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Omni-MATH / Let S be the set …h from S to itself such that there exists an a∈S such that h(a)≠a, and that for any pair of integers a∈S and b∈S such that h(a)≠a,h(b)≠b, there exists a positive …

Problem

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problem

Let SS be the set \{1,2, \ldots, 2012\}. A perfectutation is a bijective function hh from SS to itself such that there exists an a∈Sa \in S such that h(a)≠ah(a) \neq a, and that for any pair of integers a∈Sa \in S and b∈Sb \in S such that h(a)≠a,h(b)≠bh(a) \neq a, h(b) \neq b, there exists a positive integer kk such that hk(a)=bh^{k}(a)=b. Let nn be the number of ordered pairs of perfectutations (f,g)(f, g) such that f(g(i))=g(f(i))f(g(i))=g(f(i)) for all i∈Si \in S, but f≠gf \neq g. Find the remainder when nn is divided by 2011 .
Plain-text mathematical notation (without MathML)
Let S be the set \{1,2, \ldots, 2012\}. A perfectutation is a bijective function h from S to itself such that there exists an a∈S such that h(a)≠a, and that for any pair of integers a∈S and b∈S such that h(a)≠a,h(b)≠b, there exists a positive integer k such that h^(k)(a)=b. Let n be the number of ordered pairs of perfectutations (f,g) such that f(g(i))=g(f(i)) for all i∈S, but f≠g. Find the remainder when n is divided by 2011 .
Original LaTeX notation
Let $S$ be the set \{1,2, \ldots, 2012\}. A perfectutation is a bijective function $h$ from $S$ to itself such that there exists an $a \in S$ such that $h(a) \neq a$, and that for any pair of integers $a \in S$ and $b \in S$ such that $h(a) \neq a, h(b) \neq b$, there exists a positive integer $k$ such that $h^{k}(a)=b$. Let $n$ be the number of ordered pairs of perfectutations $(f, g)$ such that $f(g(i))=g(f(i))$ for all $i \in S$, but $f \neq g$. Find the remainder when $n$ is divided by 2011 .

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Source and history

task_id
a616c54d-3ce3-5844-b8d7-d7273db3a94e
task_key
test--a616c54d-3ce3-5844-b8d7-d7273db3a94e
task_revision_id
1
config / split
/ test

Official source

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