The number of permutations realized by a shift

TL;DR

This paper characterizes and enumerates permutations realizable by a shift on N symbols, revealing the structure of such permutations and identifying forbidden patterns with length N+2.

Contribution

It provides a complete characterization of permutations realized by the shift on N symbols and counts them by length, advancing understanding of shift dynamics.

Findings

Permutations realized by the shift are characterized explicitly.

Forbidden patterns have length N+2.

Enumeration of realizable permutations by length is achieved.

Abstract

A permutation p is realized by the shift on N symbols if there is an infinite word on an N-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as p. The set of realized permutations is closed under consecutive pattern containment. Permutations that cannot be realized are called forbidden patterns. It was shown in [Amigo, Elizalde, Kennel, J. Combin. Theory Ser. A 115 (2008) 485-504] that the shortest forbidden patterns of the shift on N symbols have length N+2. In this paper we give a characterization of the set of permutations that are realized by the shift on N symbols, and we enumerate them according to their length.