1 | initial version |
As you can see, d
is a kind of discrete primitive of w
. So you can compute it as follows:
sage: def primitive(w):
....: height = 0
....: d = [height]
....: for i in w:
....: if i == 1:
....: height += 1
....: d.append(height)
....: else:
....: height -= 1
....: d.append(height)
....: return d
sage: w=[1,1,1,0,0,1,0,0]
sage: primitive(w)
[0, 1, 2, 3, 2, 1, 2, 1, 0]
You can get its height as follows:
sage: max(primitive(w))
3
2 | No.2 Revision |
As you can see, d
is a kind of discrete primitive of w
. So you can compute it as follows:
sage: def primitive(w):
....: height = 0
....: d = [height]
....: for i in w:
....: if i == 1:
....: height += 1
....: d.append(height)
....: else:
....: height -= 1
....: d.append(height)
....: return d
sage: w=[1,1,1,0,0,1,0,0]
sage: primitive(w)
[0, 1, 2, 3, 2, 1, 2, 1, 0]
You can get its height as follows:
sage: max(primitive(w))
3
You can get all primitives of all Dyck words of a given length as follows:
sage: [primitive(w) for w in DyckWords(4)] [[0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 1, 0, 1, 0, 1, 2, 1, 0], [0, 1, 0, 1, 2, 1, 0, 1, 0], [0, 1, 0, 1, 2, 1, 2, 1, 0], [0, 1, 0, 1, 2, 3, 2, 1, 0], [0, 1, 2, 1, 0, 1, 0, 1, 0], [0, 1, 2, 1, 0, 1, 2, 1, 0], [0, 1, 2, 1, 2, 1, 0, 1, 0], [0, 1, 2, 1, 2, 1, 2, 1, 0], [0, 1, 2, 1, 2, 3, 2, 1, 0], [0, 1, 2, 3, 2, 1, 0, 1, 0], [0, 1, 2, 3, 2, 1, 2, 1, 0], [0, 1, 2, 3, 2, 3, 2, 1, 0], [0, 1, 2, 3, 4, 3, 2, 1, 0]]
Hence you can get the set of heights as follows:
sage: {max(primitive(w)) for w in DyckWords(4)}
{1, 2, 3, 4}
3 | No.3 Revision |
As you can see, d
is a kind of discrete primitive of w
. So you can compute it as follows:
sage: def primitive(w):
....: height = 0
....: d = [height]
....: for i in w:
....: if i == 1:
....: height += 1
....: d.append(height)
....: else:
....: height -= 1
....: d.append(height)
....: return d
sage: w=[1,1,1,0,0,1,0,0]
sage: primitive(w)
[0, 1, 2, 3, 2, 1, 2, 1, 0]
You can get its height as follows:
sage: max(primitive(w))
3
You can get all primitives of all Dyck words of a given length as follows:
sage: [primitive(w) for w in DyckWords(4)]
[[0, 1, 0, 1, 0, 1, 0, 1, 0],
[0, 1, 0, 1, 0, 1, 2, 1, 0],
[0, 1, 0, 1, 2, 1, 0, 1, 0],
[0, 1, 0, 1, 2, 1, 2, 1, 0],
[0, 1, 0, 1, 2, 3, 2, 1, 0],
[0, 1, 2, 1, 0, 1, 0, 1, 0],
[0, 1, 2, 1, 0, 1, 2, 1, 0],
[0, 1, 2, 1, 2, 1, 0, 1, 0],
[0, 1, 2, 1, 2, 1, 2, 1, 0],
[0, 1, 2, 1, 2, 3, 2, 1, 0],
[0, 1, 2, 3, 2, 1, 0, 1, 0],
[0, 1, 2, 3, 2, 1, 2, 1, 0],
[0, 1, 2, 3, 2, 3, 2, 1, 0],
[0, 1, 2, 3, 4, 3, 2, 1, Hence you can get the set of heights as follows:
sage: {max(primitive(w)) for w in DyckWords(4)}
{1, 2, 3, 4}