Physical Sciences and Mathematics Commons^™

The Number Of Permutations Realized By A Shift, Sergi Elizalde

Dartmouth Scholarship

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 …