gh-35318: fix E502 in schemes and combinat

    
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### 📚 Description

fix pycodestyle E502 warning in schemes/ and combinat/

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URL: #35318
Reported by: Frédéric Chapoton
Reviewer(s): Frédéric Chapoton, Kwankyu Lee

src/sage/combinat/designs/difference_family.py

@@ -591,8 +591,8 @@ def radical_difference_set(K, k, l=1, existence=False, check=True):
         x = K.multiplicative_generator()
         D = K.cyclotomic_cosets(x**((v-1)//k), [K.one()])
         if is_difference_family(K, D, v, k, l):
-            print("**  You found a new example of radical difference set **\n"\
-                  "**  for the parameters (v,k,l)=({},{},{}).            **\n"\
+            print("**  You found a new example of radical difference set **\n"
+                  "**  for the parameters (v,k,l)=({},{},{}).            **\n"
                   "**  Please contact sage-devel@googlegroups.com        **\n".format(v, k, l))
             if existence:
                 return True
@@ -602,8 +602,8 @@ def radical_difference_set(K, k, l=1, existence=False, check=True):
             D = K.cyclotomic_cosets(x**((v-1)//(k-1)), [K.one()])
             D[0].insert(0,K.zero())
             if is_difference_family(K, D, v, k, l):
-                print("**  You found a new example of radical difference set **\n"\
-                      "**  for the parameters (v,k,l)=({},{},{}).            **\n"\
+                print("**  You found a new example of radical difference set **\n"
+                      "**  for the parameters (v,k,l)=({},{},{}).            **\n"
                       "**  Please contact sage-devel@googlegroups.com        **\n".format(v, k, l))
                 if existence:
                     return True

src/sage/combinat/finite_state_machine_generators.py

@@ -276,7 +276,7 @@ def Word(self, word, input_alphabet=None):
         letters = list(word)
         length = len(letters)
         from sage.rings.integer_ring import ZZ
-        return Automaton([(ZZ(i), ZZ(i+1), letter)
+        return Automaton([(ZZ(i), ZZ(i + 1), letter)
                           for i, letter in enumerate(letters)],
                          initial_states=[ZZ(0)],
                          final_states=[ZZ(length)],
@@ -1241,7 +1241,7 @@ def convert_output(output):
             for ring in output_rings:
                 try:
                     return ring(output)
-                except (ValueError,TypeError):
+                except (ValueError, TypeError):
                     pass
             return output
 
@@ -1353,12 +1353,12 @@ def to_list(output):
         assert equation == parsed_equation, \
             "Parsing of %s failed for unknown reasons." % (equation,)
 
-        rule = self.RecursionRule(K=K,r=r, k=k, s=s, t=to_list(t))
+        rule = self.RecursionRule(K=K, r=r, k=k, s=s, t=to_list(t))
         return rule
 
     def Recursion(self, recursions, base, function=None, var=None,
                   input_alphabet=None, word_function=None,
-                  is_zero=None,  output_rings=[ZZ, QQ]):
+                  is_zero=None, output_rings=[ZZ, QQ]):
         r"""
         Return a transducer realizing the given recursion when reading
         the digit expansion with base ``base``.
@@ -1815,11 +1815,11 @@ def Recursion(self, recursions, base, function=None, var=None,
 
         for given_rule in rules:
             q, remainder = given_rule.r.quo_rem(base**given_rule.K)
-            rule=self.RecursionRule(K=given_rule.K,
-                                    r=remainder,
-                                    k=given_rule.k,
-                                    s=given_rule.s - base**given_rule.k*q,
-                                    t=given_rule.t)
+            rule = self.RecursionRule(K=given_rule.K,
+                                      r=remainder,
+                                      k=given_rule.k,
+                                      s=given_rule.s - base**given_rule.k * q,
+                                      t=given_rule.t)
             for m in range(max_K - rule.K + 1):
                 for ell in range(base**m):
                     R = rule.r + base**rule.K * ell
@@ -1988,9 +1988,8 @@ def f(n):
                     "Too many initial conditions, only give one of %s." %
                     cycle[1:])
             required_initial_values.update(intersection)
-            output_sum = sum([edge[2]
-                              for edge in recursion_digraph.\
-                                  outgoing_edge_iterator(cycle[1:])],
+            output_sum = sum([e[2]
+                for e in recursion_digraph.outgoing_edge_iterator(cycle[1:])],
                              [])
             if not is_zero(output_sum):
                 raise ValueError(

src/sage/combinat/gelfand_tsetlin_patterns.py

@@ -993,7 +993,7 @@ def _row_iter(self, upper_row):
             [2, 0]
             [2, 1]
         """
-        row = [x-1 for x in upper_row[1:]]
+        row = [x - 1 for x in upper_row[1:]]
         row_len = len(row)
         pos = 0
         while pos >= 0:
@@ -1002,11 +1002,11 @@ def _row_iter(self, upper_row):
                 pos -= 1
                 continue
             # If it would create an invalid entry, backstep
-            if ( pos > 0 and (row[pos] >= row[pos-1] \
-                    or (self._strict and row[pos] == row[pos-1]-1)) ) \
+            if (pos > 0 and (row[pos] >= row[pos - 1]
+                    or (self._strict and row[pos] == row[pos - 1] - 1))) \
                     or row[pos] >= upper_row[pos] \
                     or (self._k is not None and row[pos] >= self._k):
-                row[pos] = upper_row[pos+1] - 1
+                row[pos] = upper_row[pos + 1] - 1
                 pos -= 1
                 continue
             row[pos] += 1

src/sage/combinat/integer_vector.py

@@ -1455,8 +1455,10 @@ def cardinality(self):
                     return Integer(binomial(self.n + self.k - 1, self.n))
                 # do by inclusion / exclusion on the number
                 # i of parts greater than m
-                return Integer(sum( (-1)**i * binomial(self.n+self.k-1-i*(m+1), self.k-1) \
-                    * binomial(self.k,i) for i in range(self.n/(m+1)+1) ))
+                n, k = self.n, self.k
+                return Integer(sum(
+                    (-1)**i * binomial(n + k - 1 - i * (m + 1), k - 1)
+                    * binomial(k, i) for i in range(self.n // (m + 1) + 1)))
         return ZZ.sum(ZZ.one() for x in self)
 
     def __iter__(self):

src/sage/combinat/multiset_partition_into_sets_ordered.py

@@ -2787,8 +2787,8 @@ def _base_iterator(constraints):
             if min_ord:
                 min_k = max(1, min_k, min_ord // len(A))
         if infinity not in (max_k, max_ord):
-            return chain(*(_iterator_order(A, ord, range(min_k, max_k+1)) \
-                        for ord in range(min_ord, max_ord+1)))
+            return chain(*(_iterator_order(A, ord, range(min_k, max_k + 1))
+                           for ord in range(min_ord, max_ord + 1)))
     # else
     return None
 

src/sage/combinat/ncsf_qsym/generic_basis_code.py

@@ -472,8 +472,9 @@ def skew(self, x, y, side='left'):
                 x = self(x)
                 y = self.dual()(y)
                 v = 1 if side == 'left' else 0
-                return self.sum(coeff * y[IJ[1-v]] * self[IJ[v]] \
-                                for (IJ, coeff) in x.coproduct() if IJ[1-v] in y.support())
+                return self.sum(coeff * y[IJ[1-v]] * self[IJ[v]]
+                                for (IJ, coeff) in x.coproduct()
+                                if IJ[1-v] in y.support())
             else:
                 return self._skew_by_coercion(x, y, side=side)
 
@@ -710,9 +711,9 @@ def duality_pairing_matrix(self, basis, degree):
             # TODO: generalize to keys indexing the basis of the graded component
             from sage.combinat.composition import Compositions
             return matrix(self.base_ring(),
-                    [[self.duality_pairing(self[I], basis[J]) \
-                            for J in Compositions(degree)] \
-                            for I in Compositions(degree)])
+                          [[self.duality_pairing(self[I], basis[J])
+                            for J in Compositions(degree)]
+                           for I in Compositions(degree)])
 
         def counit_on_basis(self, I):
             r"""

src/sage/combinat/rigged_configurations/tensor_product_kr_tableaux.py

@@ -320,12 +320,12 @@ def __init__(self, cartan_type, B):
         FullTensorProductOfRegularCrystals.__init__(self, tensor_prod, cartan_type=cartan_type)
         # This is needed to override the module_generators set in FullTensorProductOfRegularCrystals
         self.module_generators = HighestWeightTensorKRT(self)
-        self.rename("Tensor product of Kirillov-Reshetikhin tableaux of type %s and factor(s) %s"%(\
-          cartan_type, B))
+        self.rename("Tensor product of Kirillov-Reshetikhin tableaux "
+                    f"of type {cartan_type} and factor(s) {B}")
 
     def __iter__(self):
         """
-        Returns the iterator of ``self``.
+        Return the iterator of ``self``.
 
         EXAMPLES::
 

src/sage/combinat/root_system/root_lattice_realizations.py

@@ -464,7 +464,7 @@ def simple_roots(self):
         @cached_method
         def alpha(self):
             r"""
-            Returns the family `(\alpha_i)_{i\in I}` of the simple roots,
+            Return the family `(\alpha_i)_{i\in I}` of the simple roots,
             with the extra feature that, for simple irreducible root
             systems, `\alpha_0` yields the opposite of the highest root.
 
@@ -475,13 +475,12 @@ def alpha(self):
                 alpha[1]
                 sage: alpha[0]
                 -alpha[1] - alpha[2]
-
             """
             if self.root_system.is_finite() and self.root_system.is_irreducible():
-                return Family(self.index_set(), self.simple_root, \
-                              hidden_keys = [0], hidden_function = lambda i: - self.highest_root())
-            else:
-                return self.simple_roots()
+                return Family(self.index_set(), self.simple_root,
+                              hidden_keys=[0],
+                              hidden_function=lambda i: - self.highest_root())
+            return self.simple_roots()
 
         @cached_method
         def basic_imaginary_roots(self):

src/sage/combinat/sf/sfa.py

@@ -2090,19 +2090,20 @@ def _from_cache(self, element, cache_function, cache_dict, **subs_dict):
             #   needed for the old kschur functions - TCS
             part = _Partitions(part)
             for part2, c2 in cache_dict[sum(part)][part].items():
-                if hasattr(c2,'subs'): # c3 may be in the base ring
-                    c3 = c*BR(c2.subs(**subs_dict))
+                if hasattr(c2, 'subs'):  # c3 may be in the base ring
+                    c3 = c * BR(c2.subs(**subs_dict))
                 else:
-                    c3 = c*BR(c2)
+                    c3 = c * BR(c2)
                 # c3 = c*c2
                 # if hasattr(c3,'subs'): # c3 may be in the base ring
                 #     c3 = c3.subs(**subs_dict)
-                z_elt[ part2 ] = z_elt.get(part2, zero) + BR(c3)
+                z_elt[part2] = z_elt.get(part2, zero) + BR(c3)
         return self._from_dict(z_elt)
 
-    def _invert_morphism(self, n, base_ring, self_to_other_cache, other_to_self_cache,\
-                         to_other_function=None, to_self_function=None, \
-                         upper_triangular=False, lower_triangular=False, \
+    def _invert_morphism(self, n, base_ring,
+                         self_to_other_cache, other_to_self_cache,
+                         to_other_function=None, to_self_function=None,
+                         upper_triangular=False, lower_triangular=False,
                          ones_on_diagonal=False):
         r"""
         Compute the inverse of a morphism between ``self`` and ``other``
@@ -5979,8 +5980,9 @@ def character_to_frobenius_image(self, n):
             2*s[2, 2, 1] + s[3, 1, 1] + 4*s[3, 2] + 3*s[4, 1] + 2*s[5]
         """
         p = self.parent().symmetric_function_ring().p()
-        return self.parent()(p.sum(self.eval_at_permutation_roots(rho) \
-            *p(rho)/rho.centralizer_size() for rho in Partitions(n)))
+        return self.parent()(p.sum(self.eval_at_permutation_roots(rho)
+                                   * p(rho) / rho.centralizer_size()
+                                   for rho in Partitions(n)))
 
     def principal_specialization(self, n=infinity, q=None):
         r"""

src/sage/combinat/words/paths.py

@@ -455,10 +455,10 @@ def __eq__(self, other):
             sage: W1 == W3
             False
         """
-        return self is other or (type(self) == type(other) and \
-               self.alphabet() == other.alphabet() and \
-               self.vector_space() == other.vector_space() and \
-               self.letters_to_steps() == other.letters_to_steps())
+        return self is other or (type(self) == type(other) and
+            self.alphabet() == other.alphabet() and
+            self.vector_space() == other.vector_space() and
+            self.letters_to_steps() == other.letters_to_steps())
 
     def __ne__(self, other):
         r"""

src/sage/combinat/words/suffix_trees.py

@@ -286,14 +286,13 @@ def transition_function(self, node, word):
         if word.is_empty():
             return 0
         if word.length() == 1:
-            return self._transition_function[(node,word)]
-        else:
-            return self.transition_function( \
-                    self._transition_function[(node,word[0:1])], word[1:])
+            return self._transition_function[(node, word)]
+        return self.transition_function(
+            self._transition_function[(node, word[0:1])], word[1:])
 
     def states(self):
         r"""
-        Returns the states of the automaton defined by the suffix trie.
+        Return the states of the automaton defined by the suffix trie.
 
         EXAMPLES::
 

src/sage/combinat/words/word_generators.py

@@ -139,7 +139,7 @@ class LowerChristoffelWord(FiniteWord_list):
         word: 01
     """
 
-    def __init__(self, p, q, alphabet=(0,1), algorithm='cf'):
+    def __init__(self, p, q, alphabet=(0, 1), algorithm='cf'):
         r"""
         INPUT:
 
@@ -184,7 +184,7 @@ def __init__(self, p, q, alphabet=(0,1), algorithm='cf'):
         if len(set(alphabet)) != 2:
             raise ValueError("alphabet must contain exactly two distinct elements")
         # Compute gcd of p, q; raise TypeError if not 1.
-        if gcd(p,q) != 1:
+        if gcd(p, q) != 1:
             raise ValueError("%s and %s are not relatively prime" % (p, q))
         # Compute the Christoffel word
         if algorithm == 'linear':
@@ -194,7 +194,7 @@ def __init__(self, p, q, alphabet=(0,1), algorithm='cf'):
                 w = [alphabet[0]]
             else:
                 for i in range(p + q):
-                    v = (u+p) % (p+q)
+                    v = (u + p) % (p + q)
                     new_letter = alphabet[0] if u < v else alphabet[1]
                     w.append(new_letter)
                     u = v
@@ -208,7 +208,7 @@ def __init__(self, p, q, alphabet=(0,1), algorithm='cf'):
                 cf = QQ((p, q)).continued_fraction_list()
                 u = [alphabet[0]]
                 v = [alphabet[1]]
-                #do not consider the first zero if p < q
+                # do not consider the first zero if p < q
                 start = 1 if p < q else 0
                 for i in range(start, len(cf)-1):
                     if i % 2 == 0:
@@ -1276,9 +1276,9 @@ def StandardEpisturmianWord(self, directive_word):
         """
         if not isinstance(directive_word, Word_class):
             raise TypeError("directive_word is not a word, so it cannot be used to build an episturmian word")
-        epistandard = directive_word.parent()(\
-                self._StandardEpisturmianWord_LetterIterator(directive_word),
-                datatype='iter')
+        epistandard = directive_word.parent()(
+            self._StandardEpisturmianWord_LetterIterator(directive_word),
+            datatype='iter')
         return epistandard
 
     def _StandardEpisturmianWord_LetterIterator(self, directive_word):

src/sage/schemes/affine/affine_morphism.py

@@ -377,9 +377,12 @@ def _fastpolys(self):
             if self._is_prime_finite_field:
                 prime = polys[0].base_ring().characteristic()
                 degree = max(poly_numerator.degree(), poly_denominator.degree())
-                height = max([abs(c.lift()) for c in poly_numerator.coefficients()]\
-                              + [abs(c.lift()) for c in poly_denominator.coefficients()])
-                num_terms = max(len(poly_numerator.coefficients()), len(poly_denominator.coefficients()))
+                height = max([abs(c.lift())
+                              for c in poly_numerator.coefficients()]
+                             + [abs(c.lift())
+                                for c in poly_denominator.coefficients()])
+                num_terms = max(len(poly_numerator.coefficients()),
+                                len(poly_denominator.coefficients()))
                 largest_value = num_terms * height * (prime - 1) ** degree
                 # If the calculations will not overflow the float data type use domain float
                 # Else use domain integer

src/sage/schemes/affine/affine_space.py

@@ -598,18 +598,18 @@ def __mul__(self, right):
         if isinstance(right, AffineSpace_generic):
             if self is right:
                 return self.__pow__(2)
-            return AffineSpace(self.dimension_relative() + right.dimension_relative(),\
+            return AffineSpace(self.dimension_relative() + right.dimension_relative(),
                     self.base_ring(), self.variable_names() + right.variable_names())
         elif isinstance(right, AlgebraicScheme_subscheme):
-            AS = self*right.ambient_space()
+            AS = self * right.ambient_space()
             CR = AS.coordinate_ring()
             n = self.ambient_space().coordinate_ring().ngens()
 
             phi = self.ambient_space().coordinate_ring().hom(list(CR.gens()[:n]), CR)
             psi = right.ambient_space().coordinate_ring().hom(list(CR.gens()[n:]), CR)
             return AS.subscheme([phi(t) for t in self.defining_polynomials()] + [psi(t) for t in right.defining_polynomials()])
         else:
-            raise TypeError('%s must be an affine space or affine subscheme'%right)
+            raise TypeError('%s must be an affine space or affine subscheme' % right)
 
     def change_ring(self, R):
         r"""