Geodesic
Bijections from weighted Dyck paths to Schröder paths
Journal of integer sequences, Tome 13 (2010) no. 9
Kim and Drake used generating functions to prove that the number of 2-distant noncrossing matchings, which are in bijection with little Schröder paths, is the same as the weight of Dyck paths in which downsteps from even height have weight 2. This work presents bijections from those Dyck paths to little Schröder paths, and from a similar set of Dyck paths to big Schröder paths. We show the effect of these bijections on the corresponding matchings, find generating functions for two new classes of lattice paths, and demonstrate a relationship with $231$-avoiding permutations.
Classification : 05A19, 05A15, 05A05
Keywords: lattice paths, schr$\ddot $oder numbers, matchings, 231-avoiding permutations 
@article{JIS_2010__13_9_a5,
     author = {Drake,  Dan},
     title = {Bijections from weighted {Dyck} paths to {Schr\"oder} paths},
     journal = {Journal of integer sequences},
     year = {2010},
     volume = {13},
     number = {9},
     zbl = {1298.05036},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2010__13_9_a5/}
}
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JO  - Journal of integer sequences
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LA  - en
ID  - JIS_2010__13_9_a5
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%0 Journal Article
%A Drake,  Dan
%T Bijections from weighted Dyck paths to Schröder paths
%J Journal of integer sequences
%D 2010
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%N 9
%U http://geodesic.mathdoc.fr/item/JIS_2010__13_9_a5/
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%F JIS_2010__13_9_a5
Drake,  Dan. Bijections from weighted Dyck paths to Schröder paths. Journal of integer sequences, Tome 13 (2010) no. 9. http://geodesic.mathdoc.fr/item/JIS_2010__13_9_a5/