#### Document Type

Article

#### Publication Date

2009

#### Publication Title

SIAM Journal on Discrete Mathematics

#### Department

Department of Mathematics

#### Abstract

A permutation $\pi$ 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 $\pi$. The set of realized permutations is closed under consecutive pattern containment. Permutations that cannot be realized are called forbidden patterns. It was shown in [J. M. Amigó, S. Elizalde, and M. B. Kennel, *J. Combin. Theory Ser. A*, 115 (2008), pp. 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.

#### DOI

10.1137/080726689

#### Dartmouth Digital Commons Citation

Elizalde, Sergi, "The Number of Permutations Realized By a Shift" (2009). *Dartmouth Scholarship*. 2072.

https://digitalcommons.dartmouth.edu/facoa/2072