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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?

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

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

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

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

Gian-Carlo Rota Signup and view all the answers

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

12 Signup and view all the answers

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

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

What does counting n-combinations of X correspond to?

Counting all functions N → X up to permutations of N Signup and view all the answers

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

Injective functions N → X up to permutations of N Signup and view all the answers

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

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

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

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

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

Injective function Signup and view all the answers