a few details in combinat (designs)

Author: fchapoton

src/sage/combinat/designs/group_divisible_designs.py

@@ -24,19 +24,20 @@
 ---------
 """
 
-#*****************************************************************************
+# ****************************************************************************
 # This program is free software: you can redistribute it and/or modify
 # it under the terms of the GNU General Public License as published by
 # the Free Software Foundation, either version 2 of the License, or
 # (at your option) any later version.
-#                  http://www.gnu.org/licenses/
-#*****************************************************************************
+#                  https://www.gnu.org/licenses/
+# ****************************************************************************
 
 from sage.arith.misc import is_prime_power
-from sage.misc.unknown    import Unknown
+from sage.misc.unknown import Unknown
 from .incidence_structures import IncidenceStructure
 
-def group_divisible_design(v,K,G,existence=False,check=False):
+
+def group_divisible_design(v, K, G, existence=False, check=False):
     r"""
     Return a `(v,K,G)`-Group Divisible Design.
 
@@ -90,36 +91,31 @@ def group_divisible_design(v,K,G,existence=False,check=False):
     blocks = None
 
     # from a (v+1,k,1)-BIBD
-    if (len(G) == 1 and
-        len(K) == 1 and
-        G[0]+1 in K):
+    if len(G) == 1 == len(K) and G[0] + 1 in K:
         from .bibd import balanced_incomplete_block_design
         k = K[0]
         if existence:
-            return balanced_incomplete_block_design(v+1,k,existence=True)
-        BIBD = balanced_incomplete_block_design(v+1,k)
+            return balanced_incomplete_block_design(v + 1, k, existence=True)
+        BIBD = balanced_incomplete_block_design(v + 1, k)
         groups = [[x for x in S if x != v] for S in BIBD if v in S]
-        d = {p:i for i,p in enumerate(sum(groups,[]))}
+        d = {p: i for i, p in enumerate(sum(groups, []))}
         d[v] = v
         BIBD.relabel(d)
-        groups = [list(range((k-1)*i,(k-1)*(i+1))) for i in range(v//(k-1))]
+        groups = [list(range((k - 1) * i, (k - 1) * (i + 1)))
+                  for i in range(v // (k - 1))]
         blocks = [S for S in BIBD if v not in S]
 
     # (v,{4},{2})-GDD
-    elif (v % 2 == 0   and
-          K == [4] and
-          G == [2] and
-          GDD_4_2(v//2,existence=True)):
+    elif (v % 2 == 0 and K == [4] and
+          G == [2] and GDD_4_2(v // 2, existence=True)):
         if existence:
             return True
-        return GDD_4_2(v//2,check=check)
+        return GDD_4_2(v // 2, check=check)
 
     # From a TD(k,g)
-    elif (len(G) == 1 and
-          len(K) == 1 and
-          K[0]*G[0] == v):
+    elif (len(G) == 1 == len(K) and K[0] * G[0] == v):
         from .orthogonal_arrays import transversal_design
-        return transversal_design(k=K[0],n=G[0],existence=existence)
+        return transversal_design(k=K[0], n=G[0], existence=existence)
 
     if blocks:
         return GroupDivisibleDesign(v,
@@ -134,7 +130,8 @@ def group_divisible_design(v,K,G,existence=False,check=False):
         return Unknown
     raise NotImplementedError
 
-def GDD_4_2(q,existence=False,check=True):
+
+def GDD_4_2(q, existence=False, check=True):
     r"""
     Return a `(2q,\{4\},\{2\})`-GDD for `q` a prime power with `q\equiv 1\pmod{6}`.
 
@@ -180,26 +177,27 @@ def GDD_4_2(q,existence=False,check=True):
         return True
 
     from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
-    G = GF(q,'x')
+    G = GF(q, 'x')
     w = G.primitive_element()
     e = w**((q - 1) // 3)
 
     # A first parallel class is defined. G acts on it, which yields all others.
-    first_class = [[(0,0),(1,w**i),(1,e*w**i),(1,e*e*w**i)]
+    first_class = [[(0, 0), (1, w**i), (1, e * w**i), (1, e * e * w**i)]
                    for i in range((q - 1) // 6)]
 
-    label = {p:i for i,p in enumerate(G)}
-    classes = [[[2*label[x[1]+g]+(x[0]+j) % 2 for x in S]
+    label = {p: i for i, p in enumerate(G)}
+    classes = [[[2 * label[x[1] + g] + (x[0] + j) % 2 for x in S]
                 for S in first_class]
                for g in G for j in range(2)]
 
-    return GroupDivisibleDesign(2*q,
-                                groups=[[i,i+1] for i in range(0,2*q,2)],
-                                blocks=sum(classes,[]),
-                                K=[4],
-                                G=[2],
-                                check=check,
-                                copy=False)
+    return GroupDivisibleDesign(2 * q,
+        groups=[[i, i + 1] for i in range(0, 2 * q, 2)],
+        blocks=sum(classes, []),
+        K=[4],
+        G=[2],
+        check=check,
+        copy=False)
+
 
 class GroupDivisibleDesign(IncidenceStructure):
     r"""
@@ -261,9 +259,9 @@ class GroupDivisibleDesign(IncidenceStructure):
         sage: GDD = GroupDivisibleDesign('abcdefghiklm',None,D)
         sage: sorted(GDD.groups())
         [['a', 'b', 'c'], ['d', 'e', 'f'], ['g', 'h', 'i'], ['k', 'l', 'm']]
-
     """
-    def __init__(self, points, groups, blocks, G=None, K=None, lambd=1, check=True, copy=True,**kwds):
+    def __init__(self, points, groups, blocks, G=None, K=None, lambd=1,
+                 check=True, copy=True, **kwds):
         r"""
         Constructor function
 
@@ -286,16 +284,17 @@ def __init__(self, points, groups, blocks, G=None, K=None, lambd=1, check=True,
                                     check=False,
                                     **kwds)
 
-        if (groups is None or
-            (copy is False and self._point_to_index is None)):
+        if (groups is None or (copy is False and self._point_to_index is None)):
             self._groups = groups
         elif self._point_to_index is None:
             self._groups = [g[:] for g in groups]
         else:
             self._groups = [[self._point_to_index[x] for x in g] for g in groups]
 
         if check or groups is None:
-            is_gdd = is_group_divisible_design(self._groups,self._blocks,self.num_points(),G,K,lambd,verbose=1)
+            is_gdd = is_group_divisible_design(self._groups, self._blocks,
+                                               self.num_points(), G, K,
+                                               lambd, verbose=1)
             assert is_gdd
             if groups is None:
                 self._groups = is_gdd[1]
@@ -337,7 +336,7 @@ def groups(self):
 
     def __repr__(self):
         r"""
-        Returns a string that describes self
+        Return a string that describes ``self``.
 
         EXAMPLES::
 
@@ -347,16 +346,15 @@ def __repr__(self):
             sage: GDD = GroupDivisibleDesign(40,groups,TD); GDD
             Group Divisible Design on 40 points of type 10^4
         """
+        group_sizes = [len(g) for g in self._groups]
 
-        group_sizes = [len(_) for _ in self._groups]
-
-        gdd_type = ["{}^{}".format(s,group_sizes.count(s))
-                    for s in sorted(set(group_sizes))]
+        gdd_type = ("{}^{}".format(s, group_sizes.count(s))
+                    for s in sorted(set(group_sizes)))
         gdd_type = ".".join(gdd_type)
 
         if not gdd_type:
             gdd_type = "1^0"
 
         v = self.num_points()
 
-        return "Group Divisible Design on {} points of type {}".format(v,gdd_type)
+        return "Group Divisible Design on {} points of type {}".format(v, gdd_type)

src/sage/combinat/designs/resolvable_bibd.py

@@ -140,6 +140,7 @@ def resolvable_balanced_incomplete_block_design(v,k,existence=False):
             return Unknown
         raise NotImplementedError("I don't know how to build a ({},{},1)-RBIBD!".format(v,3))
 
+
 def kirkman_triple_system(v,existence=False):
     r"""
     Return a Kirkman Triple System on `v` points.
@@ -437,6 +438,7 @@ def v_4_1_rbibd(v,existence=False):
     assert BIBD.is_resolvable()
     return BIBD
 
+
 def PBD_4_7(v,check=True, existence=False):
     r"""
     Return a `(v,\{4,7\})`-PBD
@@ -683,6 +685,7 @@ def PBD_4_7(v,check=True, existence=False):
                                   check=check,
                                   copy=False)
 
+
 def PBD_4_7_from_Y(gdd,check=True):
     r"""
     Return a `(3v+1,\{4,7\})`-PBD from a `(v,\{4,5,7\},\NN-\{3,6,10\})`-GDD.

src/sage/combinat/designs/twographs.py

@@ -59,6 +59,7 @@
 from sage.combinat.designs.incidence_structures import IncidenceStructure
 from itertools import combinations
 
+
 class TwoGraph(IncidenceStructure):
     r"""
     Two-graphs class.
@@ -199,9 +200,10 @@ def taylor_twograph(q):
     from sage.graphs.generators.classical_geometries import TaylorTwographSRG
     return TaylorTwographSRG(q).twograph()
 
-def is_twograph(T):
+
+def is_twograph(T) -> bool:
     r"""
-    Checks that the incidence system `T` is a two-graph
+    Check whether the incidence system `T` is a two-graph.
 
     INPUT:
 
@@ -237,7 +239,7 @@ def is_twograph(T):
         return False
 
     # A structure for a fast triple existence check
-    v_to_blocks = {v:set() for v in range(T.num_points())}
+    v_to_blocks = {v: set() for v in range(T.num_points())}
     for B in T._blocks:
         B = frozenset(B)
         for x in B:
@@ -248,8 +250,8 @@ def has_triple(x_y_z):
         return bool(v_to_blocks[x] & v_to_blocks[y] & v_to_blocks[z])
 
     # Check that every quadruple contains an even number of triples
-    for quad in combinations(range(T.num_points()),4):
-        if sum(map(has_triple,combinations(quad,3))) % 2 == 1:
+    for quad in combinations(range(T.num_points()), 4):
+        if sum(map(has_triple, combinations(quad, 3))) % 2 == 1:
             return False
 
     return True
@@ -296,10 +298,10 @@ def twograph_descendant(G, v, name=None):
         sage: twograph_descendant(p, 5, name=True)
         descendant of Petersen graph at 5: Graph on 9 vertices
     """
-    G = G.seidel_switching(G.neighbors(v),inplace=False)
+    G = G.seidel_switching(G.neighbors(v), inplace=False)
     G.delete_vertex(v)
     if name:
-        G.name('descendant of '+G.name()+' at '+str(v))
+        G.name('descendant of ' + G.name() + ' at ' + str(v))
     else:
         G.name('')
     return G

src/sage/combinat/matrices/dlxcpp.py

@@ -1,7 +1,7 @@
 """
 Dancing links C++ wrapper
 """
-#*****************************************************************************
+# ****************************************************************************
 #       Copyright (C) 2008 Carlo Hamalainen <[email protected]>,
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
@@ -13,14 +13,15 @@
 #
 #  The full text of the GPL is available at:
 #
-#                  http://www.gnu.org/licenses/
-#*****************************************************************************
+#                  https://www.gnu.org/licenses/
+# ****************************************************************************
 
 # OneExactCover and AllExactCovers are almost exact copies of the
 # functions with the same name in sage/combinat/dlx.py by Tom Boothby.
 
 from .dancing_links import dlx_solver
 
+
 def DLXCPP(rows):
     """
     Solves the Exact Cover problem by using the Dancing Links algorithm
@@ -85,6 +86,7 @@ def DLXCPP(rows):
     while x.search():
         yield x.get_solution()
 
+
 def AllExactCovers(M):
     """
     Solves the exact cover problem on the matrix M (treated as a dense
@@ -112,6 +114,7 @@ def AllExactCovers(M):
     for s in DLXCPP(rows):
         yield [M.row(i) for i in s]
 
+
 def OneExactCover(M):
     """
     Solves the exact cover problem on the matrix M (treated as a dense

src/sage/combinat/tuple.py

@@ -15,13 +15,14 @@
 #
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
+from itertools import product, combinations_with_replacement
 
 from sage.arith.misc import binomial
+from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
 from sage.rings.integer_ring import ZZ
 from sage.structure.parent import Parent
 from sage.structure.unique_representation import UniqueRepresentation
-from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
-from itertools import product, combinations_with_replacement
+
 
 class Tuples(Parent, UniqueRepresentation):
     """