13 Questions
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?
When there are equal cardinalities of two finite sets, n = x
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
Which mathematician is credited with the idea of the classification known as the twelvefold way?
Gian-Carlo Rota
In how many ways can the three conditions on the functions and the four equivalence relations be paired?
12
Which of the following is equivalent to counting n-permutations of X?
Counting injective functions N → X up to permutations of N
What does counting n-combinations of X correspond to?
Counting all functions N → X up to permutations of N
When n = x, counting permutations of the set X is equivalent to counting:
Injective functions N → X up to permutations of N
What is equivalent to counting partitions of the set N into x subsets?
Counting all surjective functions N → X up to permutations of X
What does counting compositions of the number n into x parts correspond to?
Counting all surjective functions N → X up to permutations of N
What is reflected by the property that any ball can go into only one box?
Injective function
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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