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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/02%3A_Enumeration/05%3A_Counting_with_RepetitionsIn counting combinations and permutations, we assumed that we were drawing from a set in which all of the elements are distinct. Of course, it is easy to come up with a scenario in which some of the e...In counting combinations and permutations, we assumed that we were drawing from a set in which all of the elements are distinct. Of course, it is easy to come up with a scenario in which some of the elements are indistinguishable. We need to know how to count the solutions to problems like this, also.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/03%3A_Graph_Theory/15%3A_Planar_Graphs/15.01%3A_Planar_GraphsVisually, there is always a risk of confusion when a graph is drawn in such a way that some of its edges cross over each other. Also, in physical realisations of a network, such a configuration can le...Visually, there is always a risk of confusion when a graph is drawn in such a way that some of its edges cross over each other. Also, in physical realisations of a network, such a configuration can lead to extra costs (think about building an overpass as compared with building an intersection). It is therefore helpful to be able to work out whether or not a particular graph can be drawn in such a way that no edges cross.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/04%3A_Design_TheoryThumbnail: The Fano plane is an example of a finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geom...Thumbnail: The Fano plane is an example of a finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study. (Public Domain; Gunther via Wikipedia)
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/02%3A_Enumeration/09%3A_Some_Important_Recursively-Defined_Sequences/9.01%3A_DerangementsA derangement of a list of objects is a permutation of the objects, in which no object is left in its original position. A classic example of this is a situation in which you write letters to ten peo...A derangement of a list of objects is a permutation of the objects, in which no object is left in its original position. A classic example of this is a situation in which you write letters to ten people, address envelopes to each of them, and then put them in the envelopes, but accidentally end up with none of the letters in the correct envelope.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/01%3A_IntroductionIn Combinatorics, we focus on combinations and arrangements of discrete structures. There are five major branches of combinatorics that we will touch on in this course: enumeration, graph theory, Rams...In Combinatorics, we focus on combinations and arrangements of discrete structures. There are five major branches of combinatorics that we will touch on in this course: enumeration, graph theory, Ramsey Theory, design theory, and coding theory. (The related topic of cryptography can also be studied in combinatorics, but we will not touch on it in this course.) We will focus on enumeration, graph theory, and design theory, but will briefly introduce the other two topics.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/04%3A_Design_Theory/19%3A_Designs_and_Codes/19.02%3A_Error-Correcting_CodesIn order to be able to correct errors in transmission, we agree to send only strings that are in a certain set C of codewords. (So the information we wish to send will need to be “encoded” as one of...In order to be able to correct errors in transmission, we agree to send only strings that are in a certain set C of codewords. (So the information we wish to send will need to be “encoded” as one of the codewords.) The set C is called a code. Choosing the code cleverly will enable us to successfully correct transmission errors. When a transmission is received, the recipient will assume that the sender transmitted the codeword that is “closest” to the string that was received.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/03%3A_Graph_Theory/12%3A_Moving_Through_Graphs/12.01%3A_Directed_GraphsSome networks include connections that only allow travel in one direction. These can be modeled using directed graphs. When drawing a digraph, we draw an arrow on each arc so that it points from the f...Some networks include connections that only allow travel in one direction. These can be modeled using directed graphs. When drawing a digraph, we draw an arrow on each arc so that it points from the first vertex of the ordered pair to the second vertex. Like multigraphs, we will not study digraphs in this course, but you should be aware of the basic definition. Many of the results we will cover in this course, generalise to the context of digraphs.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/03%3A_Graph_Theory/14%3A_Graph_Coloring/14.01%3A_Edge_ColoringSuppose you have been given the job of scheduling a round-robin tennis tournament with n players. One way to approach the problem is to model it as a graph: the vertices of the graph represent the pla...Suppose you have been given the job of scheduling a round-robin tennis tournament with n players. One way to approach the problem is to model it as a graph: the vertices of the graph represent the players and the edges represent the matches that need to be played. Since it is a round-robin tournament, every player must play every other player so the graph will be complete. Creating the schedule amounts to assigning a time to each of the edges, representing the time that the match will be played.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/01%3A_Introduction/01%3A_What_is_Combinatorics/1.05%3A_Coding_TheoryCoding theory is the study of encoding information into different symbols. When someone uses a code in an attempt to make a message that only certain other people can read, this becomes cryptography. ...Coding theory is the study of encoding information into different symbols. When someone uses a code in an attempt to make a message that only certain other people can read, this becomes cryptography. In coding theory, we ignore the question of who has access to the code and how secret it may be. Instead, one of the primary concerns becomes our ability to detect and correct errors in the code.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/zz%3A_Back_Matter/21%3A_List_of_NotationThis page contains the list of notation used throughout this textbook.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/02%3A_Enumeration/09%3A_Some_Important_Recursively-Defined_Sequences/9.04%3A_SummaryThis page contains the summary of the topics covered in Chapter 9.
