def _shifted_rpp_horizontal_strips(cls, mu): # noqa
assert Partition.is_strict_partition(mu)
if mu == ():
return [(mu, set())]
def remove_box(nu, i):
if i < len(nu) and nu[i] > 0:
nu = nu[:i] + (nu[i] - 1, ) + nu[i + 1:]
while nu and nu[-1] == 0:
nu = nu[:-1]
if all(nu[j] > nu[j + 1] for j in range(len(nu) - 1)):
yield nu
def remove_all_boxes(nu, i):
queue = [nu]
while queue:
nu, queue = queue[0], queue[1:]
yield nu
for x in remove_box(nu, i):
queue.append(x)
ans = set()
queue = [(mu, len(mu) - 1)]
while queue:
nu, i = queue[0]
queue = queue[1:]
if i >= 0:
for nu in remove_all_boxes(nu, i):
ans.add(nu)
queue.append((nu, i - 1))
return [(nu, Partition.skew(mu, nu, shifted=True)) for nu in ans]
def _shifted_rpp_vertical_strips(cls, mu): # noqa
assert Partition.is_strict_partition(mu)
if mu == ():
return [(mu, set())]
def remove_box(nu, i):
for j in range(len(nu) - 1, -1, -1):
if j + nu[j] == i + 1:
nu = nu[:j] + (nu[j] - 1, ) + nu[j + 1:]
while nu and nu[-1] == 0:
nu = nu[:-1]
yield nu
return
def remove_all_boxes(nu, i):
queue = [nu]
while queue:
nu, queue = queue[0], queue[1:]
yield nu
for x in remove_box(nu, i):
queue.append(x)
ans = set()
queue = [(mu, (mu[0] if mu else 0) - 1)]
while queue:
nu, i = queue[0]
queue = queue[1:]
if i >= 0:
for nu in remove_all_boxes(nu, i):
ans.add(nu)
queue.append((nu, i - 1))
return [(nu, Partition.skew(mu, nu, shifted=True)) for nu in ans]
def is_reducible(nu, mu):
if len(nu) == 0:
return True
if any(nu[i] == mu[i] for i in range(len(mu))):
return True
if sum(nu) - sum(mu) <= 3:
return True
# if any(nu[i] + 1 < mu[i - 1] for i in range(1, len(nu))):
# return True
# if nu == mu:
# return True
# if Partition.get(nu, 1) <= Partition.get(mu, 1) + 1:
# return True
skew = Partition.skew(nu, mu, shifted=True)
(i, j) = min(skew, key=lambda ij: (-ij[0], ij[1]))
while True:
if (i - 1, j) in skew:
(i, j) = (i - 1, j)
elif (i, j + 1) in skew:
(i, j) = (i, j + 1)
else:
break
if (i, j) != max(skew, key=lambda ij: (-ij[0], ij[1])):
return True
def _rpp_horizontal_strips(cls, mu, lam): # noqa
if mu == lam:
return [(mu, set())]
if not Partition.contains(mu, lam):
return []
def remove_box(nu, i):
if i < len(nu) and nu[i] > 0:
nu = nu[:i] + (nu[i] - 1, ) + nu[i + 1:]
while nu and nu[-1] == 0:
nu = nu[:-1]
if all(nu[j] >= nu[j + 1] for j in range(len(nu) - 1)):
if Partition.contains(nu, lam):
yield nu
def remove_all_boxes(nu, i):
queue = [nu]
while queue:
nu, queue = queue[0], queue[1:]
yield nu
for x in remove_box(nu, i):
queue.append(x)
ans = set()
queue = [(mu, len(mu) - 1)]
while queue:
nu, i = queue[0]
queue = queue[1:]
if i >= 0:
for nu in remove_all_boxes(nu, i):
ans.add(nu)
queue.append((nu, i - 1))
return [(nu, Partition.skew(mu, nu)) for nu in ans]
def from_shifted_growth_diagram(cls, growth, edges, corners):
def shdiff(nu, lam):
return next(iter(Partition.skew(nu, lam, shifted=True)))
p, q = Tableau(), Tableau()
n, m = len(growth) - 1, len(growth[0]) - 1 if growth else 0
for i in range(1, n + 1):
mu, nu = growth[i][m], growth[i - 1][m]
for a, b in Partition.skew(mu, nu, shifted=True):
p = p.add(a, b, i)
for i in range(1, m + 1):
mu, nu = growth[n][i], growth[n][i - 1]
v = -i if edges[n][i] else i
j = corners[n][i]
if mu != nu:
a, b = shdiff(mu, nu)
q = q.add(a, b, v)
elif v < 0:
x = q.last_box_in_column(j)
q = q.add(x, j, v)
else:
x = q.last_box_in_row(j)
q = q.add(j, x, v)
return p, q
def split(nu, mu):
if (nu, mu) not in SPLIT_CACHE:
i, j = None, None
for lam in lshapes(nu, mu):
term = Partition.rim_terminal_part(left_rim(nu, mu, lam))
i = max(([] if i is None else [i]) + [a for (a, _) in term])
j = min(([] if j is None else [j]) + [b for (_, b) in term])
for lam in rshapes(nu, mu):
term = Partition.rim_terminal_part(right_rim(nu, mu, lam))
i = max(([] if i is None else [i]) + [a for (a, _) in term])
j = min(([] if j is None else [j]) + [b for (_, b) in term])
skew = Partition.skew(nu, mu, shifted=True)
# new_nu = Partition.trim(tuple(min(nu[t], j - t - 1) for t in range(i)) + nu[i:])
# new_mu = Partition.trim(tuple(min(mu[t], j - t - 1) for t in range(i)) + mu[i:])
while (i + 1, j) in skew:
i += 1
rest = {(a, b) for (a, b) in skew if (a <= i or b >= j)}
split_nu, split_mu = Partition.from_skew(rest, shifted=False)
diagonal = any(a == b for (a, b) in skew)
SPLIT_CACHE[(nu, mu)] = (split_nu, split_mu, diagonal)
return SPLIT_CACHE[(nu, mu)]
def shdiff(nu, lam):
return next(iter(Partition.skew(nu, lam, shifted=True)))
def KLG_counts(cls, nu, mu, p): # noqa
rim = tuple(sorted(Partition.skew(nu, mu, shifted=True)))
return cls._KLG_count_helper(p, rim)
def columns(lam, mu):
return len({j for (i, j) in Partition.skew(lam, mu, True)})
def right_rim(nu, mu, lam):
return Partition.skew(lam, mu, shifted=True)
def left_rim(nu, mu, lam):
return Partition.skew(nu, lam, shifted=True)