Geodesic
Bijections for Permutation Tableaux
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008).

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In this paper we propose a new bijection between permutation tableaux and permutations. This bijection shows how natural statistics on the tableaux are equidistributed to classical statistics on permutations: descents, RL-minima and pattern enumerations. We then use the bijection, and a related encoding of tableaux by words, to prove results about the enumeration of permutations with a fixed number of 31-2 patterns, and to define subclasses of permutation tableaux that are in bijection with set partitions. An extended version of this work is available in [6]. 
@article{DMTCS_2008_special_255_a40,
     author = {Corteel, Sylvie and Nadeau, Philippe},
     title = {Bijections for {Permutation} {Tableaux}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)},
     year = {2008},
     doi = {10.46298/dmtcs.3632},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3632/}
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Corteel, Sylvie; Nadeau, Philippe. Bijections for Permutation Tableaux. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008). doi : 10.46298/dmtcs.3632. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3632/

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