# Combinatorics of Compositions and Words (Discrete Mathematics and Its Applications) What will be asked in the quizzes? Will it be surprise or announced? Quizzes will test your understanding of the concepts, it will be of short duration and straightforward questions. Anyone who has sat through the lectures can easily answer them. There will be 1 to 2 quizzes every week each of 5 to 10 mins duration. More details about the quiz will be shared on the first day of the class. The marking will not be boolean, you will get marks for partial attempts as described in the scheme of valuation.

I love programming! We feel the course is already heavy for 4 credits, including a programming component will tax the student a bit too much. We will be giving you interesting programming questions and exploratory assignments to try by yourself, these will not be graded. How are the grade letters assigned in this course? It has been our sincere observation from our experience so far that it is tough to decide the grading mechanism a priori and do justice to students.

The instructor will decide on the grade letter breakups after observing the final performance of the students. The grading will be fair, in the sense that, it will distinguish students on the basis of their understanding and efforts. Chapter 1 Grimaldi 2 5 Jan This lecture covered two problems: finding the number of binary search trees and the number of valid parentheses. Catalan numbers were used to solve them. Chapter 1 Grimaldi 3 6 Jan The concept of Catalan numbers was discussed in detail.

Additionally, the concept of Pigeonhole principle was discussed and used in understanding Erdos-Szekeres sequence.

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## Combinatorics - Wikipedia

Chapter 1 Grimaldi and class notes 4 6 Jan This lecture covered some problems covering basic topics like sum rule, product rule, permutations, combinations and binomial theorem along with combinations with repetitions. Chapter 1 Grimaldi 5 11 Jan In this lecture, we revisited the Erdos-Szekeres sequence and saw the complete proof. We also saw how the claim does not hold for a smaller length integer sequence. We also took a look at Binomial theorem and supplied a combinatorial proof for the expansion. We saw that the proof can be easily extended to a multinomial expansion.

Chapter 1 Grimaldi and class notes 6 12 Jan We looked at the different instances where we would come across n C 2 and saw quite a few seemingly unrelated instances. We also observed how the same question when viewed from two differnt perspectives can have different answers and still have both the answers correct. This was the case with the ice-cream parlour problem. We supplied the proof of correctness for both the answers. We proved this by establishing a bijection to a seemingly easier set of objects and gave a count of the number of objects in this new set.

Chapter 1 Grimaldi 7 13 Jan The main focus of this lecture was to understand mathematical induction. It was left as an exercise to think about the induction that goes in the proof of the maximum contigious subarray problem. Chapter 4 Grimaldi 8 18 Jan We formalized the notion of mathematical induction, defined the statement and took a look at its proof. In its proof we defined and saw the use of the principle of well ordering. We discussed on the non-trivial examples where we encounter mathenatical induction: the monks; Konigsberg bridge; maximum contigious subarray; etc.

We briefly discussed another problem of showing that every integer greater than equal to 14 can be written as the sum of 8s and 3s. We then began with the chapter on functions and relations. We defined a function, one-one function, and onto functions.

### Combinatorics

Chapter 4 Grimaldi 9 19 Jan We had a few warm up exercises of different functions that qualify as either one-one or onto, or both or neither. We saw how establishing one-one and onto nature of functions help in making counting arguments. In this regard, we saw the example of Catalan numbers. Chapter 5 Grimaldi 10 20 Jan We solved a few problems that involved counting the different kind of functions given a finite domain and co-domain. Chapter 5 Grimaldi 11 20 Jan We continued with the questions asked in the previous class. We saw a general form to count the number of onto functions.

We proved a recurrence to find out the stirling's number and ended answering the question of finding the total number of ways one can factor the number into its factors. Chapter 5 Grimaldi 12 24 Jan Is it indeed possible to compress data and be able to retrieve the data? That is, is compression even possible, in the literal sense? This is the question we began with and saw how one can use Pigeonhole principle to state the answer. We saw a variety of problems whose proof involves the pigeonhole principle.

Chapter 5 Grimaldi 13 25 Jan We were introduced to the concept of extended Pigeonhole principle and we looked at few interesting problems that used this concept.

The list of problems that were discussed in the class is included in the tutorial section. Chapter 5 Grimaldi 14 27 Jan We discussed the solutions to the second half of the problems given in the assignment 1. Assignment 1 15 27 Jan We discussed the solutions to the remaining questions given in the assignment 1. Assignment 1 16 1 Feb In today's class we observed the need for having composition of functions.

As an example, we took a look at Collatz conjecture and posed a couple of problems on the same. We also stated that the abstraction is important, although it may not have any direct impact, functions are an important concept used in computer science. With this, we began questioning the nature of functions and asked if two functions are one-one, is their composition also one-one? Similarly, if two functions are known to be onto, what can we say about their composition? Chapter 5 Grimaldi 17 2 Feb We revisited the Catalan numbers and saw that along with a closed form, the nth Catalan number can also be given in the form of a recurrence relation.

We established the recurrence relation through the example of the balanced paranthesis problem. We also solved the polygon triangulation problem using this recurrence. Chapter 1 Class notes 18 3 Feb Test 1 - 19 3 Feb We took a look at function compositions in detail and showed how inverse of a function, if it exists, is always unique.

We also took a look at the similarities in the behaviour of set of all bijective functions and integers. We showed that properties such as associativity, identity, inverses holds in both spaces. Chapter 5 Grimaldi 20 8 Feb We began by defining the notion of a relation. We saw that we can classify the differnt types of relations conditioned on whether the defined relation respects some properties. In this regard, we saw when we define a relation to be reflexive, symmetric and transitive.

Through various examples, we saw how a given relation can exhibit all possible combinations of thes properties. We ended the class making the observation that when a relation is reflexive, symmetric and transitive at the same time, the set on which the relation is defined can be partitioned into equivalence classes. Chapter 7 Grimaldi 21 9 Feb We revised the concept of reflexive, transitive and symmetric relations through examples.

We also counted the number of relexive relations, symmetric relations on a given set. As per the request of students, we took a look at a special kind of relation - one that is not reflexive but is symmetric and transitive. Despite the fact that, at first it seems impossible to construct such a relation, we showed the existence of many such relations.

Through these examples, we were better able to appreciate the definition of symmetric and transitive relations. Chapter 7 Grimaldi 22 10 Feb Revisiting the notion of function composition, we questioned the composition of relations. Noting that a relation is nothing but a subset of the cross products of sets, we saw that composition of relations also defines another relation.

We saw how we can capture a relation in a matrix and saw how matrix multiplication translates to composition of relations. Chapter 7 Grimaldi 23 10 Feb We observed the operation of matrices in order to capture composition of relations. We also saw why this operation is true. We also questioned about the properties of a matrix that represents a reflexive relation, transitive relation. We also defined what an antisymmetric relation is and how the matrix of such a relation would look like.

Chapter 7 Grimaldi 24 15 Feb We watched a video that discussed the notion of a Hasse diagram. We discussed the concept of a sink and a source and observed that a Poset always has a sink and a source. We learnt how to represent a poset using a Hasse diagram. We saw the importance to have different ways to represent a poset. We briefly saw the need for a technique like that of Topological sorting. Chapter 7 Grimaldi 25 16 Feb We had previously encountered Equivalence relations. Today we took a look at the detailed proof that shows that every equivalence relation partitions the set into disjoint equivalence classes and vice versa as well.

Chapter 7 Grimaldi 26 17 Feb We began learning the concepts of a new chapter - Graph Theory. We undersood what a graph is and its constituents of vertces and edges. We also took a look at what one means by the compliment of a graph. We saw the notion of connected versus disconnected graphs. We asked whether the complement of a disconnected graph is always disconnected and vice versa.

## Combinatorics

Chapter 11 Grimaldi 27 17 Feb We briefly discuused the different notions of a graph such as a walk, cycle, trail, path, circuit, etc. Then we were introduced to the notion of an Eulerian circuit and we took a look at the proof for the presence of an Eulerian circuit in a graph. Chapter 11 Grimaldi 28 22 Feb In today's class we saw the concept of Bell numbers and the notion of graph isomorphism.

We also completed the proof for the condition for the relation matrix M R to hold in order for the relation R to be transitive. Chapter 7,10,11 Grimaldi 29 23 Feb We formally defined the notion of graph isomorphism and solved the puzzle of determining if two graphs are isomorphic or not, which was given in the previous class. We majorly solve the first 14 questions of Assignment 2. Assignment 2,3 31 24 Feb We solved more exercise problems and quickly took a recap of all the topics that have been covered so far and the portions for minor exams.

Assignment 3 32 9 Mar We formally took a look at the Konigsberg problem and showed that there cannot be an eulerian circuit on it. We saw this with the example of a zoo problem and proved it using induction. The new concepts learnt was that of wheel graphs, subgraphs, induced subgraphs, spanning subgraphs.

Tutorial 5 34 10 Mar - Today we discussed planarity of graphs and the condition required for a graph to be planar. We discussed the non-planarity of K5, K 3,3 and the Peterson's graph. Chapter 11 Grimaldi 35 15 Mar - We studied a special kind of graph known as a tournament grpahs and we saw that proof for the fact that there is always a Hamilton path on such a graph.

Additionally, we saw that any graph on n vertices that satisfies the property that sum of the degree of any two vertices is greater than n-1 is alwaus connected. Chapter 11 Grimaldi 36 16 Mar - Borrowing ideas from the previous class, we completed the proof for the existence of a Hamiltonian path on any graph on n vertices that satisfies the property that sum of the degree of any two vertices is greater than n Chapter 11 Grimaldi 37 17 Mar - This class focused on introducing the concept of graph colouring chromatic number of a graph. We took a look at the chromatic number of a complete graph and that of a cycle. We also took a look at the derivation of the chromatic polynomial. Chapter 11 Grimaldi 38 17 Mar - This was a tutorial session where we took a look at some more problems on hamiltonian paths, minimum cut of a graph, etc. Tutorial 6 39 29 Mar - We discussed the first three units from the text. We discussed the idea of truth tables, tautology, contradiction and satisfiability. We also saw the proofs for well known statements such as "primes are infinite" and "square root of 2 is irrational". We observed the application of the laws of inference in the proofs.

We ended the class by solving a few problems on inference rules.

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Chapter 2 40 30 Mar - Today we discussed dual of Boolean expression, absorption law, De Morgan's law, proof by contradiction, proof by contraposition and practised some examples. Chapter 2 41 31 Mar - We discussed qunatifiers, namely 'forall' and 'there exists'. We observed the way to create negation of a given expression. We noted that the definition of convergence of a sequence can be captured using quantifiers and thus one can obtain the negation of the statement. Chapter 2 42 31 Mar - Class Committee Meeting - 43 5 April - We began with a motivating example of counting the number of ways in which every letter is put inside a wrong envelope.

With this in mind we took a look at the Principle of Inclusion and Exclusion. We also saw the proof of the same. As another application of the principle of inclusion and exclusion, we counted the number of possible graphs with no isolated vertex.

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Chapter 8 44 6 April - The main focus of today's class was to count the number of ways in which we can place rooks on a chess board so that none of them take each other. In this context, we studies the rook polynomial that captures the possible ways of achieving this. We further saw the benefit of representing it as a polynomial, when we can break the given chess board configuration into smaller independent subproblems. We ended the class by observing how one can recursively construct the rook polynomial of any chess board configuration.

Chapter 8 45 7 April - In this class, we discussed an interesting problem of finding the number of one-to-one functions from a set A to set B, given certain constraints such as some of the elements of the set A can not be associated with some of the elements of the set B. Electronic health care records are kept as parts of databases, and there is a lot of discrete mathematics involved in the efficient and effective design of databases. Compact discs store a lot of data, which is encoded using a modified Reed-Solomon code a binary code, and thus discrete math to automatically correct transmission errors.

Voting systems : There are different methods for voting not just the common cast-a-ballot-for-exactly-one-candidate method. The study of possible voting methods and how well their outcomes reflect the intent of the voters uses discrete mathematics.

Cell phone communications : Making efficient use of the broadcast spectrum for mobile phones uses linear algebra and information theory. Assigning frequencies so that there is no interference with nearby phones can use graph theory or can use discrete optimization. Digital image processing uses discrete mathematics to merge images or apply filters. Methods of encoding data and reducing the error in data transmission such as are used in bar codes , UPC s, data matrices , and QR codes are discrete mathematics.

Hidden Markov models, which are part of linear algebra, are used for large vocabulary continuous speech recognition. Food Webs : A food web describes the ways in which a set of species eat and don't eat each other. They can be studied using graph theory. Delivery Route Problems: If you need to leave home, visit a sequence of locations each exactly once and then return homesuch as might happen with a newspaper delivery route or scheduling bread to be delivered from a bakery to grocery stores this is known as the traveling salesperson problem , or TSP. There is a definitive source on the history of, and state-of-the-art work on TSP.

Spatio-temporal optimization is a type of algorithm design that has been applied to reducing poaching of endangered animals. Logistics deals with managing inventory and supply chains , as well as transporting goods and people to where and when they are needed. Many of the problems involved use discrete optimization. Graph theory is used in cybersecurity to identify hacked or criminal servers and generally for network security.

Discrete math is used in choosing the most on-time route for a given train trip in the UK. The software determines the probability of a given train trip being completed on time in the UK uses Markov chains. Archaeology uses discrete math to construct 3D images from scans of archaeological sites.

Determining voting districts, a process known as redistricting , is rife with problems and influenced by politics. Many researchers in various fields work on methods for fair redistricting, and some use lots of discrete math. Network flows, a part of discrete math, can be used to help protect endangered species from the threat of global warming see the abstract for this paper. Power grids : Graph theory is used in finding the most vulnerable aspects of an electric grid. Graph theory and linear algebra are used in power grid simulations.

Robot arms are a type of linkage , the study of which is part of discrete geometry. Modeling possible fingertip movements and forces uses linear algebra. Graph theory is involved in routing concrete trucks. Voting theory see earlier on this page can be used to decide how to prioritize among biodiversity conservation sites see the abstract for this paper.

## Combinatorics of Compositions and Words

Graph theory is used in kidney donor matching bonus: the speaker on the podcast has given Daily Gathers at MathILy. Determining how best to add streets to congested areas of cities uses graph theory and in fact an area of graph theory taught in one of the MathILy branch classes! Matching medical-school graduates to hospital residencies is solved using an algorithm that is provably optimal. Here are two articles that describe the discrete mathematics involved and what happens when this is extended to the problem of matching middle-school students to high schools.

Measuring the evolutionary distance between genomes can be done using permutations, as described here. Graph theory is involved in searching for terrorist groups sending covert messages on public fora. Data compression , reduction of noise in data , and automated recommendations of movies all use the same tool from linear algebra. The spread of infectious disease is affected by personal contacts and by behaviors influenced by information.

One model of epidemics uses graph theory by encoding personal contacts and behaviors as layers in a large network. We can model a crystal structure based on a set of electron microscope images using discrete tomography. Linear programming can be used in discrete tomography. Discrete tomography can also be used in medical imaging , to reconstruct an image of an organ from just a few x-ray images. Graph theory and linear algebra can be used in speeding up Facebook performance.

Assessing risk in heart-attack patients , categorizing species using few characteristics , and data mining analytics all use the same discrete math. Chemistry : Balancing chemical equations uses linear algebra, and understanding molecular structure uses graph theory. We can straighten an image taken by a misaligned camera using linear algebra. Many ways of producing rankings use both linear algebra and graph theory. Combinatorics of Compositions and Words (Discrete Mathematics and Its Applications) Combinatorics of Compositions and Words (Discrete Mathematics and Its Applications) Combinatorics of Compositions and Words (Discrete Mathematics and Its Applications) Combinatorics of Compositions and Words (Discrete Mathematics and Its Applications) Combinatorics of Compositions and Words (Discrete Mathematics and Its Applications)

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