def e_ik(itab, ktab, star=0):
"""
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import e_ik
sage: e_ik([[1,2,3]], [[1,2,3]])
[1, 2, 3] + [1, 3, 2] + [2, 1, 3] + [2, 3, 1] + [3, 1, 2] + [3, 2, 1]
sage: e_ik([[1,2,3]], [[1,2,3]], star=1)
[1, 2] + [2, 1]
"""
it = Tableau(itab)
kt = Tableau(ktab)
if star:
it = it.restrict(it.size() - star)
kt = kt.restrict(kt.size() - star)
if it.shape() != kt.shape():
raise ValueError, "the two tableaux must be of the same shape"
mult = permutation_options['mult']
permutation_options['mult'] = 'l2r'
if kt == it:
res = e(it)
elif (it, kt) in e_ik_cache:
res = e_ik_cache[(it,kt)]
else:
pi = pi_ik(it,kt)
e_ik_cache[(it,kt)] = e(it)*pi
res = e_ik_cache[(it,kt)]
permutation_options['mult'] = mult
return res
def epsilon(tab, star=0):
"""
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import epsilon
sage: epsilon([[1,2]])
1/2*[1, 2] + 1/2*[2, 1]
sage: epsilon([[1],[2]])
1/2*[1, 2] - 1/2*[2, 1]
"""
t = Tableau(tab)
if star:
t = t.restrict(t.size() - star)
mult = permutation_options['mult']
permutation_options['mult'] = 'l2r'
if t in epsilon_cache:
res = epsilon_cache[t]
else:
if t.size() == 2:
epsilon_cache[t] = e(t)*(1/kappa(t.shape()))
res = epsilon_cache[t]
elif t == Tableau([[1]]):
epsilon_cache[t] = e(t)
res = epsilon_cache[t]
else:
epsilon_cache[t] = epsilon(t, 1)*e(t)*epsilon(t,1)*( 1 / kappa(t.shape()))
res = epsilon_cache[t]
permutation_options['mult'] = mult
return res
def construct(self):
for x in range(4):
self._freecells.append(FreeCell())
self._homecells.append(HomePile())
self._tableaux.append(Tableau())
for x in range(4):
self._tableaux.append(Tableau())
def epsilon_ik(itab, ktab, star=0):
"""
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import epsilon_ik
sage: epsilon_ik([[1,2],[3]], [[1,3],[2]])
1/4*[1, 3, 2] - 1/4*[2, 3, 1] + 1/4*[3, 1, 2] - 1/4*[3, 2, 1]
sage: epsilon_ik([[1,2],[3]], [[1,3],[2]], star=1)
Traceback (most recent call last):
...
ValueError: the two tableaux must be of the same shape
"""
it = Tableau(itab)
kt = Tableau(ktab)
if star:
it = it.restrict(it.size() - star)
kt = kt.restrict(kt.size() - star)
if it.shape() != kt.shape():
raise ValueError, "the two tableaux must be of the same shape"
mult = permutation_options['mult']
permutation_options['mult'] = 'l2r'
if kt == it:
res = epsilon(itab)
elif (it, kt) in epsilon_ik_cache:
res = epsilon_ik_cache[(it,kt)]
else:
epsilon_ik_cache[(it,kt)] = epsilon(it, star+1)*e_ik(it,kt,star)*epsilon(kt, star+1) * (1/kappa(it.shape()))
res = epsilon_ik_cache[(it,kt)]
permutation_options['mult'] = mult
return res
def check_exist(self, formula, init_states):
tableau = Tableau(formula, self._atomic_str)
prod = tableau.product(self._model)
# return False
return prod.has_fair_path(tableau.initial_states & init_states,
tableau.fairness_constraints)
def run_game():
# Create deck, hand, tableau.
deck = Deck()
hand = Hand(deck)
tableau = Tableau(tableau_limit)
trade = Trade()
game_end = False
# Main game loop.
while game_end is False:
# Choose action.
# phase = gf.choose_action()
# Play phases.
for phase in range(0, 5):
gf.play_phase(phase, deck, hand, tableau, trade)
# Check hand limit
hand.hand_limit(deck)
# Check for game end.
game_end = gf.end_check(tableau, scoreboard)
return scoreboard.get_vp_total(tableau)
def e_hat(tab, star=0):
"""
The Young projection operator, an idempotent in the rational group algebra.
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import e_hat
sage: e_hat([[1,2,3]])
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
sage: e_hat([[1],[2]])
1/2*[1, 2] - 1/2*[2, 1]
There are differing conventions for the order of the symmetrizers
and antisymmetrizers. This example illustrates our conventions::
sage: e_hat([[1,2],[3]])
1/3*[1, 2, 3] + 1/3*[2, 1, 3] - 1/3*[3, 1, 2] - 1/3*[3, 2, 1]
"""
t = Tableau(tab)
if star:
t = t.restrict(t.size()-star)
if t in ehat_cache:
res = ehat_cache[t]
else:
res = (1/kappa(t.shape()))*e(t)
return res
def e_hat(tab, star=0):
"""
The Young projection operator corresponding to the Young tableau
``tab`` (which is supposed to contain every integer from `1` to
its size precisely once, but may and may not be standard). This
is an idempotent in the rational group algebra.
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import e_hat
sage: e_hat([[1,2,3]])
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
sage: e_hat([[1],[2]])
1/2*[1, 2] - 1/2*[2, 1]
There are differing conventions for the order of the symmetrizers
and antisymmetrizers. This example illustrates our conventions::
sage: e_hat([[1,2],[3]])
1/3*[1, 2, 3] + 1/3*[2, 1, 3] - 1/3*[3, 1, 2] - 1/3*[3, 2, 1]
"""
t = Tableau(tab)
if star:
t = t.restrict(t.size() - star)
if t in ehat_cache:
res = ehat_cache[t]
else:
res = (1 / kappa(t.shape())) * e(t)
return res
def b(tableau, star=0):
r"""
The column projection operator corresponding to the Young tableau
``tableau`` (which is supposed to contain every integer from
`1` to its size precisely once, but may and may not be standard).
This is the signed sum (in the group algebra of the relevant
symmetric group over `\QQ`) of all the permutations which
preserve the column of ``tableau`` (where the signs are the usual
signs of the permutations). It is called `b_{\text{tableau}}` in
[EtRT]_, Section 4.2.
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import b
sage: b([[1,2]])
[1, 2]
sage: b([[1],[2]])
[1, 2] - [2, 1]
sage: b([])
[]
sage: b([[1, 2, 4], [5, 3]])
[1, 2, 3, 4, 5] - [1, 3, 2, 4, 5] - [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1]
With the `l2r` setting for multiplication, the unnormalized
Young symmetrizer ``e(tableau)`` should be the product
``b(tableau) * a(tableau)`` for every ``tableau``. Let us check
this on the standard tableaux of size 5::
sage: from sage.combinat.symmetric_group_algebra import a, b, e
sage: all( e(t) == b(t) * a(t) for t in StandardTableaux(5) )
True
"""
t = Tableau(tableau)
if star:
t = t.restrict(t.size() - star)
cs = t.column_stabilizer().list()
n = t.size()
# This all should be over ZZ, not over QQ, but symmetric group
# algebras don't seem to preserve coercion (the one over ZZ
# doesn't coerce into the one over QQ even for the same n),
# and the QQ version of this method is more important, so let
# me stay with QQ.
# TODO: Fix this.
sgalg = SymmetricGroupAlgebra(QQ, n)
one = QQ.one()
P = permutation.Permutation
# Ugly hack for the case of an empty tableau, due to the
# annoyance of Permutation(Tableau([]).row_stabilizer()[0])
# being [1] rather than [] (which seems to have its origins in
# permutation group code).
# TODO: Fix this.
if len(tableau) == 0:
return sgalg.one()
cd = dict((P(v), v.sign() * one) for v in cs)
return sgalg._from_dict(cd)
def __init__(self, strategy, visual=False, screen=None):
S = getattr(strategies, strategy)
strategy = S()
self.tableau = Tableau()
self.redeals = 2
self.strategy = strategy
self.visual = visual
self.screen = screen
def a(tableau, star=0):
r"""
The row projection operator corresponding to the Young tableau
``tableau`` (which is supposed to contain every integer from
`1` to its size precisely once, but may and may not be standard).
This is the sum (in the group algebra of the relevant symmetric
group over `\QQ`) of all the permutations which preserve
the rows of ``tableau``. It is called `a_{\text{tableau}}` in
[EtRT]_, Section 4.2.
REFERENCES:
.. [EtRT] Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai
Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina,
"Introduction to representation theory",
:arXiv:`0901.0827v5`.
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import a
sage: a([[1,2]])
[1, 2] + [2, 1]
sage: a([[1],[2]])
[1, 2]
sage: a([])
[]
sage: a([[1, 5], [2, 3], [4]])
[1, 2, 3, 4, 5] + [1, 3, 2, 4, 5] + [5, 2, 3, 4, 1] + [5, 3, 2, 4, 1]
"""
t = Tableau(tableau)
if star:
t = t.restrict(t.size() - star)
rs = t.row_stabilizer().list()
n = t.size()
# This all should be over ZZ, not over QQ, but symmetric group
# algebras don't seem to preserve coercion (the one over ZZ
# doesn't coerce into the one over QQ even for the same n),
# and the QQ version of this method is more important, so let
# me stay with QQ.
# TODO: Fix this.
sgalg = SymmetricGroupAlgebra(QQ, n)
one = QQ.one()
P = permutation.Permutation
# Ugly hack for the case of an empty tableau, due to the
# annoyance of Permutation(Tableau([]).row_stabilizer()[0])
# being [1] rather than [] (which seems to have its origins in
# permutation group code).
# TODO: Fix this.
if len(tableau) == 0:
return sgalg.one()
rd = dict((P(h), one) for h in rs)
return sgalg._from_dict(rd)
def epsilon_ik(self, itab, ktab, star=0):
"""
Return the seminormal basis element of ``self`` corresponding to the
pair of tableaux ``itab`` and ``ktab``.
EXAMPLES::
sage: QS3 = SymmetricGroupAlgebra(QQ, 3)
sage: a = QS3.epsilon_ik([[1,2,3]], [[1,2,3]]); a
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
sage: QS3.dft()*vector(a)
(1, 0, 0, 0, 0, 0)
sage: a = QS3.epsilon_ik([[1,2],[3]], [[1,2],[3]]); a
1/3*[1, 2, 3] - 1/6*[1, 3, 2] + 1/3*[2, 1, 3] - 1/6*[2, 3, 1] - 1/6*[3, 1, 2] - 1/6*[3, 2, 1]
sage: QS3.dft()*vector(a)
(0, 0, 0, 0, 1, 0)
"""
it = Tableau(itab)
kt = Tableau(ktab)
stn = StandardTableaux_size(self.n)
if it not in stn:
raise TypeError("it must be a standard tableau of size %s" %
self.n)
if kt not in stn:
raise TypeError("kt must be a standard tableau of size %s" %
self.n)
if it.shape() != kt.shape():
raise ValueError("it and kt must be of the same shape")
BR = self.base_ring()
z_elts = {}
epik = epsilon_ik(it, kt, star=star)
for m, c in epik._monomial_coefficients.iteritems():
z_elts[m] = BR(c)
z = self._from_dict(z_elts)
if permutation_options['mult'] == 'l2r':
return z
else:
return z.map_support(lambda x: x.inverse())
def pi_ik(itab, ktab):
"""
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import pi_ik
sage: pi_ik([[1,3],[2]], [[1,2],[3]])
[1, 3, 2]
"""
it = Tableau(itab)
kt = Tableau(ktab)
p = [None]*kt.size()
for i in range(len(kt)):
for j in range(len(kt[i])):
p[ it[i][j] -1 ] = kt[i][j]
QSn = SymmetricGroupAlgebra(QQ, it.size())
p = permutation.Permutation(p)
return QSn(p)
def _build_tableau(self):
ncons = self.lp.A_fpi.shape[0]
opmat = np.vstack((np.zeros((1, ncons)), np.identity(ncons)))
opmat = np.asmatrix(opmat, np.float64)
lpmat = np.vstack((np.hstack((self.lp.c_fpi.T * (-1), [[0.0]])),
np.hstack((self.lp.A_fpi, self.lp.b_fpi))))
tableau = np.asmatrix(np.hstack((opmat, lpmat)))
return Tableau(tableau)
def setUp(self):
# variables
pre_var = [(0, True), (0, False), (0, False), (0, True), (0, True)]
self.vars = [Variable(*v) for v in pre_var]
# row lines
pre_row = [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [1, 1, -1, 0, 0],
[2, -1, 0, -1, 0], [-1, 2, 0, 0, -1]]
self.rows = [RowLine(r) for r in pre_row]
# tableau
self.table = Tableau(self.vars, self.rows)
def print_full_typology(rankings):
"""Print the full set of results: ranking, winner, and winner type."""
for ranking in rankings:
constraint_ranking = ConstraintSet(ranking)
print("----------------------------------")
print(constraint_ranking)
print("----------------------------------")
for inputs in gen.inputs():
tableau = Tableau(inputs, constraint_ranking,
gen.candidates(inputs))
print(inputs, tableau.winners, tableau.typology)
def creation_tableau_joueur(dimension: int, nom_joueur: str):
""" Fonction servant a creer un objet Tableau et un objet Joueur et attribuer a Joueur le tableau
PRE:
- dimension = int
- nom_joueur = str
POST :
- création objet plateau_joueur de classe Tableau(dimension)
- creation d'un tableau dans l'objet plateau_joueur
- creation d'un objet Joueur possedant un nom et un objet Tableau
- retourne l'objet de classe Joueur
"""
plateau_joueur = Tableau(dimension)
plateau_joueur.creation_tableau()
return Joueur(nom_joueur, plateau_joueur)
def get_exists_nodes(self, formula):
tableau = Tableau(formula, self._atomic_str)
prod = tableau.product(self._model)
print('#relations', prod.count_relations())
print(
'#reachable states',
prod.count_reachable_states(tableau.initial_states
& self._model.atomic))
states = prod.find_fair_nodes(tableau.initial_states,
tableau.fairness_constraints)
states = bdd_utils.only_consider_prims(states, self._model.msb)
bdd_utils.print_debug_bdd('states', states)
return states
def solve(self, verbose = False):
if verbose:
print("The input linear program is:\n")
print("Maximize %s" % vect_vars(self.c))
print("Such that:")
for k, row in enumerate(self.A):
print("\t%s <= %s" % (vect_vars(row), str(self.b[k])))
print("\t" + ", ".join(["x%i" % (i + 1) for i in range(self.n)]) + " are non-negative\n")
T = self.T = Tableau(self.c, self.A, self.b, self.pivot)
phase_one_objective = T.initialize_basis()
# PHASE 1
if T.nb_additional_vars > 0:
if verbose:
print("=== PHASE 1 ===")
print("Added %i new positive variables." % T.nb_additional_vars)
print("Now trying to maximize: %s" % vect_vars(phase_one_objective))
T.set_objective_vector(phase_one_objective)
T.phase(verbose)
if T.opt != 0:
print("This linear program is INFEASIBLE.")
self.log()
return
else:
T.remove_additional_vars()
if verbose:
print("Found a basic feasible solution.: %s" % T.get_solution())
print("Removed additional variables from the tableau.\n")
print("=== PHASE 2 ===")
# PHASE 2
T.set_objective_vector(self.c)
R = T.phase(verbose)
if R == FeasibleResult.BOUNDED:
print("This linear problem is FEASIBLE and BOUNDED.")
print("One optimal solution is: %s" % T.get_solution())
print("The value of the objective for this solution is: %s" % str(-T.opt))
else:
print("This linear problem is FEASIBLE but UNBOUNDED.")
self.log()
def e(tableau, star=0):
"""
The unnormalized Young projection operator.
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import e
sage: e([[1,2]])
[1, 2] + [2, 1]
sage: e([[1],[2]])
[1, 2] - [2, 1]
There are differing conventions for the order of the symmetrizers
and antisymmetrizers. This example illustrates our conventions::
sage: e([[1,2],[3]])
[1, 2, 3] + [2, 1, 3] - [3, 1, 2] - [3, 2, 1]
"""
t = Tableau(tableau)
if star:
t = t.restrict(t.size()-star)
mult = permutation_options['mult']
permutation_options['mult'] = 'l2r'
if t in e_cache:
res = e_cache[t]
else:
rs = t.row_stabilizer().list()
cs = t.column_stabilizer().list()
n = t.size()
QSn = SymmetricGroupAlgebra(QQ, n)
one = QQ(1)
P = permutation.Permutation
rd = dict((P(h), one) for h in rs)
sym = QSn._from_dict(rd)
cd = dict((P(v), v.sign()*one) for v in cs)
antisym = QSn._from_dict(cd)
res = antisym*sym
e_cache[t] = res
permutation_options['mult'] = mult
return res
def test1():
print 'Test 1'
r1 = [2, 1, -2, '<', 8]
r2 = [4, -1, 2, '>', 2]
r3 = [2, 3, -1, '>', 4]
r = list()
r.append(r1)
r.append(r2)
r.append(r3)
f = [-2, 1, -1]
tableau = Tableau(f, r)
simplex = Simplex(tableau)
simplex.execute()
print 'Solution: ' + str(simplex.solution)
def _build_aux_tableau(self):
num_cons = self._tableau.A.shape[0]
aux_c = np.zeros_like(self._tableau.ct)
aux_c = np.asmatrix(
np.hstack((np.zeros((1, num_cons)), aux_c, np.ones(
(1, num_cons)))), np.float64)
b_neglines = np.where(float_comp(self._tableau.b, 0.0, less=True))[0]
opmat = np.asmatrix(np.identity(num_cons), np.float64)
aux_opA = np.hstack((opmat, self._tableau.A))
aux_opA[b_neglines, :] = aux_opA[b_neglines, :] * (-1)
aux_opA = np.hstack((aux_opA, np.identity(aux_opA.shape[0])))
aux_b = np.copy(self._tableau.b)
aux_b[b_neglines, :] = aux_b[b_neglines, :] * (-1)
aux_tab_mat = np.vstack((np.hstack(
(aux_c, [[0.0]])), np.hstack((aux_opA, aux_b))))
aux_tab = Tableau(aux_tab_mat)
return aux_tab
def main():
logging.debug(
'\n ======================== \n = Reading Data = \n ========================'
)
vars_and_restrictions = input()
vars_and_restrictions = vars_and_restrictions.split()
r = int(vars_and_restrictions[0])
v = int(vars_and_restrictions[1])
logging.debug("Number of restrictions: {}".format(r))
logging.debug("Number of variables: {}".format(v))
c_input = input()
c_list = list(c_input.split())
c = []
for i in range(len(c_list)):
c.append(float(c_list[i]))
logging.debug("Costs vector: {}".format(c))
a = []
b = []
for i in range(r):
line = input().split()
a_row = []
for j in range(len(line)):
if j < v:
a_row.append(float(line[j]))
else:
b.append(float(line[j]))
a.append(a_row)
logging.debug("Matrix A[]: {}".format(a))
logging.debug("Vector B[]: {}".format(b))
tableau = Tableau(r, v, c, a, b)
Tableau.print_tableau(tableau.matrix_tableau)
Simplex.do_simplex(tableau)
def solucaoIlimitada():
print 'Solucao Ilimitada'
r1 = [4, 1, '>=', 20]
r2 = [1, 2, '>=', 10]
r3 = [1, 0, '>=', 2]
r = list()
r.append(r1)
r.append(r2)
r.append(r3)
f = [-1, -2]
tableau = Tableau(f, r)
simplex = Simplex(tableau)
simplex.execute()
print 'Solution: ' + str(simplex.solution)
print simplex.tableau
def input1():
print ''
r1 = [1, 1, '>=', 2]
r2 = [1, 2, '>=', 5]
# r3 = [5,3,'<=',15]
r = list()
r.append(r1)
r.append(r2)
# r.append(r3)
f = [-1, -1]
tableau = Tableau(f, r)
simplex = Simplex(tableau)
simplex.execute()
print 'Solution: ' + str(simplex.solution)
print simplex.tableau
def degenerationExample():
print 'Exemplo de degeneracao'
r1 = [1, 0, '<=', 3]
r2 = [0, 1, '<=', 4]
r3 = [4, 3, '<=', 12]
r = list()
r.append(r1)
r.append(r2)
r.append(r3)
f = [-5, -2]
tableau = Tableau(f, r)
simplex = Simplex(tableau)
simplex.execute()
print 'Solution: ' + str(simplex.solution)
print simplex.tableau
def test6():
print 'Test 6'
r1 = [1, 2, 4, -1, '=', 6]
r2 = [2, 3, -1, 4, '<=', 12]
r3 = [1, 0, 1, 1, '<=', 4]
r = list()
r.append(r1)
r.append(r2)
r.append(r3)
f = [-2, -1, -5, 3]
tableau = Tableau(f, r)
simplex = Simplex(tableau)
simplex.execute()
print 'Solution: ' + str(simplex.solution)
print simplex.tableau
def test5():
print 'Test 3'
r1 = [1, 0, 0, '<', 1]
r2 = [20, 1, 0, '<', 100]
r3 = [200, 20, 1, '<', 10000]
r = list()
r.append(r1)
r.append(r2)
r.append(r3)
f = [-100, -10, -1]
tableau = Tableau(f, r)
simplex = Simplex(tableau)
simplex.execute()
print 'Solution: ' + str(simplex.solution)
print simplex.tableau
def test4():
print 'Test 3'
r1 = [4, 1, '<', 21]
r2 = [2, 3, '>', 13]
r3 = [-1, 1, '=', 1]
r = list()
r.append(r1)
r.append(r2)
r.append(r3)
f = [-6, 1]
tableau = Tableau(f, r)
simplex = Simplex(tableau)
simplex.execute()
print 'Solution: ' + str(simplex.solution)
print simplex.tableau
def test3():
print 'Test 3'
r1 = [1, 2, '<', 6]
r2 = [-2, 1, '<', 4]
r3 = [5, 3, '<', 15]
r = list()
r.append(r1)
r.append(r2)
r.append(r3)
f = [-5, -4]
tableau = Tableau(f, r)
simplex = Simplex(tableau)
simplex.execute()
print 'Solution: ' + str(simplex.solution)
print simplex.tableau