A congruential recurrence characterizes the inverses of S\'{o}s permutations

TL;DR

This paper proves that permutations satisfying a specific congruential recurrence are exactly the inverses of Sós permutations, establishing a precise characterization and providing methods to generate higher-degree inverses.

Contribution

It establishes a one-to-one correspondence between permutations satisfying a congruential recurrence and inverses of Sós permutations, resolving an open problem and introducing a new lifting procedure.

Findings

Permutations satisfying the congruential recurrence are exactly the inverses of Sós permutations.

Upper bounds for certain permutation sets are tight and match the cardinality of Sós permutations.

A procedure is provided to increase the degree of inverses without relying on underlying parameters.

Abstract

In a proof of the three gaps theorem, a class of permutations known as the S\'{o}s permutations was introduced. It is known that a S\'{o}s permutation, as a sequence, satisfies a certain recurrence (S\'{o}s's recurrence), however, whether the converse holds remains unknown. On the other hand, the inverses of S\'{o}s permutations have been studied also. It has been reported that such a permutation satisfies a congruential recurrence as a sequence. The converse problem of this fact, i.e., whether a permutation satisfying the congruential recurrence is the inverse of a S\'{o}s permutation, is also unsolved, except for a finite number of the degrees of the permutations. This paper relates the set of permutations satisfying the congruential recurrence to other sets of permutations and gives upper bounds for their cardinalities. The upper bounds are in fact tight. In particular, the set of permutations satisfying the congruential recurrence has the same cardinality as that of S\'{o}s permutations, giving the affirmative answer to the above unsolved problem as a corollary. As another corollary, it is shown that all permutation, which is regarded as a congruential quasi-progression of diameter 1 in that the set formed by the first order differences modulo the degree of the permutation is a singleton or a set of two successive integers, is the inverse of a S\'{o}s permutation with an elementary operation called as shift applied. As an application of these facts, we present a procedure that lifts the set of the inverses of the S\'{o}s permutations of a given degree to the set of the same kind with the degree increased by one, without referring to the underlying parameters defining the S\'{o}s permutations or the Farey sequence associated to them.