The Twelvefold Way in Combinatorics
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The Twelvefold Way in Combinatorics

Explore the systematic classification of 12 related enumerative problems concerning permutations, combinations, multisets, and partitions. Learn about the idea of the classification credited to Gian-Carlo Rota and the suggested name by Joel Spencer.

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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

What does it mean when a function f is injective?

Each value for a in N must be distinct from every other

What is the condition for a function to be surjective?

For each b in X there must be at least one a in N such that f(a) = b

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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