Exercise
M is the monoid of all words in the alphabet { [ , ] }, with concatenation. Thus, words are bracketings. Consider the subset of all "well positioned" bracketings, that is, those words for which the number of "[" (opening brackets) is at least the number of "]" in each left subword. This is a submonoid of M. What is the number of words of length 2d?
d2.
2d.
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No, there are more when d is large.
No, this is the total number of words in M of length d.
Yes; these are the so-called Catalan numbers.