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Introduction

There are three principal themes to our course: Discrete Structures Graphs, digraphs, networks, designs, posets, strings, patterns, distributions, coverings, and partitions.Enumeration Permutations, combinations…

Introduction

There are three principal themes to our course:

  • Discrete Structures

    Graphs, digraphs, networks, designs, posets, strings, patterns, distributions, coverings, and partitions.

  • Enumeration

    Permutations, combinations, inclusion/exclusion, generating functions, recurrence relations, and Pólya counting.

  • Algorithms and Optimization

    Sorting, eulerian circuits, hamiltonian cycles, planarity testing, graph coloring, spanning trees, shortest paths, network flows, bipartite matchings, and chain partitions.

To illustrate the accessible, concrete nature of combinatorics and to motivate topics that we will study, this preliminary chapter provides a first look at combinatorial problems, choosing examples from enumeration, graph theory, number theory, and optimization. The discussion is very informalbut this should serve to explain why we have to be more precise at later stages. We ask lots of questions, but at this stage, you'll only be able to answer a few. Later, you'll be able to answer many more but as promised earlier, most likely you'll never be able to answer them all. And if we're wrong in making that statement, then you're certain to become very famous. Also, you'll get an A\(++\) in the course and maybe even a Ph.D. too.

አሁን ምንም ዓይነት መሳሪያ ይህን አይቆጣጠርም፤ ነገር ግን ክፍሎቹ ሊቆጠሩ ይችላሉ። በታች ያለውን ይሞክሩ ወይም የራሳችሁን ይጻፉ።

የራስዎን ስራዎች ይያዙ

ነጻ የሆኑት መተግበሪያዎች በየክፍል ውስጥ ማስታወሻዎችን ያካትታሉ፣ የተጠናቀቁትን ነገሮች መዝገብ፣ የተፈቱ ችግሮችን በአንድ ቦታ፣ ስለዚህ ገጽ መጠየቅ የሚችሉትን ረዳት ያካትታሉ፡፡ የቁጥር ትምህርት ለሁሉም ሰው ክፍት ነው፣ ተቀባይነት ያለው ወይም የለም፤

ምዝገባ መግባቱ

ፊደላት

ለሙሉ መግለጫ፣ ምስል፣ እና በውስጡ ያለውን ፊደል ትርጓሜ ለማየት ማንኛውንም ምልክት ይጫኑ።

ጥያቄዎች

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

የዚህ ገጽ ክፍሎች ከ Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). እዚህ የተቀረጸና የተተረጎመ፤ ስህተቶች የእኛ ናቸው፡፡

በ Combinatorics & Graph Theory

The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs