sage.misc.misc, sage.combinat: Modularization fixes

src/sage/algebras/quantum_clifford.py

@@ -27,7 +27,7 @@
 from sage.rings.fraction_field import FractionField
 from sage.sets.finite_enumerated_set import FiniteEnumeratedSet
 from itertools import product
-from sage.misc.misc import powerset
+from sage.combinat.subset import powerset
 
 class QuantumCliffordAlgebra(CombinatorialFreeModule):
     r"""

src/sage/arith/misc.py

@@ -20,7 +20,6 @@
 import math
 from collections.abc import Iterable
 
-from sage.misc.misc import powerset
 from sage.misc.misc_c import prod
 
 from sage.structure.element import parent
@@ -6042,6 +6041,8 @@ def squarefree_divisors(x):
         sage: list(squarefree_divisors(mpz(12)))
         [1, 2, 3, 6]
     """
+    from sage.combinat.subset import powerset
+
     for a in powerset(prime_divisors(x)):
         yield prod(a, ZZ.one())
 

src/sage/categories/coxeter_groups.py

@@ -1327,7 +1327,10 @@ def _test_coxeter_relations(self, **options):
             one = self.one()
             for si in s:
                 tester.assertEqual(si**2, one)
-            cox_mat = self.coxeter_matrix()
+            try:
+                cox_mat = self.coxeter_matrix()
+            except ImportError:
+                return
             I = cox_mat.index_set()
             for ii, i in enumerate(I):
                 for j in I[ii + 1:]:

src/sage/categories/finite_coxeter_groups.py

@@ -10,6 +10,8 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
+import sage.rings.abc
+
 from sage.misc.cachefunc import cached_method, cached_in_parent_method
 from sage.misc.lazy_attribute import lazy_attribute
 from sage.categories.category_with_axiom import CategoryWithAxiom
@@ -746,13 +748,14 @@ def permutahedron(self, point=None, base_ring=None):
                 from sage.rings.integer_ring import ZZ
                 point = [ZZ.one()] * n
             v = sum(point[i-1] * weights[i] for i in weights.keys())
-            from sage.geometry.polyhedron.constructor import Polyhedron
-            from sage.rings.qqbar import AA, QQbar
-            from sage.rings.universal_cyclotomic_field import UniversalCyclotomicField
             vertices = [v*w for w in self]
-            if base_ring is None and v.base_ring() in [UniversalCyclotomicField(), QQbar]:
-                vertices = [v.change_ring(AA) for v in vertices]
-                base_ring = AA
+            if base_ring is None:
+                if isinstance(v.base_ring(), (sage.rings.abc.UniversalCyclotomicField,
+                                              sage.rings.abc.AlgebraicField_common)):
+                    from sage.rings.qqbar import AA
+                    vertices = [v.change_ring(AA) for v in vertices]
+                    base_ring = AA
+            from sage.geometry.polyhedron.constructor import Polyhedron
             return Polyhedron(vertices=vertices, base_ring=base_ring)
 
     class ElementMethods:

src/sage/categories/loop_crystals.py

@@ -968,7 +968,7 @@ def energy_function(self, algorithm=None):
                     ....:     for b in hw)
                     True
                 """
-                from sage.functions.other import ceil
+                from sage.arith.misc import integer_ceil as ceil
 
                 C = self.parent().crystals[0]
                 ell = ceil(C.s()/C.cartan_type().c()[C.r()])
@@ -1101,7 +1101,7 @@ def e_string_to_ground_state(self):
                     ....:     for elt in hw)
                     True
                 """
-                from sage.functions.other import ceil
+                from sage.arith.misc import integer_ceil as ceil
 
                 ell = max(ceil(K.s()/K.cartan_type().c()[K.r()])
                           for K in self.parent().crystals)

src/sage/combinat/all.py

@@ -218,7 +218,7 @@
 lazy_import('sage.combinat.multiset_partition_into_sets_ordered',
             ['OrderedMultisetPartitionIntoSets',
              'OrderedMultisetPartitionsIntoSets'])
-from .subset import Subsets
+from .subset import Subsets, subsets, powerset, uniq
 from .necklace import Necklaces
 lazy_import('sage.combinat.dyck_word', ('DyckWords', 'DyckWord'))
 lazy_import('sage.combinat.nu_dyck_word', ('NuDyckWords', 'NuDyckWord'))

src/sage/combinat/blob_algebra.py

@@ -23,7 +23,7 @@
 from sage.structure.richcmp import richcmp
 #from sage.misc.inherit_comparison import InheritComparisonClasscallMetaclass
 from sage.misc.cachefunc import cached_method
-from sage.misc.misc import powerset
+from sage.combinat.subset import powerset
 from sage.arith.misc import binomial
 from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
 from sage.categories.algebras import Algebras

src/sage/combinat/combinat.py

@@ -64,18 +64,12 @@
 
 **Implemented in other modules (listed for completeness):**
 
-The ``sage.arith.all`` module contains the following
+The package :mod:`sage.arith` contains the following
 combinatorial functions:
 
--  binomial the binomial coefficient (wrapped from PARI)
+-  :func:`binomial` the binomial coefficient (wrapped from PARI)
 
--  factorial (wrapped from PARI)
-
--  partition (from the Python Cookbook) Generator of the list of
-   all the partitions of the integer `n`.
-
--  :func:`number_of_partitions` (wrapped from PARI) the
-   *number* of partitions:
+-  :func:`factorial` (wrapped from PARI)
 
 -  :func:`falling_factorial` Definition: for integer
    `a \ge 0` we have `x(x-1) \cdots (x-a+1)`. In all
@@ -87,7 +81,12 @@
    other cases we use the GAMMA-function:
    `\frac {\Gamma(x+a)} {\Gamma(x)}`.
 
--  gaussian_binomial the gaussian binomial
+From other modules:
+
+-  :func:`number_of_partitions` (wrapped from PARI) the
+   *number* of partitions:
+
+-  :func:`sage.combinat.q_analogues.gaussian_binomial` the Gaussian binomial
 
 .. MATH::
 
@@ -174,8 +173,6 @@
 from sage.rings.infinity import infinity
 from sage.rings.polynomial.polynomial_element import Polynomial
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.libs.pari.all import pari
-from sage.misc.prandom import randint
 from sage.misc.misc_c import prod
 from sage.misc.cachefunc import cached_function
 from sage.structure.sage_object import SageObject
@@ -187,7 +184,10 @@
 from sage.misc.classcall_metaclass import ClasscallMetaclass
 from sage.misc.inherit_comparison import InheritComparisonClasscallMetaclass
 from sage.structure.element import Element
+
 lazy_import('sage.interfaces.maxima_lib', 'maxima')
+lazy_import('sage.libs.pari.all', 'pari')
+lazy_import('sage.misc.prandom', 'randint')
 
 
 def bell_number(n, algorithm='flint', **options) -> Integer:
@@ -2892,7 +2892,7 @@ def unshuffle_iterator(a, one=1) -> Iterator:
         [(((), (3, 1)), 3/2), (((3,), (1,)), 3/2), (((1,), (3,)), -3/2),
          (((3, 1), ()), 3/2)]
     """
-    from sage.misc.misc import powerset
+    from sage.combinat.subset import powerset
     n = len(a)
     for I in powerset(range(n)):
         sorted_I = tuple(sorted(I))

src/sage/combinat/designs/bibd.py

@@ -50,12 +50,16 @@
 ---------
 """
 
+from sage.arith.misc import binomial, is_prime_power, is_square
 from sage.categories.sets_cat import EmptySetError
+from sage.misc.lazy_import import lazy_import
 from sage.misc.unknown import Unknown
+
 from .design_catalog import transversal_design  # type:ignore
-from sage.arith.misc import binomial, is_prime_power
-from .group_divisible_designs import GroupDivisibleDesign
 from .designs_pyx import is_pairwise_balanced_design
+from .group_divisible_designs import GroupDivisibleDesign
+
+lazy_import('sage.schemes.plane_conics.constructor', 'Conic')
 
 
 def biplane(n, existence=False):
@@ -413,8 +417,6 @@ def BruckRyserChowla_check(v, k, lambd):
         True
 
     """
-    from sage.arith.misc import is_square
-    from sage.schemes.plane_conics.constructor import Conic
     from sage.rings.rational_field import QQ
 
     # design is not symmetric

src/sage/combinat/designs/difference_family.py

@@ -1530,7 +1530,6 @@ def relative_difference_set_from_homomorphism(q, N, d, check=True, return_group=
         return G2, second_diff_set
     return second_diff_set
 
-
 def is_relative_difference_set(R, G, H, params, verbose=False):
     r"""
     Check if ``R`` is a difference set of ``G`` relative to ``H``, with the given parameters.
@@ -2040,9 +2039,6 @@ def skew_supplementary_difference_set_over_polynomial_ring(n, existence=False, c
         ...
         NotImplementedError: skew SDS of order 7 not yet implemented
     """
-    from sage.symbolic.ring import SymbolicRing
-    from sage.rings.finite_rings.integer_mod_ring import Zmod
-
     data = {
         81: (3, lambda x: x**4 - x**3 - 1, 16, 5,
              [1, 2, 4, 6, 8, 10, 12, 14], [1, 2, 3, 4, 10, 11, 13],
@@ -2060,9 +2056,11 @@ def skew_supplementary_difference_set_over_polynomial_ring(n, existence=False, c
 
     mod, poly, exp, order, ind1, ind2, ind3, ind4 = data[n]
 
+    from sage.rings.finite_rings.integer_mod_ring import Zmod
+
     Z3 = Zmod(mod)
-    R = SymbolicRing()
-    x = R.var('x')
+    R = ZZ['x']
+    x = R.gen()
     F = Z3.extension(poly(x))
 
     H = [F.gen() ** (exp * i) for i in range(order)]

src/sage/combinat/designs/incidence_structures.py

@@ -39,10 +39,13 @@
 #                    https://www.gnu.org/licenses/                          #
 # **************************************************************************
 from __future__ import annotations
-from sage.rings.integer import Integer
+
 from sage.misc.latex import latex
+from sage.misc.lazy_import import lazy_import
+from sage.rings.integer import Integer
 from sage.sets.set import Set
-from sage.libs.gap.libgap import libgap
+
+lazy_import('sage.libs.gap.libgap', 'libgap')
 
 
 class IncidenceStructure():
@@ -173,7 +176,6 @@ def __init__(self, points=None, blocks=None, incidence_matrix=None,
             sage: IncidenceStructure([])
             Incidence structure with 0 points and 0 blocks
         """
-        from sage.matrix.constructor import matrix
         from sage.structure.element import Matrix
 
         # Reformatting input
@@ -189,6 +191,7 @@ def __init__(self, points=None, blocks=None, incidence_matrix=None,
             assert incidence_matrix is None, "'incidence_matrix' cannot be defined when 'points' is defined"
 
         if incidence_matrix:
+            from sage.matrix.constructor import matrix
             M = matrix(incidence_matrix)
             v = M.nrows()
             self._points = list(range(v))

src/sage/combinat/diagram.py

@@ -615,7 +615,7 @@ def __iter__(self):
         """
         from sage.sets.non_negative_integers import NonNegativeIntegers
         from sage.categories.cartesian_product import cartesian_product
-        from sage.misc.misc import subsets
+        from sage.combinat.subset import subsets
         # the product of positive integers automatically implements an
         # an enumeration which allows us to get out of the first column
         N = NonNegativeIntegers()

src/sage/combinat/diagram_algebras.py

@@ -231,7 +231,7 @@ def planar_partitions_rec(X):
         if len(X) > 1:
             yield ([X[0]], [X[1]])
         return
-    from sage.misc.misc import powerset
+    from sage.combinat.subset import powerset
     from itertools import product
     for S in powerset(range(len(X)-1)):
         if not S:

src/sage/combinat/integer_vector_weighted.py

@@ -20,13 +20,15 @@
 from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
 from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
 from sage.categories.sets_with_grading import SetsWithGrading
+from sage.misc.lazy_import import lazy_import
 from sage.sets.disjoint_union_enumerated_sets import DisjointUnionEnumeratedSets
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
 from sage.combinat.integer_vector import IntegerVector
-from sage.combinat.words.word import Word
 from sage.combinat.permutation import Permutation
 
+lazy_import('sage.combinat.words.word', 'Word')
+
 
 class WeightedIntegerVectors(Parent, UniqueRepresentation):
     r"""

src/sage/combinat/interval_posets.py

@@ -3042,7 +3042,13 @@ def final_forest(element) -> TIP:
         """
         if isinstance(element, TamariIntervalPoset):
             return element.final_forest()
-        if element in DyckWords():
+
+        try:
+            DW = DyckWords()
+        except ImportError:
+            DW = ()
+
+        if element in DW:
             binary_tree = element.to_binary_tree_tamari()
         elif element in BinaryTrees() or element in LabelledBinaryTrees():
             binary_tree = element
@@ -3149,7 +3155,13 @@ def initial_forest(element) -> TIP:
         """
         if isinstance(element, TamariIntervalPoset):
             return element.initial_forest()
-        if element in DyckWords():
+
+        try:
+            DW = DyckWords()
+        except ImportError:
+            DW = ()
+
+        if element in DW:
             binary_tree = element.to_binary_tree_tamari()
         elif element in BinaryTrees() or element in LabelledBinaryTrees():
             binary_tree = element