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I asked the following question on MathStackExchange, but I have not received any answers after almost 3 days. Although it may not be a research level question, I thought I could ask it here.

*"Is there any natural approach to generalize the idea of binary bracketing (bracketing functions or any of its equivalent notions) to the continuous case, such that the well-known discrete version becomes a special case of the general theory?"

In other words, could this notion be extended to the continuous setting?

Here is the link of my question on MathStackExchange: https://math.stackexchange.com/questions/1061963/generalization-of-binary-bracketing

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How about Brownian excursions?http://en.m.wikipedia.org/wiki/Brownian_excursion Just as the number of left parentheses seen is always greater or equal the number of right ones, the excursion has always moved up more than down.

Formally too, the Catalan numbers can be used to count the discrete excursions as well as the bracketings. See for instance Shreve: Stochastic Calculus for Finance, volume I, chapter 5, and the discussion here: http://math.hawaii.edu/wordpress/bjoern/distribution-of-first-hitting-time/

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  • $\begingroup$ 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. $\endgroup$
    – Kaveh
    Commented Dec 14, 2014 at 7:02
  • $\begingroup$ OK, I edited in some references $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Dec 14, 2014 at 8:06
  • $\begingroup$ 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. $\endgroup$
    – Kaveh
    Commented Dec 14, 2014 at 17:56
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Just to get the ball rolling, I suggest the following view, which may tell you where to look.

When one "unwraps" a (binary) bracketed expression, one ends up with a system of labeled objects which is sort of a partition, but in some cases the order matters. In any case, one is led to a system of refinements of partitions, where each step in the refinement takes an "equivalence class" and breaks it into two parts.

I suggest looking at the unwrapping process using an arbitrary cardinal, and making the process uniform in the following way, using n to represent the possibly infinite cardinal: one starts with an initial object O and unwraps it into system S, where S has n objects O_i and some arrangement between the objects. One continues the unwrapping by picking exactly one of the objects O_j say, and unwrapping it into a system S' which bears an isomorphic resemblance (taking the objects as atoms) to S. Now iterate this process to your satisfaction, until the totality of objects reached is the right size or the right granularity.

This is a view that can be generalized, but applies to several processes in set theory, topology, and algebra. It suggests to me generalizations of clones in universal algebra, of unwrapping hierarchically defined subclasses of models in set theory, of certain kinds of homogeneity in topological spaces. I don't know what combinatoric insights this view will yield, but you might find it useful in constructing n-ary versions of Catalan numbers for finite n. This view might help in understanding existing generalizations of Catalan numbers, but as I don't know that literature, this view might also confound.

Gerhard "Good Luck In Your Search" Paseman, 2014.12.13

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  • $\begingroup$ Thanks for sharing your thought. I suspect you are proposing thinking about a generalization of the recursive property of Catalan objects, in the way that appears in bracketing, triangulations of a convex $n$-gons, etc. This seems to be an interesting idea. However, the first obstacle one should tackle is to set up the refinement such that it coincides with the known case when we restrict to the countable cardinal, while, for the uncountable cardinal case, all the possible refinements be congruent together. To me, this seems to be one of the most formidable obstacles. $\endgroup$
    – Kaveh
    Commented Dec 14, 2014 at 7:30

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