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just a few details in permutation.py #38156

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Jun 9, 2024
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just a few details in permutation.py #38156

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145 changes: 73 additions & 72 deletions src/sage/combinat/permutation.py
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Original file line number Diff line number Diff line change
Expand Up @@ -463,17 +463,17 @@ def __classcall_private__(cls, l, check=True):

# if l is a pair of standard tableaux or a pair of lists
elif isinstance(l, (tuple, list)) and len(l) == 2 and \
all(isinstance(x, Tableau) for x in l):
all(isinstance(x, Tableau) for x in l):
return RSK_inverse(*l, output='permutation')
elif isinstance(l, (tuple, list)) and len(l) == 2 and \
all(isinstance(x, list) for x in l):
P,Q = (Tableau(_) for _ in l)
all(isinstance(x, list) for x in l):
P, Q = (Tableau(_) for _ in l)
return RSK_inverse(P, Q, 'permutation')
# if it's a tuple or nonempty list of tuples, also assume cycle
# notation
elif isinstance(l, tuple) or \
(isinstance(l, list) and l and
all(isinstance(x, tuple) for x in l)):
all(isinstance(x, tuple) for x in l)):
if l and (isinstance(l[0], (int, Integer)) or len(l[0]) > 0):
if isinstance(l[0], tuple):
n = max(max(x) for x in l)
Expand Down Expand Up @@ -683,8 +683,8 @@ def _latex_(self):
return " ".join(f"{let}_{{{i}}}" for i in redword)
if display == "twoline":
return r"\begin{{pmatrix}} {} \\ {} \end{{pmatrix}}".format(
" & ".join("%s" % i for i in range(1, len(self._list)+1)),
" & ".join("%s" % i for i in self._list))
" & ".join("%s" % i for i in range(1, len(self._list)+1)),
" & ".join("%s" % i for i in self._list))
if display == "list":
return repr(self._list)
if display == "cycle":
Expand Down Expand Up @@ -1068,20 +1068,20 @@ def _to_cycles_set(self, singletons=True):

if not singletons:
# remove the fixed points
L = {i+1 for i,pi in enumerate(p) if pi != i+1}
L = {i for i, pi in enumerate(p, start=1) if pi != i}
else:
L = set(range(1,len(p)+1))
L = set(range(1, len(p) + 1))

# Go through until we've considered every remaining number
while L:
# take the first remaining element
cycleFirst = L.pop()
next = p[cycleFirst-1]
next = p[cycleFirst - 1]
cycle = [cycleFirst]
while next != cycleFirst:
cycle.append(next)
L.remove(next)
next = p[next-1]
next = p[next - 1]
# add the cycle
cycles.append(tuple(cycle))

Expand Down Expand Up @@ -1109,7 +1109,7 @@ def _to_cycles_list(self, singletons=True):

if not singletons:
# remove the fixed points
L = [i + 1 for i, pi in enumerate(p) if pi != i + 1]
L = [i for i, pi in enumerate(p, start=1) if pi != i]
else:
L = list(range(1, len(p) + 1))

Expand All @@ -1125,7 +1125,7 @@ def _to_cycles_list(self, singletons=True):
cycle.append(next)
# remove next from L
# we use a binary search to find it
L.pop(bisect_left(L,next))
L.pop(bisect_left(L, next))
next = p[next-1]
# add the cycle
cycles.append(tuple(cycle))
Expand Down Expand Up @@ -1164,7 +1164,7 @@ def signature(self) -> Integer:
sage: Permutation([]).sign()
1
"""
return (-1)**(len(self)-len(self.to_cycles()))
return (-1)**(len(self) - len(self.to_cycles()))

# one can also use sign as an alias for signature
sign = signature
Expand Down Expand Up @@ -1592,7 +1592,7 @@ def merge_and_countv(ivA_A, ivB_B):
else:
C.extend(B[j:])
ivC.extend(ivB[j:])
return ivC,C
return ivC, C

def base_case(L):
s = sorted(L)
Expand Down Expand Up @@ -1628,7 +1628,7 @@ def inversions(self) -> list:
"""
p = self[:]
n = len(p)
return [(i+1,j+1) for i in range(n-1) for j in range(i+1,n)
return [(i+1, j+1) for i in range(n-1) for j in range(i+1, n)
if p[i] > p[j]]

def stack_sort(self) -> Permutation:
Expand Down Expand Up @@ -1917,7 +1917,7 @@ def absolute_length(self) -> Integer:
"""
return self.size() - len(self.cycle_type())

@combinatorial_map(order=2,name='inverse')
@combinatorial_map(order=2, name='inverse')
def inverse(self) -> Permutation:
r"""
Return the inverse of ``self``.
Expand All @@ -1932,8 +1932,8 @@ def inverse(self) -> Permutation:
[3, 1, 5, 2, 4]
"""
w = list(range(len(self)))
for i, j in enumerate(self):
w[j - 1] = i + 1
for i, j in enumerate(self, start=1):
w[j - 1] = i
return Permutations()(w)

__invert__ = inverse
Expand Down Expand Up @@ -3081,7 +3081,6 @@ def number_of_fixed_points(self) -> Integer:
sage: Permutation([1,2,3,4]).number_of_fixed_points()
4
"""

return len(self.fixed_points())

def is_derangement(self) -> bool:
Expand Down Expand Up @@ -3778,7 +3777,7 @@ def weak_excedences(self) -> list:
sage: Permutation([1,4,3,2,5]).weak_excedences()
[1, 4, 3, 5]
"""
return [pi for i, pi in enumerate(self) if pi >= i + 1]
return [pi for i, pi in enumerate(self, start=1) if pi >= i]

def bruhat_inversions(self) -> list:
r"""
Expand Down Expand Up @@ -5370,7 +5369,7 @@ def shifted_shuffle(self, other):
True
"""
return self.shifted_concatenation(other, "right").\
right_permutohedron_interval(self.shifted_concatenation(other, "left"))
right_permutohedron_interval(self.shifted_concatenation(other, "left"))

def nth_roots(self, n):
r"""
Expand Down Expand Up @@ -5480,8 +5479,8 @@ def rewind(L, n):
b = True
poss = [P.identity()]
for pa in partition:
poss = [p*q for p in poss
for q in merging_cycles([rewind(cycles[m][i-1], n//len(pa)) for i in pa])]
poss = [p*q for p in poss
for q in merging_cycles([rewind(cycles[m][i-1], n//len(pa)) for i in pa])]
possibilities[i] += poss
if not b:
return
Expand Down Expand Up @@ -5613,6 +5612,7 @@ def number_of_nth_roots(self, n):

return result


def _tableau_contribution(T):
r"""
Get the number of SYT of shape(``T``).
Expand Down Expand Up @@ -5856,21 +5856,21 @@ def __classcall_private__(cls, n=None, k=None, **kwargs):
if len(a) == 1 and a[0] != 1:
a = a[0]
if a in StandardPermutations_all():
if a == [1,2]:
if a == [1, 2]:
return StandardPermutations_avoiding_12(n)
elif a == [2,1]:
elif a == [2, 1]:
return StandardPermutations_avoiding_21(n)
elif a == [1,2,3]:
elif a == [1, 2, 3]:
return StandardPermutations_avoiding_123(n)
elif a == [1,3,2]:
elif a == [1, 3, 2]:
return StandardPermutations_avoiding_132(n)
elif a == [2,1,3]:
elif a == [2, 1, 3]:
return StandardPermutations_avoiding_213(n)
elif a == [2,3,1]:
elif a == [2, 3, 1]:
return StandardPermutations_avoiding_231(n)
elif a == [3,1,2]:
elif a == [3, 1, 2]:
return StandardPermutations_avoiding_312(n)
elif a == [3,2,1]:
elif a == [3, 2, 1]:
return StandardPermutations_avoiding_321(n)
else:
return StandardPermutations_avoiding_generic(n, (a,))
Expand Down Expand Up @@ -5970,36 +5970,36 @@ class options(GlobalOptions):
NAME = 'Permutations'
module = 'sage.combinat.permutation'
display = {'default': "list",
'description': "Specifies how the permutations should be printed",
'values': {'list': "the permutations are displayed in list notation"
" (aka 1-line notation)",
'cycle': "the permutations are displayed in cycle notation"
" (i. e., as products of disjoint cycles)",
'singleton': "the permutations are displayed in cycle notation"
" with singleton cycles shown as well",
'reduced_word': "the permutations are displayed as reduced words"},
'alias': {'word': "reduced_word", 'reduced_expression': "reduced_word"},
'case_sensitive': False}
latex = {'default': "list",
'description': "Specifies how the permutations should be latexed",
'values': {'list': "latex as a list in one-line notation",
'twoline': "latex in two-line notation",
'cycle': "latex in cycle notation",
'singleton': "latex in cycle notation with singleton cycles shown as well",
'reduced_word': "latex as reduced words"},
'alias': {'word': "reduced_word", 'reduced_expression': "reduced_word", 'oneline': "list"},
'description': "Specifies how the permutations should be printed",
'values': {'list': "the permutations are displayed in list notation"
" (aka 1-line notation)",
'cycle': "the permutations are displayed in cycle notation"
" (i. e., as products of disjoint cycles)",
'singleton': "the permutations are displayed in cycle notation"
" with singleton cycles shown as well",
'reduced_word': "the permutations are displayed as reduced words"},
'alias': {'word': "reduced_word", 'reduced_expression': "reduced_word"},
'case_sensitive': False}
latex = {'default': "list",
'description': "Specifies how the permutations should be latexed",
'values': {'list': "latex as a list in one-line notation",
'twoline': "latex in two-line notation",
'cycle': "latex in cycle notation",
'singleton': "latex in cycle notation with singleton cycles shown as well",
'reduced_word': "latex as reduced words"},
'alias': {'word': "reduced_word", 'reduced_expression': "reduced_word", 'oneline': "list"},
'case_sensitive': False}
latex_empty_str = {'default': "1",
'description': 'The LaTeX representation of a reduced word when said word is empty',
'checker': lambda char: isinstance(char,str)}
'description': 'The LaTeX representation of a reduced word when said word is empty',
'checker': lambda char: isinstance(char, str)}
generator_name = {'default': "s",
'description': "the letter used in latexing the reduced word",
'checker': lambda char: isinstance(char,str)}
'description': "the letter used in latexing the reduced word",
'checker': lambda char: isinstance(char, str)}
mult = {'default': "l2r",
'description': "The multiplication of permutations",
'values': {'l2r': r"left to right: `(p_1 \cdot p_2)(x) = p_2(p_1(x))`",
'r2l': r"right to left: `(p_1 \cdot p_2)(x) = p_1(p_2(x))`"},
'case_sensitive': False}
'description': "The multiplication of permutations",
'values': {'l2r': r"left to right: `(p_1 \cdot p_2)(x) = p_2(p_1(x))`",
'r2l': r"right to left: `(p_1 \cdot p_2)(x) = p_1(p_2(x))`"},
'case_sensitive': False}


class Permutations_nk(Permutations):
Expand Down Expand Up @@ -6082,7 +6082,8 @@ def __iter__(self) -> Iterator[Permutation]:
sage: [p for p in Permutations(3,4)]
[]
"""
for x in itertools.permutations(range(1,self.n+1), int(self._k)):
for x in itertools.permutations(range(1, self.n + 1),
int(self._k)):
yield self.element_class(self, x, check=False)

def cardinality(self) -> Integer:
Expand Down Expand Up @@ -7751,8 +7752,8 @@ def inverse(self):
[4, 2, 1, 3]
"""
w = list(range(len(self)))
for i, j in enumerate(self):
w[j - 1] = i + 1
for i, j in enumerate(self, start=1):
w[j - 1] = i
return self.__class__(self.parent(), w)

__invert__ = inverse
Expand Down Expand Up @@ -7868,8 +7869,8 @@ def from_rank(n, rank):
"""
# Find the factoradic of rank
factoradic = [None] * n
for j in range(1,n+1):
factoradic[n-j] = Integer(rank % j)
for j in range(1, n + 1):
factoradic[n - j] = Integer(rank % j)
rank = int(rank) // j

return from_lehmer_code(factoradic, Permutations(n))
Expand All @@ -7896,8 +7897,8 @@ def from_inversion_vector(iv, parent=None):
"""
p = iv[:]
open_spots = list(range(len(iv)))
for i, ivi in enumerate(iv):
p[open_spots.pop(ivi)] = i + 1
for i, ivi in enumerate(iv, start=1):
p[open_spots.pop(ivi)] = i

if parent is None:
parent = Permutations()
Expand Down Expand Up @@ -8182,16 +8183,16 @@ def bistochastic_as_sum_of_permutations(M, check=True):

while G.size() > 0:
matching = G.matching(use_edge_labels=True)
matching = [(min(u,v), max(u, v), w) for u,v,w in matching]
matching = [(min(u, v), max(u, v), w) for u, v, w in matching]

# This minimum is strictly larger than 0
minimum = min([x[2] for x in matching])

for (u,v,l) in matching:
for u, v, l in matching:
if minimum == l:
G.delete_edge((u,v,l))
G.delete_edge((u, v, l))
else:
G.set_edge_label(u,v,l-minimum)
G.set_edge_label(u, v, l - minimum)

matching.sort(key=lambda x: x[0])
value += minimum * CFM(P([x[1]-n+1 for x in matching]))
Expand Down Expand Up @@ -8481,9 +8482,9 @@ def descents_composition_last(dc):
raise TypeError("The argument must be of type Composition")
s = 0
res = []
for i in reversed(range(len(dc))):
res = list(range(s+1,s+dc[i]+1)) + res
s += dc[i]
for dci in reversed(dc):
res = list(range(s + 1, s + dci + 1)) + res
s += dci

return Permutations()(res)

Expand Down Expand Up @@ -8756,7 +8757,7 @@ def from_major_code(mc, final_descent=False):
# the letter i into the word w^(i+1) in such a way that
# maj(w^i)-maj(w^(i+1)) = mc[i]

for i in reversed(range(1,len(mc))):
for i in reversed(range(1, len(mc))):
# Lemma 2.2 in Skandera

# Get the descents of w and place them in reverse order
Expand Down
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