Geodesic
Continued fractions and Catalan problems
The electronic journal of combinatorics, Tome 7 (2000)
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We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also expressed as a continued fraction. Among these problems is the enumeration of $(132)$-pattern avoiding permutations that have a given number of increasing patterns of length $k$. This extends and illuminates a result of Robertson, Wilf and Zeilberger for the case $k=3$.
DOI : 10.37236/1523
Classification : 05A15, 11A55, 11B75 
@article{10_37236_1523,
     author = {Mahendra Jani and Robert G. Rieper},
     title = {Continued fractions and {Catalan} problems},
     journal = {The electronic journal of combinatorics},
     year = {2000},
     volume = {7},
     doi = {10.37236/1523},
     zbl = {1047.05500},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1523/}
}
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%A Robert G. Rieper
%T Continued fractions and Catalan problems
%J The electronic journal of combinatorics
%D 2000
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Mahendra Jani; Robert G. Rieper. Continued fractions and Catalan problems. The electronic journal of combinatorics, Tome 7 (2000). doi: 10.37236/1523

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