Geodesic
Permutations Containing and Avoiding $\textit{123}$ and $\textit{132}$ Patterns
Discrete mathematics & theoretical computer science, Tome 3 (1998-1999) no. 4.

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We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern, equals $(n-2)2^{n-3}$, for $n \ge 3$. We then give a bijection onto the set of permutations which avoid 123-patterns and have exactly one 132-pattern. Finally, we show that the number of permutations which contain exactly one 123-pattern and exactly one 132-pattern is $(n-3)(n-4)2^{n-5}$, for $n \ge 5$. 
@article{DMTCS_1999_3_4_a1,
     author = {Robertson, Aaron},
     title = {Permutations {Containing} and {Avoiding} $\textit{123}$ and $\textit{132}$ {Patterns}},
     journal = {Discrete mathematics & theoretical computer science},
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     volume = {3},
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     year = {1998-1999},
     doi = {10.46298/dmtcs.261},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.261/}
}
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Robertson, Aaron. Permutations Containing and Avoiding $\textit{123}$ and $\textit{132}$ Patterns. Discrete mathematics & theoretical computer science, Tome 3 (1998-1999) no. 4. doi : 10.46298/dmtcs.261. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.261/

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