Basic Examples
Basic Examples
Here is the simplest nontrivial example, with 2 pairs of brackets:
@2
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{[[]],[][]}
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Here are the ways to nest 3 pairs of brackets::
@3
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{[[[]]],[[][]],[[]][],[][[]],[][][]}
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There are five ways to do that:
Length@%
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5
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Dyck words are counted by the Catalan numbers:
CatalanNumber@Range@10
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{1,2,5,14,42,132,429,1430,4862,16796}
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Scope
Scope
Define the helper function:
DyckPointsFromDyckWord@word_:=Prepend[Accumulate[StringSplit@StringReplace[word,{"[""[ ","]""] "}]/.{"["{1,1},"]"{1,-1}}],{0,0}]
A Dyck path of length units starts at the origin, goes NE or SE one unit, and finishes on the x axis:
2n
MapListLinePlot@*DyckPointsFromDyckWord,
[3]
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,
,
,
,
,,,,Out[]=