Intereting Posts

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A and B disjoint, A compact, and B closed implies there is positive distance between both sets
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Intuition or figure for Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$ (Abbott p 11 q1.2.5)
How does one show that $\int_{0}^{\pi/4}\sin(2x)\ln{(\ln^2{\cot{x}})}\mathrm dx=\ln{\pi\over4}-\gamma?$
Compute $P(X>40\; |\; X>10)$ where $X$ has an exponential distribution
How does one know that a theorem is strong enough to publish?
Is there a first-order-logic for calculus?
Cardinal equality: $\;\left|\{0,1\}^{\Bbb N}\right|=\left|\{0,1,2,3\}^{\Bbb N}\right|$
Picard group and cohomology
Do there exist equations that cannot be solved in $\mathbb{C}$, but can be solved in $\mathbb{H}$?
The derivative of a function of a variable with respect to a function of the same variable
A maximal ideal among those avoiding a multiplicative set is prime
MVT for integrals: strict inequality not needed before applying IVT?
Proving Caratheodory measurability if and only if the measure of a set summed with the measure of its complement is the measure of the whole space.

Suppose I fix $n$ and let $\sigma_k$ represent the $k$th permutation of $S_n$ with respect to some ordering (whatever ordering might serve my purpose). Is there an elementary formula for $\sigma_k(i)$ which requires only $i, k,$ and $n$?

Is one known for small $n$, perhaps even as small as 4?

- Xmas Combinatorics 2014
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- sum of all the numbers that can be formed using the digits 2,3,3,4,4,4..

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- Proof that no permutation can be expressed both as the product of an even number of transpositions and as a product of an odd number of transpositions
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- Combinatoric in binary sequence
- Symmetric Group $S_n$ is complete
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- Endomorphism rings and torsion subgroups.
- Solvability of a group with order $p^n$

It depends upon what you mean by elementary. If the ordering is lexicographic, the first element of the $k^{\text(th)}$ permutation of $S_n$ is $\lfloor \frac{k}{(n-1)!}\rfloor$ (assuming the first element of the set is $0$).This leads to an easy recursive function to find the whole $k^{\text(th)}$ permutation.

The magic words are “ranking permutations”; see for example Wilf’s lecture notes.

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