1,1) P(2, 0) P(2,1) P(2, 2) P(3, 0) P(3,1) P(3,