Thanks for the link. I understand how you want to count them say for ()()(). But what is the question you are asking?

I do see the formula you are conjecturing on page 5 of that pdf.

]]>I found this: http://www-math.mit.edu/~djk/18.310/18. … theses.pdf

]]>I think I understand the problem now. I have notes on this bracketing problem somewhere. I am looking for them now.

]]>L(1) = 1, R(1) = 0 -- taking the first bracket

L(2) = 2, R(2) = 0 -- taking the first two brackets

L(3) = 2, R(3) = 1 -- taking the first three brackets

L(4) = 3, R(4) = 1 -- taking the first four brackets

L(5) = 3, R(5) = 2 -- taking the first five brackets

L(6) = 3, R(6) = 3 -- taking all six brackets

In this case, we see that the only value of t for which L(t) = R(t) is t = 6.

Yes, they are the Catalan numbers. But I'd like to know how to get that result combinatorially.

]]>I do not understand about R(t), L(t), can you explain further.

The numbers you are suggesting are every other Catalan number.

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