Let n ≥2 be an integer. Prove that the total number of...

Let $n \geq 2$ be an integer. Prove that the total number of inversions of all $n !$ permutations of $1,2, \ldots, n$ equals $$ \frac{1}{2} n !\left(\begin{array}{l} n \\ 2 \end{array}\right)=n ! \frac{n(n-1)}{4} $$ (Hint: Pair up the permutations so that the number of inversions in each pair is $n(n-1) / 2 .)$

Let $n \geq 2$ be an integer. Prove that the total number of inversions of all $n !$ permutations of $1,2, \ldots, n$ equals
$$
\frac{1}{2} n !\left(\begin{array}{l}
n \\
2
\end{array}\right)=n ! \frac{n(n-1)}{4}
$$
(Hint: Pair up the permutations so that the number of inversions in each pair is $n(n-1) / 2 .)$

Key Concepts

Double Counting and Symmetry

Double counting is a powerful combinatorial tool where a quantity is counted in two different ways, often revealing an intrinsic symmetry. By pairing each permutation with another whose inversion count complements it to a constant total, one can sum over the entire set more efficiently. This approach underlines the importance of symmetry in solving combinatorial enumeration problems.

Combinatorial Pairing Technique

The combinatorial pairing technique involves pairing elements, in this case, permutations, in such a way that each pair exhibits a symmetric property. When dealing with inversion counts, the permutations can be paired so that the total number of inversions in each pair is constant. This method simplifies the calculation of a sum over a large set by leveraging symmetry between paired objects.

Inversions

An inversion in a permutation is a pair of elements that are out of their natural order. More formally, in a permutation of numbers, an inversion is a pair (i, j) such that i < j but the number in position i is greater than the number in position j. Inversions are a measure of the 'disorder' within a permutation and play a critical role in analyzing sorting algorithms and other combinatorial properties.

Permutations

A permutation is an ordered arrangement of elements where the order matters. In combinatorics, permutations of a set are commonly studied objects, and counting properties such as inversions across all permutations gives insights into the overall structure and complexity of the permutation space.

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