Coverage for local/lib/python2.7/site-packages/sage/combinat/species/empty_species.py : 24%

""" 

Empty Species 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2008 Florent Hivert <Florent.Hivert@univ-rouen,fr>, 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from .species import GenericCombinatorialSpecies 

from .series_order import inf 

from sage.structure.unique_representation import UniqueRepresentation 

class EmptySpecies(GenericCombinatorialSpecies, UniqueRepresentation): 

""" 

Returns the empty species. This species has no structure at all. 

It is the zero of the semi-ring of species. 

EXAMPLES:: 

sage: X = species.EmptySpecies(); X 

Empty species 

sage: X.structures([]).list() 

[] 

sage: X.structures([1]).list() 

[] 

sage: X.structures([1,2]).list() 

[] 

sage: X.generating_series().coefficients(4) 

[0, 0, 0, 0] 

sage: X.isotype_generating_series().coefficients(4) 

[0, 0, 0, 0] 

sage: X.cycle_index_series().coefficients(4) 

[0, 0, 0, 0] 

The empty species is the zero of the semi-ring of species. 

The following tests that it is neutral with respect to addition:: 

sage: Empt = species.EmptySpecies() 

sage: S = species.CharacteristicSpecies(2) 

sage: X = S + Empt 

sage: X == S # TODO: Not Implemented 

True 

sage: (X.generating_series().coefficients(4) == 

....: S.generating_series().coefficients(4)) 

True 

sage: (X.isotype_generating_series().coefficients(4) == 

....: S.isotype_generating_series().coefficients(4)) 

True 

sage: (X.cycle_index_series().coefficients(4) == 

....: S.cycle_index_series().coefficients(4)) 

True 

The following tests that it is the zero element with respect to 

multiplication:: 

sage: Y = Empt*S 

sage: Y == Empt # TODO: Not Implemented 

True 

sage: Y.generating_series().coefficients(4) 

[0, 0, 0, 0] 

sage: Y.isotype_generating_series().coefficients(4) 

[0, 0, 0, 0] 

sage: Y.cycle_index_series().coefficients(4) 

[0, 0, 0, 0] 

TESTS:: 

sage: Empt = species.EmptySpecies() 

sage: Empt2 = species.EmptySpecies() 

sage: Empt is Empt2 

True 

""" 

def __init__(self, min=None, max=None, weight=None): 

""" 

Initializer for the empty species. 

EXAMPLES:: 

sage: F = species.EmptySpecies() 

sage: F._check() 

True 

sage: F == loads(dumps(F)) 

True 

""" 

# There is no structure at all, so we set min and max accordingly. 

GenericCombinatorialSpecies.__init__(self, weight=weight) 

self._name = "Empty species" 

def _gs(self, series_ring, base_ring): 

""" 

Return the generating series for self. 

EXAMPLES:: 

sage: F = species.EmptySpecies() 

sage: F.generating_series().coefficients(5) # indirect doctest 

[0, 0, 0, 0, 0] 

sage: F.generating_series().count(3) 

0 

sage: F.generating_series().count(4) 

0 

""" 

return series_ring.zero() 

_itgs = _gs 

_cis = _gs 

def _order(self): 

""" 

Returns the order of the generating series. 

EXAMPLES:: 

sage: F = species.EmptySpecies() 

sage: F._order() 

Infinite series order 

""" 

return inf 

def _structures(self, structure_class, labels): 

""" 

Thanks to the couting optimisation, this is never called... Otherwise 

this should return an empty iterator. 

EXAMPLES:: 

sage: F = species.EmptySpecies() 

sage: F.structures([]).list() # indirect doctest 

[] 

sage: F.structures([1,2,3]).list() # indirect doctest 

[] 

""" 

assert False, "This should never be called" 

_default_structure_class = 0 

_isotypes = _structures 

def _equation(self, var_mapping): 

""" 

Returns the right hand side of an algebraic equation satisfied by 

this species. This is a utility function called by the 

algebraic_equation_system method. 

EXAMPLES:: 

sage: C = species.EmptySpecies() 

sage: Qz = QQ['z'] 

sage: R.<node0> = Qz[] 

sage: var_mapping = {'z':Qz.gen(), 'node0':R.gen()} 

sage: C._equation(var_mapping) 

0 

""" 

return 0 

EmptySpecies_class = EmptySpecies