SIAM Journal on Discrete Mathematics
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.
Elizalde, Sergi, "The Number of Permutations Realized By a Shift" (2009). Open Dartmouth: Faculty Open Access Articles. 2072.