The Twelvefold Way in Combinatorics
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The Twelvefold Way in Combinatorics

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Questions and Answers

In the twelvefold way in combinatorics, what is the general problem that is considered?

  • Enumeration of equivalence classes of functions (correct)
  • Counting permutations and combinations
  • Enumeration of subsets
  • Classification of finite sets

    • What does it mean when a function f is injective?

  • Each b in X may occur multiple times in the image of f
  • Each a in N may be sent by f to any b in X
  • Each value for a in N must be distinct from every other (correct)
  • There must be at least one a in N such that f(a) = b for each b in X

    • What is the condition for a function to be surjective?

  • Each a in N may be sent by f to any b in X
  • For each b in X there must be at least one a in N such that f(a) = b (correct)
  • Each b in X may occur multiple times in the image of f
  • Each value for a in N must be distinct from every other

    • When is 'f is bijective' considered as an option?

    <p>When there are equal cardinalities of two finite sets, n = x</p> Signup and view all the answers

    What does 'No condition' imply for the function f?

    <p>Each value for a in N may be sent by f to any b in X</p> Signup and view all the answers

    Which mathematician is credited with the idea of the classification known as the twelvefold way?

    <p>Gian-Carlo Rota</p> Signup and view all the answers

    In how many ways can the three conditions on the functions and the four equivalence relations be paired?

    <p>12</p> Signup and view all the answers

    Which of the following is equivalent to counting n-permutations of X?

    <p>Counting injective functions N → X up to permutations of N</p> Signup and view all the answers

    What does counting n-combinations of X correspond to?

    <p>Counting all functions N → X up to permutations of N</p> Signup and view all the answers

    When n = x, counting permutations of the set X is equivalent to counting:

    <p>Injective functions N → X up to permutations of N</p> Signup and view all the answers

    What is equivalent to counting partitions of the set N into x subsets?

    <p>Counting all surjective functions N → X up to permutations of X</p> Signup and view all the answers

    What does counting compositions of the number n into x parts correspond to?

    <p>Counting all surjective functions N → X up to permutations of N</p> Signup and view all the answers

    What is reflected by the property that any ball can go into only one box?

    <p>Injective function</p> Signup and view all the answers

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