def coproduct_on_basis(self, x):
r"""
Return the coproduct of `F_{\sigma}` for `\sigma` a permutation.
EXAMPLES::
sage: A = algebras.FQSym(QQ).F()
sage: x = A([1])
sage: ascii_art(A.coproduct(A.one())) # indirect doctest
1 # 1
sage: ascii_art(A.coproduct(x)) # indirect doctest
1 # F + F # 1
[1] [1]
sage: A = algebras.FQSym(QQ).F()
sage: x, y, z = A([1]), A([2,1]), A([3,2,1])
sage: A.coproduct(z)
F[] # F[3, 2, 1] + F[1] # F[2, 1] + F[2, 1] # F[1]
+ F[3, 2, 1] # F[]
"""
if not len(x):
return self.one().tensor(self.one())
return sum(
self(Word(x[:i]).standard_permutation()).tensor(
self(Word(x[i:]).standard_permutation()))
for i in range(len(x) + 1))
def __iter__(self):
"""
TESTS::
sage: WeightedIntegerVectors(7, [2,2]).list()
[]
sage: WeightedIntegerVectors(3, [2,1,1]).list()
[[1, 0, 1], [1, 1, 0], [0, 0, 3], [0, 1, 2], [0, 2, 1], [0, 3, 0]]
::
sage: ivw = [ WeightedIntegerVectors(k, [1,1,1]) for k in range(11) ]
sage: iv = [ IntegerVectors(k, 3) for k in range(11) ]
sage: all( [ sorted(iv[k].list()) == sorted(ivw[k].list()) for k in range(11) ] )
True
::
sage: ivw = [ WeightedIntegerVectors(k, [2,3,7]) for k in range(11) ]
sage: all( [ i.cardinality() == len(i.list()) for i in ivw] )
True
"""
if len(self._weights) == 0:
if self._n == 0:
yield []
return
perm = Word(self._weights).standard_permutation()
l = [x for x in sorted(self._weights)]
for x in self._recfun(self._n, l):
yield perm.action(x)
def to_word(self, use_str=True, upper_case_as_inverse=True):
"""
Convert ``self`` to a word.
A free group element is a reduced words in the generators and their
inverses. This is naturally a finite word. Some choices have to be
done:
1. ``use_str==True``: the letters of the words are strings
(possibly a single character, but not necessarily).
``use_str==False``: the letters of the word are the generators
and their inverses themselves, that is to say the letters are
FreeGroupElement.
2. ``upper_case_as_inverse==True``: the inverse of a generator is
written as the same string in upper case. This is used only with
the ``use_string==True`` option.
INPUT:
- ``use_str`` -- (default: ``True``) use strings rather than
:class:`FreeGroupElement` as letters
- ``upper_case_as_inverse`` -- (default: ``True``) use upper case
letters as inverses
OUTPUT:
A finite :class:`~sage.combinat.words.finite_word.FiniteWord_class`.
EXAMPLES::
sage: from train_track import *
sage: F = FreeGroup(3)
sage: w = F([1,-2,1,3,-1])
sage: w.to_word()
word: x0,X1,x0,x2,X0
sage: type(w.to_word()[1])
<... 'str'>
sage: w.to_word(use_str=False)
word: x0,x1^-1,x0,x2,x0^-1
sage: w.to_word(use_str=False)[1] == w[1]
True
"""
from sage.combinat.words.word import Word
if use_str and upper_case_as_inverse:
wt = self.Tietze()
A = self.parent().variable_names()
w = []
for a in wt:
if a > 0:
w.append(A[a - 1])
else:
w.append(A[-a - 1].upper())
return Word(w)
if use_str:
return Word([str(a) for a in self])
return Word(list(self))
def spanning_tree(self, verbose=False):
"""
A spanning tree.
OUPUT:
a dictionnary that maps each vertex to an edge-path from the origin vertex.
SEE ALSO:
``maximal_tree()`` that returns a list of edges of a spanning tree.
WARNING:
``self`` must be connected.
"""
A = self._alphabet
tree = {self.initial_vertex(A[0]): Word()}
done = False
while not done:
done = True
for a in A.positive_letters():
vi = self.initial_vertex(a)
vt = self.terminal_vertex(a)
if vi in tree and vt not in tree:
tree[vt] = self.reduce_path(tree[vi] * Word([a]))
done = False
elif vt in tree and vi not in tree:
tree[vi] = self.reduce_path(tree[vt] *
Word([A.inverse_letter(a)]))
done = False
return tree
def _create_rep_gen_set(self, n, p):
"""
A helper function. Creates the gen_set for has_irred_rep().
"""
import itertools
alphabet = [_to_word(self.gen(i)) for i in range(self.ngens())]
power_word = [alphabet[i]**n for i in range(self.ngens())]
for i in range(1, p + 1):
for words in itertools.product(alphabet, repeat=i):
yield_word = True
word = Word('')
for w in words:
word = word * w
rewrite = []
empty_word = Word('')
for rel in self._reduce:
if rel[0].is_factor(word):
yield_word = False
break
if yield_word:
for pw in power_word:
if pw.is_factor(word):
yield_word = False
break
if yield_word: yield word
def subdivide(self, edge_list):
"""
Subdvides each of the edges in ``edge_list`` into two edges.
An edge may appear several time in the list: it will be
subdivided as many times.
OUTPUT:
A dictionnary that maps an old edge to a path in the new
graph.
"""
A = self._alphabet
result_map = dict((e, Word([e])) for e in A)
new_edges = A.add_new_letters(len(edge_list))
new_vertices = self.new_vertices(len(edge_list))
for i, e in enumerate(edge_list):
ee = A.inverse_letter(e)
ne = result_map[e][-1]
v = new_vertices[i]
vt = self.terminal_vertex(result_map[e][-1])
f = new_edges[i][0]
ff = new_edges[i][1]
self.set_terminal_vertex(ne, v)
self.add_edge(v, vt, [f, ff])
result_map[e] = result_map[e] * Word([f])
result_map[ee] = Word([ff]) * result_map[ee]
return result_map
def __iter__(self):
"""
TESTS::
sage: WeightedIntegerVectors(7, [2,2]).list()
[]
sage: WeightedIntegerVectors(3, [2,1,1]).list()
[[1, 0, 1], [1, 1, 0], [0, 0, 3], [0, 1, 2], [0, 2, 1], [0, 3, 0]]
::
sage: ivw = [ WeightedIntegerVectors(k, [1,1,1]) for k in range(11) ]
sage: iv = [ IntegerVectors(k, 3) for k in range(11) ]
sage: all( [ sorted(iv[k].list()) == sorted(ivw[k].list()) for k in range(11) ] )
True
::
sage: ivw = [ WeightedIntegerVectors(k, [2,3,7]) for k in range(11) ]
sage: all( [ i.cardinality() == len(i.list()) for i in ivw] )
True
"""
if len(self._weights) == 0:
if self._n == 0:
yield []
return
perm = Word(self._weights).standard_permutation()
l = [x for x in sorted(self._weights)]
for x in self._recfun(self._n, l):
yield perm.action(x)
def rips_machine_moves(self,side=None,holes=None,orientation=None,verbose=False):
r"""
Return the list of possible moves of the Rips machine.
A move is an automorphism of the free group. Assume that self
is the convex core of an automorphism phi. Let psi be an
automorphism given by ``self.rips_machine_moces(side=1)``,
then ``ConvexCore(phi*psi)`` is obtained from ``self`` by a
move of the Rips machine. On the other side: if ``psi`` is
given by ``self.rips_machine_moces(side=0)``, then
``ConvexCore(psi.inverse()*phi)`` is given by a move of the
Rips machine.
INPUT:
- `side` (default 0): side of Rips moves to find, either 0, 1
or None for both sides.
OUPUT:
- A list of free group automorphisms.
"""
if holes is None:
holes=self.rips_machine_holes(side=side,orientation=orientation,verbose=verbose and verbose>1 and verbose-1)
result=[]
for i,hole in enumerate(holes):
e,b,start_letters,end_letters=hole
A=self.tree(side=b[1]).alphabet()
hole_map=dict((a,Word([a])) for a in A.positive_letters())
bb=A.inverse_letter(b[0])
if bb in start_letters:
if verbose:
print i,":",hole
print "surgery:",e[2],",",(bb,b[1])
for x in end_letters:
if A.is_positive_letter(x):
hole_map[x]=Word([bb])*hole_map[x]
else:
xx=A.inverse_letter(x)
hole_map[xx]=hole_map[xx]*Word([b[0]])
else:
if verbose:
print i,":",hole
print "surgery:",e[2],",",b
for x in start_letters:
if A.is_positive_letter(x):
hole_map[x]=Word([bb])*hole_map[x]
else:
xx=A.inverse_letter(x)
hole_map[xx]=hole_map[xx]*Word([b[0]])
result.append(hole_map)
return result
def list(self):
"""
TESTS::
sage: WeightedIntegerVectors(7, [2,2]).list()
[]
sage: WeightedIntegerVectors(3, [2,1,1]).list()
[[1, 0, 1], [1, 1, 0], [0, 0, 3], [0, 1, 2], [0, 2, 1], [0, 3, 0]]
::
sage: ivw = [ WeightedIntegerVectors(k, [1,1,1]) for k in range(11) ]
sage: iv = [ IntegerVectors(k, 3) for k in range(11) ]
sage: all( [ sorted(iv[k].list()) == sorted(ivw[k].list()) for k in range(11) ] )
True
::
sage: ivw = [ WeightedIntegerVectors(k, [2,3,7]) for k in range(11) ]
sage: all( [ i.cardinality() == len(i.list()) for i in ivw] )
True
"""
if len(self.weight) == 0:
if self.n == 0:
return [[]]
else:
return []
perm = Word(self.weight).standard_permutation()
l = [x for x in sorted(self.weight)]
return [perm.action(_) for _ in self._recfun(self.n, l)]
def list(self):
"""
TESTS::
sage: WeightedIntegerVectors(7, [2,2]).list()
[]
sage: WeightedIntegerVectors(3, [2,1,1]).list()
[[1, 0, 1], [1, 1, 0], [0, 0, 3], [0, 1, 2], [0, 2, 1], [0, 3, 0]]
::
sage: ivw = [ WeightedIntegerVectors(k, [1,1,1]) for k in range(11) ]
sage: iv = [ IntegerVectors(k, 3) for k in range(11) ]
sage: all( [ sorted(iv[k].list()) == sorted(ivw[k].list()) for k in range(11) ] )
True
::
sage: ivw = [ WeightedIntegerVectors(k, [2,3,7]) for k in range(11) ]
sage: all( [ i.cardinality() == len(i.list()) for i in ivw] )
True
"""
if len(self.weight) == 0:
if self.n == 0:
return [[]]
else:
return []
perm = Word(self.weight).standard_permutation()
l = [x for x in sorted(self.weight)]
return [perm.action(_) for _ in self._recfun(self.n,l)]
def blow_up_vertices(self, germ_components):
"""
Blow-up ``self`` according to classes of germs given in
``germ_components``.
INPUT:
``germ_components`` a list of classes of germs outgoing from a
vertex.
OUTPUT:
A dictionnay that maps an old edge to the path in the new
graph.
"""
A = self.alphabet()
result_map = dict((a, Word([a])) for a in A)
new_vertices = self.new_vertices(len(germ_components))
new_edges = A.add_new_letters(len(germ_components))
for i, c in enumerate(germ_components):
v0 = self.initial_vertex(c[0])
vc = new_vertices[i]
ec = new_edges[i]
self.add_edge(v0, vc, ec)
for a in c:
self.set_initial_vertex(a, vc)
result_map[a] = Word([ec[0]]) * result_map[a]
aa = A.inverse_letter(a)
result_map[aa] = result_map[aa] * Word([ec[1]])
return result_map
def spin_rec(t, nexts, current, part, weight, length):
"""
Routine used for constructing the spin polynomial.
INPUT:
- ``weight`` - list of non-negative integers
- ``length`` - the length of the ribbons we're tiling
with
- ``t`` - the variable
EXAMPLES::
sage: from sage.combinat.ribbon_tableau import spin_rec
sage: sp = SkewPartition
sage: t = ZZ['t'].gen()
sage: spin_rec(t, [], [[[], [3, 3]]], sp([[2, 2, 2], []]), [2], 3)
[t^4]
sage: spin_rec(t, [[0], [t^4]], [[[2, 1, 1, 1, 1], [0, 3]], [[2, 2, 2], [3, 0]]], sp([[2, 2, 2, 2, 1], []]), [2, 1], 3)
[t^5]
sage: spin_rec(t, [], [[[], [3, 3, 0]]], sp([[3, 3], []]), [2], 3)
[t^2]
sage: spin_rec(t, [[t^4], [t^3], [t^2]], [[[2, 2, 2], [0, 0, 3]], [[3, 2, 1], [0, 3, 0]], [[3, 3], [3, 0, 0]]], sp([[3, 3, 3], []]), [2, 1], 3)
[t^6 + t^4 + t^2]
sage: spin_rec(t, [[t^5], [t^4], [t^6 + t^4 + t^2]], [[[2, 2, 2, 2, 1], [0, 0, 3]], [[3, 3, 1, 1, 1], [0, 3, 0]], [[3, 3, 3], [3, 0, 0]]], sp([[3, 3, 3, 2, 1], []]), [2, 1, 1], 3)
[2*t^7 + 2*t^5 + t^3]
"""
from sage.combinat.words.word import Word
R = ZZ['t']
if current == []:
return [R(0)]
tmp = []
partp = part[0].conjugate()
#compute the contribution of the ribbons added at
#the current node
for perms in [current[i][1] for i in range(len(current))]:
perm = [
partp[i] + len(partp) - (i + 1) - perms[i]
for i in range(len(partp))
]
perm.reverse()
perm = Word(perm).standard_permutation()
tmp.append((weight[-1] * (length - 1) - perm.number_of_inversions()))
if nexts != []:
return [
sum([
sum([t**tmp[i] * nexts[i][j] for j in range(len(nexts[i]))])
for i in range(len(tmp))
])
]
else:
return [sum([t**tmp[i] for i in range(len(tmp))])]
def splitting(i, A):
"""
The ``MarkedMetricGraph`` that corresponds to the splitting
``F(A)=F(A[:i])*F(A[i:])``.
This is a graph with two vertices linked by an edge e and a
loop for each letter in A. Letters in A[:i] are attached to
the first vertex while letters in A[:i] are attached to the
second vertex.
All loops have length 0, the splitting edge ``e`` has length 1.
INPUT:
- ``i`` -- integer correspond to the splitting index
- ``A`` -- an alphabet
OUTPUT:
The ``MarkedMetricGraph`` corresponding to splitting.
EXAMPLES::
sage: from train_track import *
sage: from train_track.marked_graph import MarkedMetricGraph
sage: A = AlphabetWithInverses(5)
sage: print(MarkedMetricGraph.splitting(2,A))
Marked graph: a: 0->0, b: 0->0, c: 1->1, d: 1->1, e:
1->1, f: 0->1
Marking: a->a, b->b, c->fcF, d->fdF, e->feF
Length: a:0, b:0, c:0, d:0, e:0, f:1
"""
graph = dict()
length = dict()
marking = dict()
B = A.copy()
[e, ee] = B.add_new_letter()
for j in range(i):
a = A[j]
graph[a] = (0, 0)
length[a] = 0
marking[a] = Word([a])
for j in range(i, len(A)):
a = A[j]
graph[a] = (1, 1)
length[a] = 0
marking[a] = Word([e, a, ee])
graph[e] = (0, 1)
length[e] = 1
return MarkedMetricGraph(graph, marking, length, B, A)
def spin_rec(t, nexts, current, part, weight, length):
"""
Routine used for constructing the spin polynomial.
INPUT:
- ``weight`` - list of non-negative integers
- ``length`` - the length of the ribbons we're tiling
with
- ``t`` - the variable
EXAMPLES::
sage: from sage.combinat.ribbon_tableau import spin_rec
sage: sp = SkewPartition
sage: t = ZZ['t'].gen()
sage: spin_rec(t, [], [[[], [3, 3]]], sp([[2, 2, 2], []]), [2], 3)
[t^4]
sage: spin_rec(t, [[0], [t^4]], [[[2, 1, 1, 1, 1], [0, 3]], [[2, 2, 2], [3, 0]]], sp([[2, 2, 2, 2, 1], []]), [2, 1], 3)
[t^5]
sage: spin_rec(t, [], [[[], [3, 3, 0]]], sp([[3, 3], []]), [2], 3)
[t^2]
sage: spin_rec(t, [[t^4], [t^3], [t^2]], [[[2, 2, 2], [0, 0, 3]], [[3, 2, 1], [0, 3, 0]], [[3, 3], [3, 0, 0]]], sp([[3, 3, 3], []]), [2, 1], 3)
[t^6 + t^4 + t^2]
sage: spin_rec(t, [[t^5], [t^4], [t^6 + t^4 + t^2]], [[[2, 2, 2, 2, 1], [0, 0, 3]], [[3, 3, 1, 1, 1], [0, 3, 0]], [[3, 3, 3], [3, 0, 0]]], sp([[3, 3, 3, 2, 1], []]), [2, 1, 1], 3)
[2*t^7 + 2*t^5 + t^3]
"""
from sage.combinat.words.word import Word
R = ZZ['t']
if current == []:
return [R(0)]
tmp = []
partp = part[0].conjugate()
#compute the contribution of the ribbons added at
#the current node
for perms in [current[i][1] for i in range(len(current))]:
perm = [partp[i] + len(partp)-(i+1)-perms[i] for i in range(len(partp))]
perm.reverse()
perm = Word(perm).standard_permutation()
tmp.append( (weight[-1]*(length-1)-perm.number_of_inversions()) )
if nexts != []:
return [sum([sum([t**tmp[i]*nexts[i][j] for j in range(len(nexts[i]))]) for i in range(len(tmp))])]
else:
return [sum([t**tmp[i] for i in range(len(tmp))])]
def succ_product_on_basis(self, x, y):
r"""
Return the `\succ` product of two permutations.
This is the shifted shuffle of `x` and `y` with the additional
condition that the first letter of the result comes from `y`.
The usual symbol for this operation is `\succ`.
.. SEEALSO::
- :meth:`product_on_basis`, :meth:`prec_product_on_basis`
EXAMPLES::
sage: A = algebras.FQSym(QQ).F()
sage: x = Permutation([1,2])
sage: A.succ_product_on_basis(x, x)
F[3, 1, 2, 4] + F[3, 1, 4, 2] + F[3, 4, 1, 2]
sage: y = Permutation([])
sage: A.succ_product_on_basis(x, y) == 0
True
sage: A.succ_product_on_basis(y, x) == A(x)
True
TESTS::
sage: u = A.one().support()[0]
sage: A.succ_product_on_basis(u, u)
Traceback (most recent call last):
...
ValueError: products | < | and | > | are not defined
"""
if not y:
if not x:
raise ValueError(
"products | < | and | > | are not defined")
else:
return self.zero()
basis = self.basis()
if not x:
return basis[y]
K = basis.keys()
n = len(x)
shy = Word([a + n for a in y])
shy0 = shy[0]
return self.sum(basis[K([shy0] + list(u))]
for u in Word(x).shuffle(Word(shy[1:])))
def subdivide(self, edge_list):
"""
Subdvides each of the edges in ``edge_list`` into two edges.
An edge may appear several time in the list: it will be
subdivided as many times.
INPUT:
- ``edge_list`` -- list of edges
OUTPUT:
A dictionnary that maps an old edge to a path in the new
graph.
EXAMPLES::
sage: from train_track.inverse_graph import GraphWithInverses
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,1,'c']])
sage: G.subdivide(['a','c'])
{'A': word: DA,
'B': word: B,
'C': word: EC,
'a': word: ad,
'b': word: b,
'c': word: ce}
sage: print(G)
a: 0->2, b: 0->1, c: 1->3, d: 2->0, e: 3->1
"""
A = self._alphabet
result_map = dict((e, Word([e])) for e in A)
new_edges = A.add_new_letters(len(edge_list))
new_vertices = self.new_vertices(len(edge_list))
for i, e in enumerate(edge_list):
ee = A.inverse_letter(e)
ne = result_map[e][-1]
v = new_vertices[i]
vt = self.terminal_vertex(result_map[e][-1])
f = new_edges[i][0]
ff = new_edges[i][1]
self.set_terminal_vertex(ne, v)
self.add_edge(v, vt, [f, ff])
result_map[e] = result_map[e] * Word([f])
result_map[ee] = Word([ff]) * result_map[ee]
return result_map
def reverse_sorting_permutation(
t): # TODO: put "stable sorting" as keyword somewhere
r"""
Return a permutation `p` such that is decreasing
INPUT:
- `t` -- a list/tuple/... of numbers
OUTPUT:
a minimal permutation `p` such that `w \circ p` is sorted decreasingly
EXAMPLES::
sage: t = [3, 3, 1, 2]
sage: s = reverse_sorting_permutation(t); s
[1, 2, 4, 3]
sage: [t[s[i]-1] for i in range(len(t))]
[3, 3, 2, 1]
sage: t = [4, 2, 3, 2, 1, 3]
sage: s = reverse_sorting_permutation(t); s
[1, 3, 6, 2, 4, 5]
sage: [t[s[i]-1] for i in range(len(t))]
[4, 3, 3, 2, 2, 1]
"""
return ~(Word([-i for i in t]).standard_permutation())
def to_word(self, alphabet=[1, 2, 3]):
r"""
Return the billiard word.
INPUT:
- ``alphabet`` - list
EXAMPLES::
sage: from slabbe import BilliardCube
sage: b = BilliardCube((1,pi,sqrt(2)))
sage: b.to_word()
word: 2321232212322312232123221322231223212322...
::
sage: B = BilliardCube((sqrt(3),sqrt(5),sqrt(7)))
sage: B.to_word()
word: 3213213231232133213213231232132313231232...
"""
steps = map(vector, ((1, 0, 0), (0, 1, 0), (0, 0, 1)))
for step in steps:
step.set_immutable()
coding = dict(zip(steps, alphabet))
it = (coding[step] for step in self.step_iterator())
#WP = WordPaths(alphabet, [(1,0,0),(0,1,0),(0,0,1)])
return Word(it, alphabet=alphabet)
def rose_marked_graph(alphabet):
"""
The rose on ``alphabet`` marked with the identity.
INPUT:
- ``alphabet`` -- an alphabet
OUTPUT:
The rose on ``alphabet`` marked with the identity.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: from train_track.marked_graph import MarkedGraph
sage: print(MarkedGraph.rose_marked_graph(AlphabetWithInverses(2)))
Marked graph: a: 0->0, b: 0->0
Marking: a->a, b->b
"""
marking = dict((a, Word([a])) for a in alphabet.positive_letters())
return MarkedGraph(graph=GraphWithInverses.rose_graph(alphabet),
marking=marking,
marking_alphabet=alphabet)
def __iter__(self):
"""
TESTS::
sage: WeightedIntegerVectors(7, [2,2]).list()
[]
sage: WeightedIntegerVectors(3, [2,1,1]).list()
[[1, 0, 1], [1, 1, 0], [0, 0, 3], [0, 1, 2], [0, 2, 1], [0, 3, 0]]
::
sage: ivw = [ WeightedIntegerVectors(k, [1,1,1]) for k in range(11) ]
sage: iv = [ IntegerVectors(k, 3) for k in range(11) ]
sage: all(sorted(map(list, iv[k])) == sorted(map(list, ivw[k])) for k in range(11))
True
::
sage: ivw = [ WeightedIntegerVectors(k, [2,3,7]) for k in range(11) ]
sage: all(i.cardinality() == len(i.list()) for i in ivw)
True
"""
if not self._weights:
if self._n == 0:
yield self.element_class(self, [])
return
perm = Word(self._weights).standard_permutation()
perm = [len(self._weights) - i for i in perm]
l = [x for x in sorted(self._weights, reverse=True)]
for x in iterator_fast(self._n, l):
yield self.element_class(self, [x[i] for i in perm])
def to_labelling_permutation(self):
r"""
Returns the labelling of the support Dyck path of the parking function.
INPUT:
- ``self`` -- parking function word
OUTPUT:
- returns the labelling of the Dyck path
EXAMPLES::
sage: PF = ParkingFunction([6, 1, 5, 2, 2, 1, 5])
sage: PF.to_labelling_permutation()
[2, 6, 4, 5, 3, 7, 1]
::
sage: ParkingFunction([3,1,1,4]).to_labelling_permutation()
[2, 3, 1, 4]
sage: ParkingFunction([4,1,1,1]).to_labelling_permutation()
[2, 3, 4, 1]
sage: ParkingFunction([2,1,4,1]).to_labelling_permutation()
[2, 4, 1, 3]
"""
from sage.combinat.words.word import Word
return Word(self).standard_permutation().inverse()
def __iter__(self):
"""
TESTS::
sage: [ p for p in OrderedSetPartitions([1,2,3,4], [2,1,1]) ]
[[{1, 2}, {3}, {4}],
[{1, 2}, {4}, {3}],
[{1, 3}, {2}, {4}],
[{1, 4}, {2}, {3}],
[{1, 3}, {4}, {2}],
[{1, 4}, {3}, {2}],
[{2, 3}, {1}, {4}],
[{2, 4}, {1}, {3}],
[{3, 4}, {1}, {2}],
[{2, 3}, {4}, {1}],
[{2, 4}, {3}, {1}],
[{3, 4}, {2}, {1}]]
"""
comp = self.c
lset = [x for x in self._set]
l = len(self.c)
dcomp = [-1] + comp.descents(final_descent=True)
p = []
for j in range(l):
p += [j + 1] * comp[j]
for x in permutation.Permutations(p):
res = permutation.Permutation(range(
1, len(lset))) * Word(x).standard_permutation().inverse()
res = [lset[x - 1] for x in res]
yield self.element_class(
self,
[Set(res[dcomp[i] + 1:dcomp[i + 1] + 1]) for i in range(l)])
def discrete_curve(nb_equipes,
max_points=100,
K=1,
R=2,
base=2,
verbose=False):
r"""
INPUT:
- ``nb_equipes`` -- integer
- ``max_points`` -- integer
- ``K`` -- the value at ``p = nb_equipes``
- ``R`` -- real (default: ``2``), curve parameter
- ``base`` -- 2
- ``verbose`` - bool
EXEMPLES::
sage: from slabbe.ranking_scale import discrete_curve
sage: A = discrete_curve(64, 100,verbose=True)
First difference sequence is
[1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1,
2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 4, 4, 4]
::
sage: A = discrete_curve(64, 100)
sage: B = discrete_curve(32, 50)
sage: C = discrete_curve(16, 25)
::
sage: A
[100, 96, 92, 88, 85, 82, 79, 77, 74, 71, 69, 67, 64, 62, 60, 58,
56, 54, 52, 50, 49, 47, 45, 44, 42, 41, 39, 38, 36, 35, 33, 32, 31,
29, 28, 27, 26, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12,
11, 10, 9, 8, 8, 7, 6, 5, 5, 4, 3, 2, 2, 1]
sage: B
[50, 46, 43, 40, 37, 35, 33, 31, 29, 27, 25, 23, 21, 20, 18, 17,
16, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 2, 1]
sage: C
[25, 22, 19, 17, 15, 13, 11, 9, 8, 7, 6, 5, 4, 3, 2, 1]
::
sage: A = discrete_curve(64+2, 100) #Movember, Bye Bye, Cdf, MA
sage: B = discrete_curve(32+2, 70) # la flotte
sage: C = discrete_curve(16+2, 40) # october fest, funenuf, la viree
"""
from sage.misc.functional import round
from sage.rings.integer_ring import ZZ
from sage.combinat.words.word import Word
fn_normalise = curve(nb_equipes, max_points, K=K, R=R, base=base)
L = [ZZ(round(fn_normalise(p=i))) for i in range(1, nb_equipes + 1)]
if verbose:
print "First difference sequence is"
print list(Word(L).reversal().finite_differences())
return L
def lie_polynomial(self, w):
"""
Return the Lie polynomial associated to the Lyndon word ``w``. If
``w`` is not Lyndon, then return the product of Lie polynomials of the
Lyndon factorization of ``w``.
INPUT:
- ``w`` -- a word or an element of the free monoid
EXAMPLES::
sage: F = FreeAlgebra(QQ, 3, 'x,y,z')
sage: M.<x,y,z> = FreeMonoid(3)
sage: F.lie_polynomial(x*y)
x*y - y*x
sage: F.lie_polynomial(y*x)
y*x
sage: F.lie_polynomial(x^2*y*x)
x^2*y*x - x*y*x^2
sage: F.lie_polynomial(y*z*x*z*x*z)
y*z*x*z*x*z - y*z*x*z^2*x - y*z^2*x^2*z + y*z^2*x*z*x
- z*y*x*z*x*z + z*y*x*z^2*x + z*y*z*x^2*z - z*y*z*x*z*x
TESTS:
We test some corner cases and alternative inputs::
sage: F.lie_polynomial(Word('xy'))
x*y - y*x
sage: F.lie_polynomial('xy')
x*y - y*x
sage: F.lie_polynomial(M.one())
1
sage: F.lie_polynomial(Word([]))
1
sage: F.lie_polynomial('')
1
"""
if not w:
return self.one()
M = self._basis_keys
if len(w) == 1:
return self(M(w))
ret = self.one()
# We have to be careful about order here.
# Since the Lyndon factors appear from left to right
# we must multiply from left to right as well.
for factor in Word(w).lyndon_factorization():
if len(factor) == 1:
ret = ret * self(M(factor))
continue
x,y = factor.standard_factorization()
x = M(x)
y = M(y)
ret = ret * (self(x * y) - self(y * x))
return ret
def __contains__(self, x):
"""
EXAMPLES::
sage: from sage.combinat.words.shuffle_product import ShuffleProduct_w1w2
sage: W = Words("abcd")
sage: w = W("ab")
sage: u = W("cd")
sage: S = ShuffleProduct_w1w2(w,u)
sage: w*u in S
True
sage: all(w.is_subword_of(x) for x in S)
True
sage: w in S
False
We check that :trac:`14121` is solved::
sage: w = W('ab')
sage: x = W('ac')
sage: x*w in w.shuffle(x)
True
"""
from sage.combinat.words.word import Word
if not isinstance(x, Word_class):
return False
if x.length() != self._w1.length() + self._w2.length():
return False
w1 = list(self._w1)
w2 = list(self._w2)
wx = list(x)
for _ in range(len(wx)):
try:
letter = wx.pop(0)
except IndexError:
return False
if w1 and w2 and letter == w1[0] == w2[0]:
return (Word(wx) in self._w1[1:].shuffle(self._w2) or
Word(wx) in self._w1.shuffle(self._w2[1:]))
if w1 and letter == w1[0]:
w1.pop(0)
elif w2 and letter == w2[0]:
w2.pop(0)
else:
return False
return not wx
def spanning_tree(self, verbose=False):
"""
Return a spanning tree of ``self``.
The origin vertex of the spanning tree is the initial vertex
of the edge labeled by ``self._alphabet[0]``.
OUPUT:
a dictionnary that maps each vertex to an edge-path
from the origin vertex.
.. SEEALSO::
:meth:`maximal_tree()` that returns a list of edges
of a spanning tree.
.. WARNING::
``self`` must be connected.
EXAMPLES::
sage: from train_track.inverse_graph import GraphWithInverses
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,0,'c']])
sage: G.spanning_tree()
{0: word: , 1: word: b}
"""
A = self._alphabet
tree = {self.initial_vertex(A[0]): Word()}
done = False
while not done:
done = True
for a in A.positive_letters():
vi = self.initial_vertex(a)
vt = self.terminal_vertex(a)
if vi in tree and vt not in tree:
tree[vt] = self.reduce_path(tree[vi] * Word([a]))
done = False
elif vt in tree and vi not in tree:
tree[vi] = self.reduce_path(tree[vt] *
Word([A.inverse_letter(a)]))
done = False
return tree
def contract_forest(self, forest):
"""
Contract the forest.
Each tree of the forest is contracted to the initial vertex of its first
edge.
INPUT:
``forest`` is a list of disjoint subtrees each given as
lists of edges.
OUTPUT:
A dictionnary that maps an old edge to its image in the new
graph.
"""
A = self._alphabet
edge_map = dict((e, Word([e])) for e in A)
vertex_map = {}
for tree in forest:
first = True
for e in tree:
if first:
vtree = self.initial_vertex(e)
first = False
vertex_map[self.initial_vertex(e)] = vtree
vertex_map[self.terminal_vertex(e)] = vtree
edge_map[e] = Word([])
edge_map[A.inverse_letter(e)] = Word([])
self.remove_edge(e)
for e in A:
v = self.initial_vertex(e)
if v in vertex_map and v != vertex_map[v]:
self.set_initial_vertex(e, vertex_map[self.initial_vertex(e)])
for v in vertex_map:
if v != vertex_map[v]:
self.remove_vertex(v)
return edge_map
def blow_up_vertices(self, germ_components):
"""
Blow-up ``self`` according to classes of germs given in
``germ_components``.
INPUT:
- ``germ_components`` -- a list of classes of germs outgoing
from a vertex
OUTPUT:
A dictionnary that maps an old edge to the path in the new graph.
EXAMPLES::
sage: from train_track import *
sage: from train_track.inverse_graph import GraphWithInverses
sage: G = GraphWithInverses.rose_graph(AlphabetWithInverses(2))
sage: G.blow_up_vertices([['a','A'],['b'],['B']])
{'A': word: cAC, 'B': word: eBD, 'a': word: caC, 'b': word: dbE}
sage: print(G)
a: 1->1, b: 2->3, c: 0->1, d: 0->2, e: 0->3
"""
A = self.alphabet()
result_map = dict((a, Word([a])) for a in A)
new_vertices = self.new_vertices(len(germ_components))
new_edges = A.add_new_letters(len(germ_components))
for i, c in enumerate(germ_components):
v0 = self.initial_vertex(c[0])
vc = new_vertices[i]
ec = new_edges[i]
self.add_edge(v0, vc, ec)
for a in c:
self.set_initial_vertex(a, vc)
result_map[a] = Word([ec[0]]) * result_map[a]
aa = A.inverse_letter(a)
result_map[aa] = result_map[aa] * Word([ec[1]])
return result_map
def subdivide_domain(self, e):
"""
Subdivide an edge in the domain graph.
The edge ``e`` is subdivided into ``l`` edges where ``l`` is
the length of the image of ``e`` by ``self``.
Update the edge map of ``self``.
Intended to be used by Stalling's folding algorithm to get an
immersion.
SEE ALSO::
GraphWithInverses.subdivide_edge()
"""
G = self.domain()
A = G.alphabet()
result_map = dict((a, Word([a])) for a in A)
w = self.image(e)
new_edges = A.add_new_letters(len(w) - 1)
new_vertices = G.new_vertices(len(w) - 1)
d = {(a, self.image(a)) for a in A.positive_letters()}
for i, a in enumerate(new_edges):
v = new_vertices[i]
if i == 0:
vi = G.initial_vertex(e)
vt = G.terminal_vertex(e)
f = new_edges[i][0]
ee = A.inverse_letter(e)
ff = new_edges[i][1]
G.set_terminal_vertex(e, v)
G.add_edge(v, vt, [f, ff])
result_map[e] = result_map[e] * Word([f])
result_map[ee] = Word([ff]) * result_map[ee]
d[a[0]] = w[i + 1]
else:
vi = self._domain.initial_vertex(new_edges[i - 1][0])
vt = self._domain.terminal_vertex(new_edges[i - 1][0])
f = new_edges[i][0]
#ee=A.inverse_letter(e) #already done
ff = new_edges[i][1]
self._domain.set_terminal_vertex(new_edges[i - 1][0], v)
self._domain.add_edge(v, vt, [f, ff])
result_map[e] = result_map[e] * Word([f])
result_map[ee] = Word([ff]) * result_map[ee]
d[a[0]] = w[i + 1]
# updating self.edge_map after subdivision
if A.is_positive_letter(e):
d[e] = Word([self.image(e)[0]])
else:
d[ee] = Word([self.image(ee)[-1]])
self.set_edge_map(d)
return result_map
def skeleton_cycles(self):
r"""
EXAMPLES::
sage: from slabbe import DoubleRootedTree
sage: edges = [(0,5),(1,2),(2,6),(3,2),(4,1),(5,7),(7,1),(8,2),(9,4)]
sage: D = DoubleRootedTree(edges, 6, 0)
sage: D.skeleton()
[0, 5, 7, 1, 2, 6]
sage: D.skeleton_cycles()
[(0,), (1, 5), (2, 7, 6)]
"""
skeleton = self.skeleton()
sorted_skeleton = sorted(skeleton)
w = Word(skeleton)
cycles_standard = w.standard_permutation().to_cycles()
cycles = []
for cycle_standard in cycles_standard:
t = tuple(sorted_skeleton[c-1] for c in cycle_standard)
cycles.append(t)
return cycles
def is_littlewood_richardson(t, heights):
"""
Return whether semistandard tableau ``t`` is Littleword-Richardson
with respect to ``heights``.
A tableau is Littlewood-Richardson with respect to ``heights`` given
by `(h_1, h_2, \ldots)` if each subtableau with respect to the
alphabets `\{1, 2, \ldots, h_1\}`, `\{h_1+1, \ldots, h_1+h_2\}`,
etc. is Yamanouchi.
EXAMPLES::
sage: from sage.combinat.lr_tableau import is_littlewood_richardson
sage: t = Tableau([[1,1,2,3,4],[2,3,3],[3]])
sage: is_littlewood_richardson(t,[2,2])
False
sage: t = Tableau([[1,1,3],[2,3],[4,4]])
sage: is_littlewood_richardson(t,[2,2])
True
sage: t = Tableau([[7],[8]])
sage: is_littlewood_richardson(t,[2,3,3])
False
sage: is_littlewood_richardson([[2],[3]],[3,3])
False
"""
from sage.combinat.words.word import Word
partial = [
sum(heights[i] for i in range(j)) for j in range(len(heights) + 1)
]
try:
w = t.to_word()
except AttributeError: # Not an instance of Tableau
w = sum(reversed(t), [])
for i in range(len(heights)):
subword = Word([j for j in w if partial[i] + 1 <= j <= partial[i + 1]],
alphabet=list(range(partial[i] + 1,
partial[i + 1] + 1)))
if not subword.is_yamanouchi():
return False
return True
def reading_word(self):
"""
Return the reading word of ``self``, obtained by reading the boxes
entries of self from right to left, starting in the upper right.
EXAMPLES::
sage: a = AugmentedLatticeDiagramFilling([[1,6],[2],[3,4,2],[],[],[5,5]])
sage: a.reading_word()
word: 25465321
"""
w = [self[i, j] for i, j in self.reading_order() if j > 0]
return Word(w)
def counit(self, S):
"""
Return the counit of ``S``.
EXAMPLES::
sage: F = ShuffleAlgebra(QQ,'ab')
sage: S = F.an_element(); S
B[word: ] + 2*B[word: a] + 3*B[word: b] + B[word: bab]
sage: F.counit(S)
1
"""
return S.monomial_coefficients().get(Word(), 0)
def is_littlewood_richardson(t, heights):
r"""
Return whether semistandard tableau ``t`` is Littleword-Richardson
with respect to ``heights``.
A tableau is Littlewood-Richardson with respect to ``heights`` given
by `(h_1, h_2, \ldots)` if each subtableau with respect to the
alphabets `\{1, 2, \ldots, h_1\}`, `\{h_1+1, \ldots, h_1+h_2\}`,
etc. is Yamanouchi.
EXAMPLES::
sage: from sage.combinat.lr_tableau import is_littlewood_richardson
sage: t = Tableau([[1,1,2,3,4],[2,3,3],[3]])
sage: is_littlewood_richardson(t,[2,2])
False
sage: t = Tableau([[1,1,3],[2,3],[4,4]])
sage: is_littlewood_richardson(t,[2,2])
True
sage: t = Tableau([[7],[8]])
sage: is_littlewood_richardson(t,[2,3,3])
False
sage: is_littlewood_richardson([[2],[3]],[3,3])
False
"""
from sage.combinat.words.word import Word
partial = [sum(heights[i] for i in range(j)) for j in range(len(heights)+1)]
try:
w = t.to_word()
except AttributeError: # Not an instance of Tableau
w = sum(reversed(t), [])
for i in range(len(heights)):
subword = Word([j for j in w if partial[i]+1 <= j <= partial[i+1]],
alphabet=list(range(partial[i]+1,partial[i+1]+1)))
if not subword.is_yamanouchi():
return False
return True