Establishing a Natural Bijection between Derangements and Permutations with Specific Properties

In this article, we will explore the relationship between derangements of length n and permutations sigma; of the set {0, 1, ..., n} that satisfy specific conditions.

Derangements

A derangement of length n is a permutation of the elements {1, 2, ..., n} such that no element appears in its original position. Mathematically, for a derangement π of the set {1, 2, ..., n}, the condition is:

π(i) ≠ i for all i 1, 2, ..., n.

Permutations with Specific Properties

Next, let's consider the permutations sigma; of the set {0, 1, ..., n}. For the purposes of establishing a bijection, we can specify the following properties for sigma;:

Fixing 0: The first element sigma;(0) must be equal to 0. Derangement of the Remaining Elements: The elements {1, 2, ..., n} must form a derangement. That is, for all i 1, 2, ..., n, sigma(l) ≠ l.

Establishing the Bijection

To establish a natural bijection between derangements of length n and the specified permutations sigma of {0, 1, ..., n}, we can follow these steps:

From Derangement to Permutation

Let π be a derangement of {1, 2, ..., n. Construct sigma such that: sigma;(0) 0 and sigma;(i) π(i) for i 1, 2, ..., n.

This construction ensures that sigma; fixes 0 and that the remaining elements form a derangement.

From Permutation to Derangement

Conversely, given a permutation sigma that satisfies the properties (fixing 0 and deranging {1, 2, ..., n}), we can derive a derangement π by taking: π(i) sigma;(i) for i 1, 2, ..., n.

This gives us a valid derangement as sigma;(i) ≠ i for all i, ensuring no element appears in its original position.

Conclusion

Thus, we have defined a natural bijection between the set of derangements of length n and the set of permutations sigma of {0, 1, ..., n}. Each derangement corresponds uniquely to a permutation meeting the specified properties, and vice versa.