Geodesic
Extending from bijections between marked occurrences of patterns to all occurrences of patterns
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) (2012).

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We consider two recent open problems stating that certain statistics on various sets of combinatorial objects are equidistributed. The first, posed by Anders Claesson and Svante Linusson, relates nestings in matchings on $\{1,2,\ldots,2n\}$ to occurrences of a certain pattern in permutations in $S_n$. The second, posed by Miles Jones and Jeffrey Remmel, relates occurrences of a large class of consecutive permutation patterns to occurrences of the same pattern in the cycles of permutations. We develop a general method that solves both of these problems and many more. We further employ the Garsia-Milne involution principle to obtain purely bijective proofs of these results. 
@article{DMTCS_2012_special_263_a84,
     author = {Remmel, Jeffrey and Tiefenbruck, Mark},
     title = {Extending from bijections between marked occurrences of patterns to all occurrences of patterns},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)},
     year = {2012},
     doi = {10.46298/dmtcs.3098},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3098/}
}
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%D 2012
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Remmel, Jeffrey; Tiefenbruck, Mark. Extending from bijections between marked occurrences of patterns to all occurrences of patterns. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) (2012). doi : 10.46298/dmtcs.3098. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3098/

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