Basic Examples

Here is the simplest nontrivial example, with 2 pairs of brackets:
[◼]
DyckWords
@2
In[]:=
{[[]],[][]}
Out[]=
Here are the ways to nest 3 pairs of brackets::
[◼]
DyckWords
@3
In[]:=
{[[[]]],[[][]],[[]][],[][[]],[][][]}
Out[]=
There are five ways to do that:
Length@%
In[]:=
5
Out[]=
Dyck words are counted by the Catalan numbers:
CatalanNumber@Range@10
In[]:=
{1,2,5,14,42,132,429,1430,4862,16796}
Out[]=

Scope

Define the helper function:
DyckPointsFromDyckWord@word_:=​​Prepend[​​Accumulate[​​StringSplit@StringReplace[​​word,​​{"[""[ ","]""] "}​​]/.{"["{1,1},"]"{1,-1}}​​],​​{0,0}​​]
A Dyck path of length
2n
units starts at the origin, goes NE or SE one unit, and finishes on the x axis:
MapListLinePlot@*DyckPointsFromDyckWord,
[◼]
DyckWords
[3]
In[]:=
,,,,
Out[]=