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composition.py
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composition.py
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r"""
Integer compositions
A composition `c` of a nonnegative integer `n` is a list of positive integers
(the *parts* of the composition) with total sum `n`.
This module provides tools for manipulating compositions and enumerated
sets of compositions.
EXAMPLES::
sage: Composition([5, 3, 1, 3])
[5, 3, 1, 3]
sage: list(Compositions(4))
[[1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [1, 3], [2, 1, 1], [2, 2], [3, 1], [4]]
AUTHORS:
- Mike Hansen, Nicolas M. Thiery
- MuPAD-Combinat developers (algorithms and design inspiration)
- Travis Scrimshaw (2013-02-03): Removed ``CombinatorialClass``
"""
#*****************************************************************************
# Copyright (C) 2007 Mike Hansen <mhansen@gmail.com>
# 2009 Nicolas M. Thiery <nthiery at users.sf.net>
#
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#*****************************************************************************
from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.structure.unique_representation import UniqueRepresentation
from sage.structure.parent import Parent
from sage.rings.all import ZZ
from combinat import CombinatorialElement
from cartesian_product import CartesianProduct
from integer_list import IntegerListsLex
import __builtin__
from sage.rings.integer import Integer
from sage.combinat.combinatorial_map import combinatorial_map
class Composition(CombinatorialElement):
r"""
Integer compositions
A composition of a nonnegative integer `n` is a list
`(i_1, \ldots, i_k)` of positive integers with total sum `n`.
EXAMPLES:
The simplest way to create a composition is by specifying its
entries as a list, tuple (or other iterable)::
sage: Composition([3,1,2])
[3, 1, 2]
sage: Composition((3,1,2))
[3, 1, 2]
sage: Composition(i for i in range(2,5))
[2, 3, 4]
You can also create a composition from its code. The *code* of
a composition `(i_1, i_2, \ldots, i_k)` of `n` is a list of length `n`
that consists of a `1` followed by `i_1-1` zeros, then a `1` followed
by `i_2-1` zeros, and so on.
::
sage: Composition([4,1,2,3,5]).to_code()
[1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0]
sage: Composition(code=_)
[4, 1, 2, 3, 5]
sage: Composition([3,1,2,3,5]).to_code()
[1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0]
sage: Composition(code=_)
[3, 1, 2, 3, 5]
You can also create the composition of `n` corresponding to a subset of
`\{1, 2, \ldots, n-1\}` under the bijection that maps the composition
`(i_1, i_2, \ldots, i_k)` of `n` to the subset
`\{i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + \cdots + i_{k-1}\}`
(see :meth:`to_subset`)::
sage: Composition(from_subset=({1, 2, 4}, 5))
[1, 1, 2, 1]
sage: Composition([1, 1, 2, 1]).to_subset()
{1, 2, 4}
The following notation equivalently specifies the composition from the
set `\{i_1 - 1, i_1 + i_2 - 1, i_1 + i_2 + i_3 - 1, \dots, i_1 + \cdots
+ i_{k-1} - 1, n-1\}` or `\{i_1 - 1, i_1 + i_2 - 1, i_1 + i_2 + i_3
- 1, \dots, i_1 + \cdots + i_{k-1} - 1\}` and `n`. This provides
compatibility with Python's `0`-indexing.
::
sage: Composition(descents=[1,0,4,8,11])
[1, 1, 3, 4, 3]
sage: Composition(descents=[0,1,3,4])
[1, 1, 2, 1]
sage: Composition(descents=([0,1,3],5))
[1, 1, 2, 1]
sage: Composition(descents=({0,1,3},5))
[1, 1, 2, 1]
EXAMPLES::
sage: C = Composition([3,1,2])
sage: TestSuite(C).run()
"""
@staticmethod
def __classcall_private__(cls, co=None, descents=None, code=None, from_subset=None):
"""
This constructs a list from optional arguments and delegates the
construction of a :class:`Composition` to the ``element_class()`` call
of the appropriate parent.
EXAMPLES::
sage: Composition([3,2,1])
[3, 2, 1]
sage: Composition(from_subset=({1, 2, 4}, 5))
[1, 1, 2, 1]
sage: Composition(descents=[1,0,4,8,11])
[1, 1, 3, 4, 3]
sage: Composition([4,1,2,3,5]).to_code()
[1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0]
sage: Composition(code=_)
[4, 1, 2, 3, 5]
"""
if descents is not None:
if isinstance(descents, tuple):
return Compositions().from_descents(descents[0], nps=descents[1])
else:
return Compositions().from_descents(descents)
elif code is not None:
return Compositions().from_code(code)
elif from_subset is not None:
return Compositions().from_subset(*from_subset)
elif isinstance(co, Composition):
return co
else:
return Compositions()(list(co))
def _ascii_art_(self):
"""
TESTS::
sage: ascii_art(Compositions(4).list())
[ * ]
[ * ** * * ]
[ * * ** *** * ** * ]
[ *, * , * , * , **, ** , ***, **** ]
sage: Partitions.global_options(diagram_str='#', convention="French")
sage: ascii_art(Compositions(4).list())
[ # ]
[ # # # ## ]
[ # # ## # # ## ### ]
[ #, ##, #, ###, #, ##, #, #### ]
"""
from sage.typeset.ascii_art import ascii_art
return ascii_art(self.to_skew_partition())
def __setstate__(self, state):
r"""
In order to maintain backwards compatibility and be able to unpickle a
old pickle from ``Composition_class`` we have to override the default
``__setstate__``.
EXAMPLES::
sage: loads("x\x9ck`J.NLO\xd5K\xce\xcfM\xca\xccK,\x011\n\xf2\x8b3K2\xf3\xf3\xb8\x9c\x11\xec\xf8\xe4\x9c\xc4\xe2b\xaeBF\xcd\xc6B\xa6\xdaBf\x8dP\xd6\xf8\x8c\xc4\xe2\x8cB\x16? +'\xb3\xb8\xa4\x905\xb6\x90M\x03bZQf^z\xb1^f^Ijzj\x11Wnbvj<\x8cS\xc8\x1e\xcah\xd8\x1aT\xc8\x91\x01d\x18\x01\x19\x9c\x19P\x11\xae\xd4\xd2$=\x00eW0g")
[1, 2, 1]
sage: loads(dumps( Composition([1,2,1]) )) # indirect doctest
[1, 2, 1]
"""
if isinstance(state, dict): # for old pickles from Composition_class
self._set_parent(Compositions())
self.__dict__ = state
else:
self._set_parent(state[0])
self.__dict__ = state[1]
@combinatorial_map(order=2, name='conjugate')
def conjugate(self):
r"""
Return the conjugate of the composition ``self``.
The conjugate of a composition `I` is defined as the
complement (see :meth:`complement`) of the reverse composition
(see :meth:`reversed`) of `I`.
An equivalent definition of the conjugate goes by saying that
the ribbon shape of the conjugate of a composition `I` is the
conjugate of the ribbon shape of `I`. (The ribbon shape of a
composition is returned by :meth:`to_skew_partition`.)
This implementation uses the algorithm from mupad-combinat.
EXAMPLES::
sage: Composition([1, 1, 3, 1, 2, 1, 3]).conjugate()
[1, 1, 3, 3, 1, 3]
The ribbon shape of the conjugate of `I` is the conjugate of
the ribbon shape of `I`::
sage: all( I.conjugate().to_skew_partition()
....: == I.to_skew_partition().conjugate()
....: for I in Compositions(4) )
True
TESTS::
sage: parent(list(Compositions(1))[0].conjugate())
Compositions of 1
sage: parent(list(Compositions(0))[0].conjugate())
Compositions of 0
"""
comp = self
if comp == []:
return self
n = len(comp)
coofcp = [sum(comp[:j])-j+1 for j in range(1,n+1)]
cocjg = []
for i in range(n-1):
cocjg += [i+1 for _ in range(0, (coofcp[n-i-1]-coofcp[n-i-2]))]
cocjg += [n for j in range(coofcp[0])]
return self.parent()([cocjg[0]] + [cocjg[i]-cocjg[i-1]+1 for i in range(1,len(cocjg))])
@combinatorial_map(order=2, name='reversed')
def reversed(self):
r"""
Return the reverse composition of ``self``.
The reverse composition of a composition `(i_1, i_2, \ldots, i_k)`
is defined as the composition `(i_k, i_{k-1}, \ldots, i_1)`.
EXAMPLES::
sage: Composition([1, 1, 3, 1, 2, 1, 3]).reversed()
[3, 1, 2, 1, 3, 1, 1]
"""
return self.parent()(reversed(self))
@combinatorial_map(order=2, name='complement')
def complement(self):
r"""
Return the complement of the composition ``self``.
The complement of a composition `I` is defined as follows:
If `I` is the empty composition, then the complement is the empty
composition as well. Otherwise, let `S` be the descent set of `I`
(that is, the subset
`\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}`
of `\{ 1, 2, \ldots, |I|-1 \}`, where `I` is written as
`(i_1, i_2, \ldots, i_k)`). Then, the complement of `I` is
defined as the composition of size `|I|` whose descent set is
`\{ 1, 2, \ldots, |I|-1 \} \setminus S`.
The complement of a composition `I` also is the reverse
composition (:meth:`reversed`) of the conjugate
(:meth:`conjugate`) of `I`.
EXAMPLES::
sage: Composition([1, 1, 3, 1, 2, 1, 3]).conjugate()
[1, 1, 3, 3, 1, 3]
sage: Composition([1, 1, 3, 1, 2, 1, 3]).complement()
[3, 1, 3, 3, 1, 1]
"""
return self.conjugate().reversed()
def __add__(self, other):
"""
Return the concatenation of two compositions.
EXAMPLES::
sage: Composition([1, 1, 3]) + Composition([4, 1, 2])
[1, 1, 3, 4, 1, 2]
TESTS::
sage: Composition([]) + Composition([]) == Composition([])
True
"""
return Compositions()(list(self)+list(other))
def size(self):
"""
Return the size of ``self``, that is the sum of its parts.
EXAMPLES::
sage: Composition([7,1,3]).size()
11
"""
return sum(self)
@staticmethod
def sum(compositions):
"""
Return the concatenation of the given compositions.
INPUT:
- ``compositions`` -- a list (or iterable) of compositions
EXAMPLES::
sage: Composition.sum([Composition([1, 1, 3]), Composition([4, 1, 2]), Composition([3,1])])
[1, 1, 3, 4, 1, 2, 3, 1]
Any iterable can be provided as input::
sage: Composition.sum([Composition([i,i]) for i in [4,1,3]])
[4, 4, 1, 1, 3, 3]
Empty inputs are handled gracefully::
sage: Composition.sum([]) == Composition([])
True
"""
return sum(compositions, Compositions()([]))
def near_concatenation(self, other):
r"""
Return the near-concatenation of two nonempty compositions
``self`` and ``other``.
The near-concatenation `I \odot J` of two nonempty compositions
`I` and `J` is defined as the composition
`(i_1, i_2, \ldots , i_{n-1}, i_n + j_1, j_2, j_3, \ldots , j_m)`,
where `(i_1, i_2, \ldots , i_n) = I` and
`(j_1, j_2, \ldots , j_m) = J`.
This method returns ``None`` if one of the two input
compositions is empty.
EXAMPLES::
sage: Composition([1, 1, 3]).near_concatenation(Composition([4, 1, 2]))
[1, 1, 7, 1, 2]
sage: Composition([6]).near_concatenation(Composition([1, 5]))
[7, 5]
sage: Composition([1, 5]).near_concatenation(Composition([6]))
[1, 11]
TESTS::
sage: Composition([]).near_concatenation(Composition([]))
<BLANKLINE>
sage: Composition([]).near_concatenation(Composition([2, 1]))
<BLANKLINE>
sage: Composition([3, 2]).near_concatenation(Composition([]))
<BLANKLINE>
"""
if len(self) == 0 or len(other) == 0:
return None
return Compositions()(list(self)[:-1] + [self[-1] + other[0]] + list(other)[1:])
def ribbon_decomposition(self, other, check=True):
r"""
Return a pair describing the ribbon decomposition of a composition
``self`` with respect to a composition ``other`` of the same size.
If `I` and `J` are two compositions of the same nonzero size, then
the ribbon decomposition of `I` with respect to `J` is defined as
follows: Write `I` and `J` as `I = (i_1, i_2, \ldots , i_n)` and
`J = (j_1, j_2, \ldots , j_m)`. Then, the equality
`I = I_1 \bullet I_2 \bullet \ldots \bullet I_m` holds for a
unique `m`-tuple `(I_1, I_2, \ldots , I_m)` of compositions such
that each `I_k` has size `j_k` and for a unique choice of `m-1`
signs `\bullet` each of which is either the concatenation sign
`\cdot` or the near-concatenation sign `\odot` (see
:meth:`__add__` and :meth:`near_concatenation` for the definitions
of these two signs). This `m`-tuple and this choice of signs
together are said to form the ribbon decomposition of `I` with
respect to `J`. If `I` and `J` are empty, then the same definition
applies, except that there are `0` rather than `m-1` signs.
See Section 4.8 of [NCSF1]_.
INPUT:
- ``other`` -- composition of same size as ``self``
- ``check`` -- (default: ``True``) a Boolean determining whether
to check the input compositions for having the same size
OUTPUT:
- a pair ``(u, v)``, where ``u`` is a tuple of compositions
(corresponding to the `m`-tuple `(I_1, I_2, \ldots , I_m)` in
the above definition), and ``v`` is a tuple of `0`s and `1`s
(encoding the choice of signs `\bullet` in the above definition,
with a `0` standing for `\cdot` and a `1` standing for `\odot`).
EXAMPLES::
sage: Composition([3, 1, 1, 3, 1]).ribbon_decomposition([4, 3, 2])
(([3, 1], [1, 2], [1, 1]), (0, 1))
sage: Composition([9, 6]).ribbon_decomposition([1, 3, 6, 3, 2])
(([1], [3], [5, 1], [3], [2]), (1, 1, 1, 1))
sage: Composition([9, 6]).ribbon_decomposition([1, 3, 5, 1, 3, 2])
(([1], [3], [5], [1], [3], [2]), (1, 1, 0, 1, 1))
sage: Composition([1, 1, 1, 1, 1]).ribbon_decomposition([3, 2])
(([1, 1, 1], [1, 1]), (0,))
sage: Composition([4, 2]).ribbon_decomposition([6])
(([4, 2],), ())
sage: Composition([]).ribbon_decomposition([])
((), ())
Let us check that the defining property
`I = I_1 \bullet I_2 \bullet \ldots \bullet I_m` is satisfied::
sage: def compose_back(u, v):
....: comp = u[0]
....: r = len(v)
....: if len(u) != r + 1:
....: raise ValueError("something is wrong")
....: for i in range(r):
....: if v[i] == 0:
....: comp += u[i + 1]
....: else:
....: comp = comp.near_concatenation(u[i + 1])
....: return comp
sage: all( all( all( compose_back(*(I.ribbon_decomposition(J))) == I
....: for J in Compositions(n) )
....: for I in Compositions(n) )
....: for n in range(1, 5) )
True
TESTS::
sage: Composition([3, 1, 1, 3, 1]).ribbon_decomposition([4, 3, 1])
Traceback (most recent call last):
...
ValueError: [3, 1, 1, 3, 1] is not the same size as [4, 3, 1]
AUTHORS:
- Darij Grinberg (2013-08-29)
"""
# Speaking in terms of the definition in the docstring, we have
# I = self and J = other.
if check and (sum(self) != sum(other)):
raise ValueError("{} is not the same size as {}".format(self, other))
factors = []
signs = []
I_iter = iter(self)
i = 0
for j in other:
current_factor = []
current_factor_size = 0
while True:
if i == 0:
try:
i = next(I_iter)
except StopIteration:
factors.append(Compositions()(current_factor))
return (tuple(factors), tuple(signs))
if current_factor_size + i <= j:
current_factor.append(i)
current_factor_size += i
i = 0
else:
if j == current_factor_size:
signs.append(0)
else:
current_factor.append(j - current_factor_size)
i -= j - current_factor_size
signs.append(1)
factors.append(Compositions()(current_factor))
break
return (tuple(factors), tuple(signs))
def join(self, other, check=True):
r"""
Return the join of ``self`` with a composition ``other`` of the
same size.
The join of two compositions `I` and `J` of size `n` is the
coarsest composition of `n` which refines each of `I` and `J`. It
can be described as the composition whose descent set is the
union of the descent sets of `I` and `J`. It is also the
concatenation of `I_1, I_2, \cdots , I_m`, where
`I = I_1 \bullet I_2 \bullet \ldots \bullet I_m` is the ribbon
decomposition of `I` with respect to `J` (see
:meth:`ribbon_decomposition`).
INPUT:
- ``other`` -- composition of same size as ``self``
- ``check`` -- (default: ``True``) a Boolean determining whether
to check the input compositions for having the same size
OUTPUT:
- the join of the compositions ``self`` and ``other``
EXAMPLES::
sage: Composition([3, 1, 1, 3, 1]).join([4, 3, 2])
[3, 1, 1, 2, 1, 1]
sage: Composition([9, 6]).join([1, 3, 6, 3, 2])
[1, 3, 5, 1, 3, 2]
sage: Composition([9, 6]).join([1, 3, 5, 1, 3, 2])
[1, 3, 5, 1, 3, 2]
sage: Composition([1, 1, 1, 1, 1]).join([3, 2])
[1, 1, 1, 1, 1]
sage: Composition([4, 2]).join([3, 3])
[3, 1, 2]
sage: Composition([]).join([])
[]
Let us verify on small examples that the join
of `I` and `J` refines both of `I` and `J`::
sage: all( all( I.join(J).is_finer(I) and
....: I.join(J).is_finer(J)
....: for J in Compositions(4) )
....: for I in Compositions(4) )
True
and is the coarsest composition to do so::
sage: all( all( all( K.is_finer(I.join(J))
....: for K in I.finer()
....: if K.is_finer(J) )
....: for J in Compositions(3) )
....: for I in Compositions(3) )
True
Let us check that the join of `I` and `J` is indeed the
conctenation of `I_1, I_2, \cdots , I_m`, where
`I = I_1 \bullet I_2 \bullet \ldots \bullet I_m` is the ribbon
decomposition of `I` with respect to `J`::
sage: all( all( Composition.sum(I.ribbon_decomposition(J)[0])
....: == I.join(J) for J in Compositions(4) )
....: for I in Compositions(4) )
True
Also, the descent set of the join of `I` and `J` is the
union of the descent sets of `I` and `J`::
sage: all( all( I.to_subset().union(J.to_subset())
....: == I.join(J).to_subset()
....: for J in Compositions(4) )
....: for I in Compositions(4) )
True
TESTS::
sage: Composition([3, 1, 1, 3, 1]).join([4, 3, 1])
Traceback (most recent call last):
...
ValueError: [3, 1, 1, 3, 1] is not the same size as [4, 3, 1]
.. SEEALSO::
:meth:`meet`, :meth:`ribbon_decomposition`
AUTHORS:
- Darij Grinberg (2013-09-05)
"""
# The following code is a slimmed down version of the
# ribbon_decomposition method. It is a lot faster than
# using to_subset() and from_subset, and also a lot
# faster than ribbon_decomposition.
# Speaking in terms of the definition in the docstring, we have
# I = self and J = other.
if check and (sum(self) != sum(other)):
raise ValueError("{} is not the same size as {}".format(self, other))
factors = []
I_iter = iter(self)
i = 0
for j in other:
current_factor_size = 0
while True:
if i == 0:
try:
i = next(I_iter)
except StopIteration:
return Compositions()(factors)
if current_factor_size + i <= j:
factors.append(i)
current_factor_size += i
i = 0
else:
if not j == current_factor_size:
factors.append(j - current_factor_size)
i -= j - current_factor_size
break
return Compositions()(factors)
sup = join
def meet(self, other, check=True):
r"""
Return the meet of ``self`` with a composition ``other`` of the
same size.
The meet of two compositions `I` and `J` of size `n` is the
finest composition of `n` which is coarser than each of `I` and
`J`. It can be described as the composition whose descent set is
the intersection of the descent sets of `I` and `J`.
INPUT:
- ``other`` -- composition of same size as ``self``
- ``check`` -- (default: ``True``) a Boolean determining whether
to check the input compositions for having the same size
OUTPUT:
- the meet of the compositions ``self`` and ``other``
EXAMPLES::
sage: Composition([3, 1, 1, 3, 1]).meet([4, 3, 2])
[4, 5]
sage: Composition([9, 6]).meet([1, 3, 6, 3, 2])
[15]
sage: Composition([9, 6]).meet([1, 3, 5, 1, 3, 2])
[9, 6]
sage: Composition([1, 1, 1, 1, 1]).meet([3, 2])
[3, 2]
sage: Composition([4, 2]).meet([3, 3])
[6]
sage: Composition([]).meet([])
[]
sage: Composition([1]).meet([1])
[1]
Let us verify on small examples that the meet
of `I` and `J` is coarser than both of `I` and `J`::
sage: all( all( I.is_finer(I.meet(J)) and
....: J.is_finer(I.meet(J))
....: for J in Compositions(4) )
....: for I in Compositions(4) )
True
and is the finest composition to do so::
sage: all( all( all( I.meet(J).is_finer(K)
....: for K in I.fatter()
....: if J.is_finer(K) )
....: for J in Compositions(3) )
....: for I in Compositions(3) )
True
The descent set of the meet of `I` and `J` is the
intersection of the descent sets of `I` and `J`::
sage: def test_meet(n):
....: return all( all( I.to_subset().intersection(J.to_subset())
....: == I.meet(J).to_subset()
....: for J in Compositions(n) )
....: for I in Compositions(n) )
sage: all( test_meet(n) for n in range(1, 5) )
True
sage: all( test_meet(n) for n in range(5, 9) ) # long time
True
TESTS::
sage: Composition([3, 1, 1, 3, 1]).meet([4, 3, 1])
Traceback (most recent call last):
...
ValueError: [3, 1, 1, 3, 1] is not the same size as [4, 3, 1]
.. SEEALSO::
:meth:`join`
AUTHORS:
- Darij Grinberg (2013-09-05)
"""
# The following code is much faster than using to_subset()
# and from_subset.
# Speaking in terms of the definition in the docstring, we have
# I = self and J = other.
if check and (sum(self) != sum(other)):
raise ValueError("{} is not the same size as {}".format(self, other))
factors = []
current_part = 0
I_iter = iter(self)
i = 0
for j in other:
current_factor_size = 0
while True:
if i == 0:
try:
i = next(I_iter)
except StopIteration:
factors.append(current_part)
return Compositions()(factors)
if current_factor_size + i <= j:
current_part += i
current_factor_size += i
i = 0
else:
if j == current_factor_size:
factors.append(current_part)
current_part = 0
else:
i -= j - current_factor_size
current_part += j - current_factor_size
break
return Compositions()(factors)
inf = meet
def finer(self):
"""
Return the set of compositions which are finer than ``self``.
EXAMPLES::
sage: C = Composition([3,2]).finer()
sage: C.cardinality()
8
sage: list(C)
[[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 1, 1], [1, 2, 2], [2, 1, 1, 1], [2, 1, 2], [3, 1, 1], [3, 2]]
"""
return CartesianProduct(*[Compositions(i) for i in self]).map(Composition.sum)
def is_finer(self, co2):
"""
Return ``True`` if the composition ``self`` is finer than the
composition ``co2``; otherwise, return ``False``.
EXAMPLES::
sage: Composition([4,1,2]).is_finer([3,1,3])
False
sage: Composition([3,1,3]).is_finer([4,1,2])
False
sage: Composition([1,2,2,1,1,2]).is_finer([5,1,3])
True
sage: Composition([2,2,2]).is_finer([4,2])
True
"""
co1 = self
if sum(co1) != sum(co2):
raise ValueError("compositions self (= %s) and co2 (= %s) must be of the same size"%(self, co2))
sum1 = 0
sum2 = 0
i1 = 0
for j2 in co2:
sum2 += j2
while sum1 < sum2:
sum1 += co1[i1]
i1 += 1
if sum1 > sum2:
return False
return True
def fatten(self, grouping):
r"""
Return the composition fatter than ``self``, obtained by grouping
together consecutive parts according to ``grouping``.
INPUT:
- ``grouping`` -- a composition whose sum is the length of ``self``
EXAMPLES:
Let us start with the composition::
sage: c = Composition([4,5,2,7,1])
With ``grouping`` equal to `(1, \ldots, 1)`, `c` is left unchanged::
sage: c.fatten(Composition([1,1,1,1,1]))
[4, 5, 2, 7, 1]
With ``grouping`` equal to `(\ell)` where `\ell` is the length of
`c`, this yields the coarsest composition above `c`::
sage: c.fatten(Composition([5]))
[19]
Other values for ``grouping`` yield (all the) other compositions
coarser than `c`::
sage: c.fatten(Composition([2,1,2]))
[9, 2, 8]
sage: c.fatten(Composition([3,1,1]))
[11, 7, 1]
TESTS::
sage: Composition([]).fatten(Composition([]))
[]
sage: c.fatten(Composition([3,1,1])).__class__ == c.__class__
True
"""
result = [None] * len(grouping)
j = 0
for i in range(len(grouping)):
result[i] = sum(self[j:j+grouping[i]])
j += grouping[i]
return Compositions()(result)
def fatter(self):
"""
Return the set of compositions which are fatter than ``self``.
Complexity for generation: `O(|c|)` memory, `O(|r|)` time where `|c|`
is the size of ``self`` and `r` is the result.
EXAMPLES::
sage: C = Composition([4,5,2]).fatter()
sage: C.cardinality()
4
sage: list(C)
[[4, 5, 2], [4, 7], [9, 2], [11]]
Some extreme cases::
sage: list(Composition([5]).fatter())
[[5]]
sage: list(Composition([]).fatter())
[[]]
sage: list(Composition([1,1,1,1]).fatter()) == list(Compositions(4))
True
"""
return Compositions(len(self)).map(self.fatten)
def refinement_splitting(self, J):
r"""
Return the refinement splitting of ``self`` according to ``J``.
INPUT:
- ``J`` -- A composition such that ``self`` is finer than ``J``
OUTPUT:
- the unique list of compositions `(I^{(p)})_{p=1, \ldots , m}`,
obtained by splitting `I`, such that
`|I^{(p)}| = J_p` for all `p = 1, \ldots, m`.
.. SEEALSO::
:meth:`refinement_splitting_lengths`
EXAMPLES::
sage: Composition([1,2,2,1,1,2]).refinement_splitting([5,1,3])
[[1, 2, 2], [1], [1, 2]]
sage: Composition([]).refinement_splitting([])
[]
sage: Composition([3]).refinement_splitting([2])
Traceback (most recent call last):
...
ValueError: compositions self (= [3]) and J (= [2]) must be of the same size
sage: Composition([2,1]).refinement_splitting([1,2])
Traceback (most recent call last):
...
ValueError: composition J (= [2, 1]) does not refine self (= [1, 2])
"""
I = self
if sum(I) != sum(J):
#Error: compositions are not of the same size
raise ValueError("compositions self (= %s) and J (= %s) must be of the same size"%(I, J))
sum1 = 0
sum2 = 0
i1 = -1
decomp = []
for j2 in J:
new_comp = []
sum2 += j2
while sum1 < sum2:
i1 += 1
new_comp.append(I[i1])
sum1 += new_comp[-1]
if sum1 > sum2:
raise ValueError("composition J (= %s) does not refine self (= %s)"%(I, J))
decomp.append(Compositions()(new_comp))
return decomp
def refinement_splitting_lengths(self, J):
"""
Return the lengths of the compositions in the refinement splitting of
``self`` according to ``J``.
.. SEEALSO::
:meth:`refinement_splitting` for the definition of refinement splitting
EXAMPLES::
sage: Composition([1,2,2,1,1,2]).refinement_splitting_lengths([5,1,3])
[3, 1, 2]
sage: Composition([]).refinement_splitting_lengths([])
[]
sage: Composition([3]).refinement_splitting_lengths([2])
Traceback (most recent call last):
...
ValueError: compositions self (= [3]) and J (= [2]) must be of the same size
sage: Composition([2,1]).refinement_splitting_lengths([1,2])
Traceback (most recent call last):
...
ValueError: composition J (= [2, 1]) does not refine self (= [1, 2])
"""
return Compositions()([len(_) for _ in self.refinement_splitting(J)])
def major_index(self):
"""
Return the major index of ``self``. The major index is
defined as the sum of the descents.
EXAMPLES::
sage: Composition([1, 1, 3, 1, 2, 1, 3]).major_index()
31
"""
co = self
lv = len(co)
if lv == 1:
return 0
else:
return sum([(lv-(i+1))*co[i] for i in range(lv)])
def to_code(self):
r"""
Return the code of the composition ``self``. The code of a composition
`I` is a list of length `\mathrm{size}(I)` of 1s and 0s such that
there is a 1 wherever a new part starts. (Exceptional case: When the
composition is empty, the code is ``[0]``.)
EXAMPLES::
sage: Composition([4,1,2,3,5]).to_code()
[1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0]
"""
if self == []:
return [0]
code = []
for i in self:
code += [1] + [0]*(i-1)
return code
def partial_sums(self, final=True):
r"""
The partial sums of the sequence defined by the entries of the
composition.
If `I = (i_1, \ldots, i_m)` is a composition, then the partial sums of
the entries of the composition are
`[i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_m]`.
INPUT:
- ``final`` -- (default: ``True``) whether or not to include the final
partial sum, which is always the size of the composition.
.. SEEALSO::
:meth:`to_subset`
EXAMPLES::
sage: Composition([1,1,3,1,2,1,3]).partial_sums()
[1, 2, 5, 6, 8, 9, 12]