Geodesic
Classical pattern distributions in $\mathcal{S}_{n}(132)$ and $\mathcal{S}_{n}(123)$
Discrete mathematics & theoretical computer science, Permutation Patters 2018, Tome 21 (2019) no. 2.

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Classical pattern avoidance and occurrence are well studied in the symmetric group $\mathcal{S}_{n}$. In this paper, we provide explicit recurrence relations to the generating functions counting the number of classical pattern occurrence in the set of 132-avoiding permutations and the set of 123-avoiding permutations. 
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     author = {Qiu, Dun and Remmel, Jeffrey},
     title = {Classical pattern distributions in $\mathcal{S}_{n}(132)$ and $\mathcal{S}_{n}(123)$},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {2019},
     doi = {10.23638/DMTCS-21-2-4},
     language = {en},
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Qiu, Dun; Remmel, Jeffrey. Classical pattern distributions in $\mathcal{S}_{n}(132)$ and $\mathcal{S}_{n}(123)$. Discrete mathematics & theoretical computer science, Permutation Patters 2018, Tome 21 (2019) no. 2. doi : 10.23638/DMTCS-21-2-4. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-2-4/

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