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Notation Question :

Below is the excerpt from a textbook :

$i(n,k)$ denote the number of permutation $\sigma \in S_n$ with $i(\sigma) = k$ that is $k$ inversions.

from the above statement which set does $S_n$ denote for?

[edit: additional question]

In following statement, What exactly does LHS mean?

$\sum_{\sigma \in S_n} i(\sigma) = {n!\over 2} \binom{n}{2}$

Beverlie
  • 2,645
  • $S_n$ denotes the group of permutation of the set $\left{1,2,3,...,n\right}$. Probably. – Hellen Sep 26 '17 at 12:22
  • @Hellen or, would it be more plausible to understand $S_n$ denotes every sequences made from ${1,2,3,...,n}$? – Beverlie Sep 26 '17 at 12:27
  • $\sigma$ in that sentence is being called a permutation. Permutations are bijective functions from $\left{1,2,...,n \right}$ to $\left{1,2,...,n \right}$. – Hellen Sep 26 '17 at 12:30
  • Perhaps some some books or you would like to call that a sequence, but it is more common to call sequence to a function, not necessarily bijective from $\mathbb{N}$ to $\left{1,2,...,n \right}$. – Hellen Sep 26 '17 at 12:32
  • @Hellen I had added one more additional edit to the OP. Could you give me some advice about what LHS denote for? – Beverlie Sep 26 '17 at 12:36
  • The sum of $i(\sigma)$ when $\sigma$ ranges over all $n!$ permutations of the set $\left{1,2,...,n\right}$. From the sentence in your question it looks like $i(\sigma)$ is defined to be the smallest number of inversions (permutations that only change the position of two elements) required to obtain $\sigma$. – Hellen Sep 26 '17 at 12:40

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