def from_bounce_pair(dyck, alpha):
r"""
Returns an LAC-tree in bijection with the given bounce pair. A check is
performed for validity.
INPUT:
- ``dyck``: a Dyck path in the 0,1 format
- ``alpha``: a composition indicating the bounce path
OUTPUT:
An LAC-tree in bijection with this bounce pair
"""
n = sum(alpha)
if n * 2 != len(dyck):
raise ValueError('Inconsistent sizes of parameters')
bounce = []
for a in alpha:
bounce += ([1] * a) + ([0] * a)
dwa = DyckWord(dyck)
dwb = DyckWord(bounce)
area_a = dwa.to_area_sequence()
area_b = dwb.to_area_sequence()
if not all(area_a[i] >= area_b[i] for i in range(n)):
raise ValueError('Incompatible parameters')
# count children number by counting north steps on each x-coordinate
vsteps = [0] * (n + 1)
cur = 0
x = 0
while cur < len(dyck):
while 1 == dyck[cur]:
cur += 1
vsteps[x] += 1
cur += 1
x += 1
# construct tree
l = len(alpha)
dyckpost = n - alpha[l - 1]
actives = [OrderedTree([]) for i in range(alpha[l - 1])]
for region in range(l - 2, -1, -1):
newactives = []
for i in range(alpha[region]):
newnode = OrderedTree(actives[:vsteps[dyckpost]])
actives = actives[vsteps[dyckpost]:]
newactives.append(newnode)
dyckpost -= 1
actives = newactives + actives
T = OrderedTree(actives)
# coloring with existent function
return LACTree(T, alpha)
def from_labelled_dyck_word(LDW):
r"""
Return the parking function corresponding to the labelled Dyck word.
INPUT:
- ``LDW`` -- labelled Dyck word
OUTPUT:
- the parking function corresponding to the labelled Dyck
word that is half the size of ``LDW``
EXAMPLES::
sage: from sage.combinat.parking_functions import from_labelled_dyck_word
sage: LDW = [2, 6, 0, 4, 5, 0, 0, 0, 3, 7, 0, 1, 0, 0]
sage: from_labelled_dyck_word(LDW)
[6, 1, 5, 2, 2, 1, 5]
::
sage: from_labelled_dyck_word([2, 3, 0, 0, 1, 0, 4, 0])
[3, 1, 1, 4]
sage: from_labelled_dyck_word([2, 3, 4, 0, 0, 0, 1, 0])
[4, 1, 1, 1]
sage: from_labelled_dyck_word([2, 4, 0, 1, 0, 0, 3, 0])
[2, 1, 4, 1]
"""
L = [ell for ell in LDW if ell != 0]
D = DyckWord(map(lambda x: Integer(not x.is_zero()), LDW))
return from_labelling_and_area_sequence(L, D.to_area_sequence())
def from_labelled_dyck_word(LDW):
r"""
Returns the parking function corresponding to the labelled Dyck word.
INPUT:
- ``LDW`` -- labelled Dyck word
OUTPUT:
- returns the parking function corresponding to the labelled Dyck word that is
half the size of ``LDW``
EXAMPLES::
sage: from sage.combinat.parking_functions import from_labelled_dyck_word
sage: LDW = [2, 6, 0, 4, 5, 0, 0, 0, 3, 7, 0, 1, 0, 0]
sage: from_labelled_dyck_word(LDW)
[6, 1, 5, 2, 2, 1, 5]
::
sage: from_labelled_dyck_word([2, 3, 0, 0, 1, 0, 4, 0])
[3, 1, 1, 4]
sage: from_labelled_dyck_word([2, 3, 4, 0, 0, 0, 1, 0])
[4, 1, 1, 1]
sage: from_labelled_dyck_word([2, 4, 0, 1, 0, 0, 3, 0])
[2, 1, 4, 1]
"""
L = [ell for ell in LDW if ell!=0]
D = DyckWord(map(lambda x: Integer(not x.is_zero()), LDW))
return from_labelling_and_area_sequence(L, D.to_area_sequence())
