On permutation
On permutation
For any permutation $\sigma:\{1,2,\dots,n\}\to\{1,2,\dots,n\}$ define its displacement as
$$D(\sigma)=\prod_{i=1}^n |i-\sigma(i)|$$
What is greater: the sum of displacements of even permutations or the sum of displacements of odd permutations? The answer may depend on $n$.
$$D(\sigma)=\prod_{i=1}^n |i-\sigma(i)|$$
What is greater: the sum of displacements of even permutations or the sum of displacements of odd permutations? The answer may depend on $n$.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
- Tolaso J Kos
- Administrator
- Posts: 868
- Joined: Sat Nov 07, 2015 7:12 pm
- Location: Larisa
- Contact:
Re: On permutation
The sum of $D(\sigma)$ over the even permutations minus the one over the odd permutations is the determinant of the matrix $A$ with entries $a_{i,j}=\vert i-j\vert$ and this determinant is known to be
$$\det A = (-1)^{n-1} (n-1) 2^{n-2}$$
$$\det A = (-1)^{n-1} (n-1) 2^{n-2}$$
Imagination is much more important than knowledge.
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 0 guests