Sorting 1-away permutations with underlying Fibonacci convolutions

We discuss some combinatorics associated with 1-away permutations, where an element can be displaced from its correct position by at most one location. Specifically, we look at a sorting algorithm for such permutations and analyze its number of comparisons, \(C_n\). We find that the mean is a certain combination of two-fold convolutions of Fibonacci numbers and the variance is a certain combination of three-fold convolutions of Fibonacci numbers, with corresponding asymptotics (as \(n\rightarrow \infty \)):

$${\mathbb {E}}[C_n] \sim \frac{5 + \sqrt{5}}{10}\, n, \qquad {\mathbb {V}\textrm{ar}}[C_n]\sim \frac{\sqrt{5}}{25} \, n.$$

The proofs contain finer asymptotics down to exponentially small error terms. The relatively small variance admits a weak law and a central limit theorem via a super moment generating function. In view of the special nature of the data, such a specialized algorithm outperforms general comparison-based sorting algorithms.