# Deriving the average number of inversions across all permutations

In the answer by Raphael to the question "Is there a system behind the magic of algorithm analysis?", there is an equation:

$$\qquad\displaystyle \mathbb{E}[C_{\text{swaps}}] = \frac{1}{n!} \sum_{A} \operatorname{inv}(A) \tag{1},$$

where $A$ is an array, $\mathbb{E}[C_{\text{swaps}}]$ is the expected number of swaps it takes to sort a random permutation with Bubblesort, $n$ is the length of $A$, and $\operatorname{inv}(A)$ is the number of inversions of $A$.

Then (1) is equal to the following equation:

$$\qquad\displaystyle \mathbb{E}[C_{\text{swaps}}] = \frac{1}{2} \cdot \binom{n}{2}. \tag{2}$$

But I don't understand how to prove (2) based on (1). Could anyone please give me a proof?

## 2 Answers

You can not go directly from one equation to the other; you need to add a whole proof which is separate from what I explain there. Hence my statement "it has been shown".

A boring proof using induction is here.

For a constructive proof, you can follow the idea from here:

1. Show that the maximum number of inversions is $\binom{n}{2}$.
2. Observe that every permutation and its reverse pair up: their respective sets of inversions are complement to each other, so both together add $\binom{n}{2}$ inversions to the sum.
3. Since this pairing neatly and uniquely covers all permutations, we get that the total number of inversions across all permutations is $\frac{n!}{2} \cdot \binom{n}{2}$.

The claim follows immediately.

For $i < j$ and a random permutation $A$, let $X_{ij}$ be the indicator variable for the event $A[i] > A[j]$. Clearly $\Pr[X_{ij} = 1] = 1/2$ and so $E[X_{ij}] = 1/2$. The total number of inversions is $\sum_{i<j} X_{ij}$, and so the average number of inversions is $$E\left[\sum_{i<j} X_{ij}\right] = \sum_{i<j} E[X_{ij}] = \sum_{i<j} \frac{1}{2} = \frac{1}{2} \binom{n}{2}.$$