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Dec 14, 2014 at 17:56 comment added Kaveh Thank you for adding the new reference. I found it quite illuminating and now I understand your first answer much better. For the sake of curiosity, I very much like to start thinking about the way one could translate the scope of your note on random walks to the analogous case of some of the other Catalan objects, such as noncrossing partitions or triangulations, where I could see an ordering a bit better.
Dec 14, 2014 at 8:06 comment added Bjørn Kjos-Hanssen OK, I edited in some references
Dec 14, 2014 at 8:06 history edited Bjørn Kjos-Hanssen CC BY-SA 3.0
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Dec 14, 2014 at 7:02 comment added Kaveh Thanks for sharing the link and your idea. I looked at the brief description of Brownian Excursions. In the section "Connections and Applications", it was referred to the connection between Brownian Excursions and the height of random rooted binary trees. However, it didn't seem to be a sort of generalization. In particular, I skimmed "On the total heights of rooted binary trees", by Lajos Takacs. and I suspect the question he considers in that paper is slightly different. Could you please mention any reference that focuses on the combinatorial connections you mentioned in your answer.
Dec 14, 2014 at 5:48 history answered Bjørn Kjos-Hanssen CC BY-SA 3.0