Coverage for local/lib/python2.7/site-packages/sage/combinat/composition.py: 78%

sage: loads(b"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()

"""

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([]))

sage: Composition([]).near_concatenation(Composition([2, 1]))

sage: Composition([3, 2]).near_concatenation(Composition([]))

"""

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

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`::

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()

sage: C.list()

[[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]]

sage: Composition([]).finer()

{[]}

"""

if not self:

return FiniteEnumeratedSet([self])

else:

return cartesian_product([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()

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()

"""

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]

With ``final = False``, the last partial sum is not included::

sage: Composition([1,1,3,1,2,1,3]).partial_sums(final=False)

[1, 2, 5, 6, 8, 9]

"""

s = 0

partial_sums = []

for i in self:

s += i

partial_sums.append(s)

if final is False:

partial_sums.pop()

return partial_sums

def to_subset(self, final=False):

r"""

The subset corresponding to ``self`` under the bijection (see below)

between compositions of `n` and subsets of `\{1, 2, \ldots, n-1\}`.

The bijection maps a composition `(i_1, \ldots, i_k)` of `n` to

`\{i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + \cdots + i_{k-1}\}`.

INPUT:

- ``final`` -- (default: ``False``) whether or not to include the final

partial sum, which is always the size of the composition.

.. SEEALSO::

:meth:`partial_sums`

EXAMPLES::

sage: Composition([1,1,3,1,2,1,3]).to_subset()

{1, 2, 5, 6, 8, 9}

sage: for I in Compositions(3): print(I.to_subset())

{1, 2}

{1}

{2}

{}

With ``final=True``, the sum of all the elements of the composition is

included in the subset::

sage: Composition([1,1,3,1,2,1,3]).to_subset(final=True)

{1, 2, 5, 6, 8, 9, 12}

TESTS:

We verify that ``to_subset`` is indeed a bijection for compositions of

size `n = 8`::

sage: n = 8

sage: all(Composition(from_subset=(S, n)).to_subset() == S \

....: for S in Subsets(n-1))

True

sage: all(Composition(from_subset=(I.to_subset(), n)) == I \

....: for I in Compositions(n))

True

"""

from sage.sets.set import Set

return Set(self.partial_sums(final=final))

def descents(self, final_descent=False):

r"""

This gives one fewer than the partial sums of the composition.

This is here to maintain some sort of backward compatibility, even

through the original implementation was broken (it gave the wrong

answer). The same information can be found in :meth:`partial_sums`.

.. SEEALSO::

:meth:`partial_sums`

INPUT:

- ``final_descent`` -- (Default: ``False``) a boolean integer

OUTPUT:

- the list of partial sums of ``self`` with each part

decremented by `1`. This includes the sum of all entries when

``final_descent`` is ``True``.

EXAMPLES::

sage: c = Composition([2,1,3,2])

sage: c.descents()

[1, 2, 5]

sage: c.descents(final_descent=True)

[1, 2, 5, 7]

"""

return [i - 1 for i in self.partial_sums(final=final_descent)]

def peaks(self):

"""

Return a list of the peaks of the composition ``self``. The

peaks of a composition are the descents which do not

immediately follow another descent.

EXAMPLES::

sage: Composition([1, 1, 3, 1, 2, 1, 3]).peaks()

[4, 7]

"""

descents = dict((d-1,True) for d in self.to_subset(final=True))

return [i+1 for i in range(len(self))

if i not in descents and i+1 in descents]

@combinatorial_map(name='to partition')

def to_partition(self):

"""

Return the partition obtained by sorting ``self`` into decreasing

order.

EXAMPLES::

sage: Composition([2,1,3]).to_partition()

[3, 2, 1]

sage: Composition([4,2,2]).to_partition()

[4, 2, 2]

sage: Composition([]).to_partition()

[]

"""

from sage.combinat.partition import Partition

return Partition(sorted(self, reverse=True))

def to_skew_partition(self, overlap=1):

"""

Return the skew partition obtained from ``self``. This is a

skew partition whose rows have the entries of ``self`` as their

length, taken in reverse order (so the first entry of ``self``

is the length of the lowermost row, etc.). The parameter

``overlap`` indicates the number of cells on each row that are

directly below cells of the previous row. When it is set to `1`

(its default value), the result is the ribbon shape of ``self``.

EXAMPLES::

sage: Composition([3,4,1]).to_skew_partition()

[6, 6, 3] / [5, 2]

sage: Composition([3,4,1]).to_skew_partition(overlap=0)

[8, 7, 3] / [7, 3]

sage: Composition([]).to_skew_partition()

[] / []

sage: Composition([1,2]).to_skew_partition()

[2, 1] / []

sage: Composition([2,1]).to_skew_partition()

[2, 2] / [1]

"""

from sage.combinat.skew_partition import SkewPartition

outer = []

inner = []

sum_outer = -1*overlap

for k in self[:-1]:

outer.append(k + sum_outer + overlap)

sum_outer += k - overlap

inner.append(sum_outer + overlap)

if self != []:

outer.append(self[-1] + sum_outer + overlap)

else:

return SkewPartition([[],[]])

return SkewPartition(

[ [x for x in reversed(outer) if x != 0],

[x for x in reversed(inner) if x != 0] ])

def shuffle_product(self, other, overlap=False):

r"""

The (overlapping) shuffles of ``self`` and ``other``.

Suppose `I = (i_1, \ldots, i_k)` and `J = (j_1, \ldots, j_l)` are two

compositions. A *shuffle* of `I` and `J` is a composition of length

`k + l` that contains both `I` and `J` as subsequences.

More generally, an *overlapping shuffle* of `I` and `J` is obtained by

distributing the elements of `I` and `J` (preserving the relative

ordering of these elements) among the positions of an empty list; an

element of `I` and an element of `J` are permitted to share the same

position, in which case they are replaced by their sum. In particular,

a shuffle of `I` and `J` is an overlapping shuffle of `I` and `J`.

INPUT:

- ``other`` -- composition

- ``overlap`` -- boolean (default: ``False``); if ``True``, the

overlapping shuffle product is returned.

OUTPUT:

An enumerated set (allowing for multiplicities)

EXAMPLES:

The shuffle product of `[2,2]` and `[1,1,3]`::

sage: alph = Composition([2,2])

sage: beta = Composition([1,1,3])

sage: S = alph.shuffle_product(beta); S

Shuffle product of [2, 2] and [1, 1, 3]

sage: S.list()

[[2, 2, 1, 1, 3], [2, 1, 2, 1, 3], [2, 1, 1, 2, 3], [2, 1, 1, 3, 2], [1, 2, 2, 1, 3], [1, 2, 1, 2, 3], [1, 2, 1, 3, 2], [1, 1, 2, 2, 3], [1, 1, 2, 3, 2], [1, 1, 3, 2, 2]]

The *overlapping* shuffle product of `[2,2]` and `[1,1,3]`::

sage: alph = Composition([2,2])

sage: beta = Composition([1,1,3])

sage: O = alph.shuffle_product(beta, overlap=True); O

Overlapping shuffle product of [2, 2] and [1, 1, 3]

sage: O.list()

[[2, 2, 1, 1, 3], [2, 1, 2, 1, 3], [2, 1, 1, 2, 3], [2, 1, 1, 3, 2], [1, 2, 2, 1, 3], [1, 2, 1, 2, 3], [1, 2, 1, 3, 2], [1, 1, 2, 2, 3], [1, 1, 2, 3, 2], [1, 1, 3, 2, 2], [3, 2, 1, 3], [2, 3, 1, 3], [3, 1, 2, 3], [2, 1, 3, 3], [3, 1, 3, 2], [2, 1, 1, 5], [1, 3, 2, 3], [1, 2, 3, 3], [1, 3, 3, 2], [1, 2, 1, 5], [1, 1, 5, 2], [1, 1, 2, 5], [3, 3, 3], [3, 1, 5], [1, 3, 5]]

Note that the shuffle product of two compositions can include the same

composition more than once since a composition can be a shuffle of two

compositions in several ways. For example::

sage: S = Composition([1]).shuffle_product([1]); S

Shuffle product of [1] and [1]

sage: S.list()

[[1, 1], [1, 1]]

sage: O = Composition([1]).shuffle_product([1], overlap=True); O

Overlapping shuffle product of [1] and [1]

sage: O.list()

[[1, 1], [1, 1], [2]]

TESTS::

sage: Composition([]).shuffle_product([]).list()

[[]]

"""

if overlap:

from sage.combinat.words.shuffle_product import ShuffleProduct_overlapping

return ShuffleProduct_overlapping(self, other)

else:

from sage.combinat.words.shuffle_product import ShuffleProduct_w1w2

return ShuffleProduct_w1w2(self, other)

def wll_gt(self, co2):

"""

Return ``True`` if the composition ``self`` is greater than the

composition ``co2`` with respect to the wll-ordering; otherwise,

return ``False``.

The wll-ordering is a total order on the set of all compositions

defined as follows: A composition `I` is greater than a

composition `J` if and only if one of the following conditions

holds:

- The size of `I` is greater than the size of `J`.

- The size of `I` equals the size of `J`, but the length of `I`

is greater than the length of `J`.

- The size of `I` equals the size of `J`, and the length of `I`

equals the length of `J`, but `I` is lexicographically

greater than `J`.

("wll-ordering" is short for "weight, length, lexicographic

ordering".)

EXAMPLES::

sage: Composition([4,1,2]).wll_gt([3,1,3])

True

sage: Composition([7]).wll_gt([4,1,2])

False

sage: Composition([8]).wll_gt([4,1,2])

True

sage: Composition([3,2,2,2]).wll_gt([5,2])

True

sage: Composition([]).wll_gt([3])

False

sage: Composition([2,1]).wll_gt([2,1])

False

sage: Composition([2,2,2]).wll_gt([4,2])

True

sage: Composition([4,2]).wll_gt([2,2,2])

False

sage: Composition([1,1,2]).wll_gt([2,2])

True

sage: Composition([2,2]).wll_gt([1,3])

True

sage: Composition([2,1,2]).wll_gt([])

True

"""

co1 = self

if sum(co1) > sum(co2):

return True

elif sum(co1) < sum(co2):

return False

if len(co1) > len(co2):

return True

if len(co1) < len(co2):

return False

for i in range(len(co1)):

if co1[i] > co2[i]:

return True

elif co1[i] < co2[i]:

return False

##############################################################

class Compositions(UniqueRepresentation, Parent):

r"""

Set of integer compositions.

A composition `c` of a nonnegative integer `n` is a list of

positive integers with total sum `n`.

.. SEEALSO::

- :class:`Composition`

- :class:`Partitions`

- :class:`IntegerVectors`

EXAMPLES:

There are 8 compositions of 4::

sage: Compositions(4).cardinality()

Here is the list of them::

sage: Compositions(4).list()

[[1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [1, 3], [2, 1, 1], [2, 2], [3, 1], [4]]

You can use the ``.first()`` method to get the 'first' composition of

a number::

sage: Compositions(4).first()

[1, 1, 1, 1]

You can also calculate the 'next' composition given the current

one::

sage: Compositions(4).next([1,1,2])

[1, 2, 1]

If `n` is not specified, this returns the combinatorial class of

all (non-negative) integer compositions::

sage: Compositions()

Compositions of non-negative integers

sage: [] in Compositions()

True

sage: [2,3,1] in Compositions()

True

sage: [-2,3,1] in Compositions()

False

If `n` is specified, it returns the class of compositions of `n`::

sage: Compositions(3)

Compositions of 3

sage: list(Compositions(3))

[[1, 1, 1], [1, 2], [2, 1], [3]]

sage: Compositions(3).cardinality()

The following examples show how to test whether or not an object

is a composition::

sage: [3,4] in Compositions()

True

sage: [3,4] in Compositions(7)

True

sage: [3,4] in Compositions(5)

False

Similarly, one can check whether or not an object is a composition

which satisfies further constraints::

sage: [4,2] in Compositions(6, inner=[2,2])

True

sage: [4,2] in Compositions(6, inner=[2,3])

False

sage: [4,1] in Compositions(5, inner=[2,1], max_slope = 0)

True

An example with incompatible constraints::

sage: [4,2] in Compositions(6, inner=[2,2], min_part=3)

False

The options ``length``, ``min_length``, and ``max_length`` can be used

to set length constraints on the compositions. For example, the

compositions of 4 of length equal to, at least, and at most 2 are

given by::

sage: Compositions(4, length=2).list()

[[3, 1], [2, 2], [1, 3]]

sage: Compositions(4, min_length=2).list()

[[3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]

sage: Compositions(4, max_length=2).list()

[[4], [3, 1], [2, 2], [1, 3]]

Setting both ``min_length`` and ``max_length`` to the same value is

equivalent to setting ``length`` to this value::

sage: Compositions(4, min_length=2, max_length=2).list()

[[3, 1], [2, 2], [1, 3]]

The options ``inner`` and ``outer`` can be used to set part-by-part

containment constraints. The list of compositions of 4 bounded

above by ``[3,1,2]`` is given by::

sage: list(Compositions(4, outer=[3,1,2]))

[[3, 1], [2, 1, 1], [1, 1, 2]]

``outer`` sets ``max_length`` to the length of its argument. Moreover, the

parts of ``outer`` may be infinite to clear the constraint on specific

parts. This is the list of compositions of 4 of length at most 3

such that the first and third parts are at most 1::

sage: Compositions(4, outer=[1,oo,1]).list()

[[1, 3], [1, 2, 1]]

This is the list of compositions of 4 bounded below by ``[1,1,1]``::

sage: Compositions(4, inner=[1,1,1]).list()

[[2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]

The options ``min_slope`` and ``max_slope`` can be used to set constraints

on the slope, that is the difference ``p[i+1]-p[i]`` of two

consecutive parts. The following is the list of weakly increasing

compositions of 4::

sage: Compositions(4, min_slope=0).list()

[[4], [2, 2], [1, 3], [1, 1, 2], [1, 1, 1, 1]]

Here are the weakly decreasing ones::

sage: Compositions(4, max_slope=0).list()

[[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]

The following is the list of compositions of 4 such that two

consecutive parts differ by at most one::

sage: Compositions(4, min_slope=-1, max_slope=1).list()

[[4], [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]

The constraints can be combined together in all reasonable ways.

This is the list of compositions of 5 of length between 2 and 4

such that the difference between consecutive parts is between -2

and 1::

sage: Compositions(5, max_slope=1, min_slope=-2, min_length=2, max_length=4).list()

[[3, 2], [3, 1, 1], [2, 3], [2, 2, 1], [2, 1, 2], [2, 1, 1, 1], [1, 2, 2], [1, 2, 1, 1], [1, 1, 2, 1], [1, 1, 1, 2]]

We can do the same thing with an outer constraint::

sage: Compositions(5, max_slope=1, min_slope=-2, min_length=2, max_length=4, outer=[2,5,2]).list()

[[2, 3], [2, 2, 1], [2, 1, 2], [1, 2, 2]]

However, providing incoherent constraints may yield strange

results. It is up to the user to ensure that the inner and outer

compositions themselves satisfy the parts and slope constraints.

Note that if you specify ``min_part=0``, then the objects produced may

have parts equal to zero. This violates the internal assumptions

that the composition class makes. Use at your own risk, or

preferably consider using ``IntegerVectors`` instead::

sage: Compositions(2, length=3, min_part=0).list()

doctest:...: RuntimeWarning: Currently, setting min_part=0 produces Composition objects which violate internal assumptions. Calling methods on these objects may produce errors or WRONG results!

[[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1], [0, 0, 2]]

sage: list(IntegerVectors(2, 3))

[[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1], [0, 0, 2]]

The generation algorithm is constant amortized time, and handled

by the generic tool :class:`IntegerListsLex`.

TESTS::

sage: C = Compositions(4, length=2)

sage: C == loads(dumps(C))

True

sage: Compositions(6, min_part=2, length=3)

Compositions of the integer 6 satisfying constraints length=3, min_part=2

sage: [2, 1] in Compositions(3, length=2)

True

sage: [2,1,2] in Compositions(5, min_part=1)

True

sage: [2,1,2] in Compositions(5, min_part=2)

False

sage: Compositions(4, length=2).cardinality()

sage: Compositions(4, min_length=2).cardinality()

sage: Compositions(4, max_length=2).cardinality()

sage: Compositions(4, max_part=2).cardinality()

sage: Compositions(4, min_part=2).cardinality()

sage: Compositions(4, outer=[3,1,2]).cardinality()

sage: Compositions(4, length=2).list()

[[3, 1], [2, 2], [1, 3]]

sage: Compositions(4, min_length=2).list()

[[3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]

sage: Compositions(4, max_length=2).list()

[[4], [3, 1], [2, 2], [1, 3]]

sage: Compositions(4, max_part=2).list()

[[2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]

sage: Compositions(4, min_part=2).list()

[[4], [2, 2]]

sage: Compositions(4, outer=[3,1,2]).list()

[[3, 1], [2, 1, 1], [1, 1, 2]]

sage: Compositions(3, outer = Composition([3,2])).list()

[[3], [2, 1], [1, 2]]

sage: Compositions(4, outer=[1,oo,1]).list()

[[1, 3], [1, 2, 1]]

sage: Compositions(4, inner=[1,1,1]).list()

[[2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]

sage: Compositions(4, inner=Composition([1,2])).list()

[[2, 2], [1, 3], [1, 2, 1]]

sage: Compositions(4, min_slope=0).list()

[[4], [2, 2], [1, 3], [1, 1, 2], [1, 1, 1, 1]]

sage: Compositions(4, min_slope=-1, max_slope=1).list()

[[4], [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]

sage: Compositions(5, max_slope=1, min_slope=-2, min_length=2, max_length=4).list()

[[3, 2], [3, 1, 1], [2, 3], [2, 2, 1], [2, 1, 2], [2, 1, 1, 1], [1, 2, 2], [1, 2, 1, 1], [1, 1, 2, 1], [1, 1, 1, 2]]

sage: Compositions(5, max_slope=1, min_slope=-2, min_length=2, max_length=4, outer=[2,5,2]).list()

[[2, 3], [2, 2, 1], [2, 1, 2], [1, 2, 2]]

"""

@staticmethod

def __classcall_private__(self, n=None, **kwargs):

"""

Return the correct parent based upon the input.

EXAMPLES::

sage: C = Compositions(3)

sage: C2 = Compositions(int(3))

sage: C is C2

True

"""

if n is None:

if len(kwargs) != 0:

raise ValueError("Incorrect number of arguments")

return Compositions_all()

else:

if len(kwargs) == 0:

if isinstance(n, (int,Integer)):

return Compositions_n(n)

else:

raise ValueError("n must be an integer")

else:

# FIXME: should inherit from IntegerListLex, and implement repr, or _name as a lazy attribute

kwargs['name'] = "Compositions of the integer %s satisfying constraints %s"%(n, ", ".join( ["%s=%s"%(key, kwargs[key]) for key in sorted(kwargs)] ))

kwargs['element_class'] = Composition

if 'min_part' not in kwargs:

kwargs['min_part'] = 1

elif kwargs['min_part'] == 0:

from warnings import warn

warn("Currently, setting min_part=0 produces Composition objects which violate internal assumptions. Calling methods on these objects may produce errors or WRONG results!", RuntimeWarning)

if 'outer' in kwargs:

kwargs['ceiling'] = list(kwargs['outer'])

if 'max_length' in kwargs:

kwargs['max_length'] = min(len(kwargs['outer']), kwargs['max_length'])

else:

kwargs['max_length'] = len(kwargs['outer'])

del kwargs['outer']

if 'inner' in kwargs:

inner = list(kwargs['inner'])

kwargs['floor'] = inner

del kwargs['inner']

# Should this be handled by integer lists lex?

if 'min_length' in kwargs:

kwargs['min_length'] = max(len(inner), kwargs['min_length'])

else:

kwargs['min_length'] = len(inner)

return IntegerListsLex(n, **kwargs)

def __init__(self, is_infinite=False):

"""

Initialize ``self``.

EXAMPLES::

sage: C = Compositions()

sage: TestSuite(C).run()

"""

if is_infinite:

Parent.__init__(self, category=InfiniteEnumeratedSets())

else:

Parent.__init__(self, category=FiniteEnumeratedSets())

Element = Composition

def _element_constructor_(self, lst):

"""

Construct an element with ``self`` as parent.

EXAMPLES::

sage: P = Compositions()

sage: P([3,3,1]) # indirect doctest

[3, 3, 1]

"""

if isinstance(lst, Composition):

lst = list(lst)

elt = self.element_class(self, lst)

if elt not in self:

raise ValueError("%s not in %s"%(elt, self))

return elt

def __contains__(self, x):

"""

TESTS::

sage: [2,1,3] in Compositions()

True

sage: [] in Compositions()

True

sage: [-2,-1] in Compositions()

False

sage: [0,0] in Compositions()

True

"""

if isinstance(x, Composition):

return True

elif isinstance(x, builtins.list):

for i in x:

if (not isinstance(i, (int, Integer))) and i not in ZZ:

return False

if i < 0:

return False

return True

else:

return False

def from_descents(self, descents, nps=None):

"""

Return a composition from the list of descents.

INPUT:

- ``descents`` -- an iterable

- ``nps`` -- (default: ``None``) an integer or ``None``

OUTPUT:

- The composition of ``nps`` whose descents are listed in

``descents``, assuming that ``nps`` is not ``None`` (otherwise,

the last element of ``descents`` is removed from ``descents``, and

``nps`` is set to be this last element plus 1).

EXAMPLES::

sage: [x-1 for x in Composition([1, 1, 3, 4, 3]).to_subset()]

[0, 1, 4, 8]

sage: Compositions().from_descents([1,0,4,8],12)

[1, 1, 3, 4, 3]

sage: Compositions().from_descents([1,0,4,8,11])

[1, 1, 3, 4, 3]

"""

d = [x+1 for x in sorted(descents)]

if nps is None:

nps = d.pop()

return self.from_subset(d, nps)

def from_subset(self, S, n):

r"""

The composition of `n` corresponding to the subset ``S`` 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:`Composition.to_subset`).

INPUT:

- ``S`` -- an iterable, a subset of `\{1, 2, \ldots, n-1\}`

- ``n`` -- an integer

EXAMPLES::

sage: Compositions().from_subset([2,1,5,9], 12)

[1, 1, 3, 4, 3]

sage: Compositions().from_subset({2,1,5,9}, 12)

[1, 1, 3, 4, 3]

sage: Compositions().from_subset([], 12)

[12]

sage: Compositions().from_subset([], 0)

[]

TESTS::

sage: Compositions().from_subset([2,1,5,9],9)

Traceback (most recent call last):

...

ValueError: S (=[1, 2, 5, 9]) is not a subset of {1, ..., 8}

"""

d = sorted(S)

if d == []:

if n == 0:

return self.element_class(self, [])

else:

return self.element_class(self, [n])

if n <= d[-1]:

raise ValueError("S (=%s) is not a subset of {1, ..., %s}" % (d,n-1))

else:

d.append(n)

co = [d[0]]

for i in range(len(d)-1):

co.append(d[i+1]-d[i])

return self.element_class(self, co)

def from_code(self, code):

r"""

Return the composition from its code. The code of a composition

`I` is a list of length `\mathrm{size}(I)` consisting 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]

sage: Compositions().from_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: Compositions().from_code(_)

[3, 1, 2, 3, 5]

"""

if code == [0]:

return []

L = [x for x in range(len(code)) if code[x]==1] #the positions of the letter 1

return self.element_class(self, [L[i]-L[i-1] for i in range(1, len(L))] + [len(code)-L[-1]])

# Allows to unpickle old constrained Compositions_constraints objects.

class Compositions_constraints(IntegerListsLex):

def __setstate__(self, data):

"""

TESTS::

# This is the unpickling sequence for Compositions(4, max_part=2) in sage <= 4.1.1

sage: pg_Compositions_constraints = unpickle_global('sage.combinat.composition', 'Compositions_constraints')

sage: si = unpickle_newobj(pg_Compositions_constraints, ())

sage: pg_make_integer = unpickle_global('sage.rings.integer', 'make_integer')

sage: unpickle_build(si, {'constraints':{'max_part':pg_make_integer('2')}, 'n':pg_make_integer('4')})

sage: si

Integer lists of sum 4 satisfying certain constraints

sage: si.list()

[[2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]

"""

n = data['n']

self.__class__ = IntegerListsLex

constraints = {'min_part' : 1,

'element_class' : Composition}

constraints.update(data['constraints'])

self.__init__(n, **constraints)

class Compositions_all(Compositions):

"""

Class of all compositions.

"""

def __init__(self):

"""

Initialize ``self``.

TESTS::

sage: C = Compositions()

sage: TestSuite(C).run()

"""

Compositions.__init__(self, True)

def _repr_(self):

"""

Return a string representation of ``self``.

TESTS::

sage: repr(Compositions())

'Compositions of non-negative integers'

"""

return "Compositions of non-negative integers"

def subset(self, size=None):

"""

Return the set of compositions of the given size.

EXAMPLES::

sage: C = Compositions()

sage: C.subset(4)

Compositions of 4

sage: C.subset(size=3)

Compositions of 3

"""

if size is None:

return self

return Compositions(size)

def __iter__(self):

"""

Iterate over all compositions.

TESTS::

sage: C = Compositions()

sage: it = C.__iter__()

sage: [next(it) for i in range(10)]

[[], [1], [1, 1], [2], [1, 1, 1], [1, 2], [2, 1], [3], [1, 1, 1, 1], [1, 1, 2]]

"""

n = 0

while True:

for c in Compositions(n):

yield self.element_class(self, list(c))

n += 1

class Compositions_n(Compositions):

"""

Class of compositions of a fixed `n`.

"""

@staticmethod

def __classcall_private__(cls, n):

"""

Standardize input to ensure a unique representation.

EXAMPLES::

sage: C = Compositions(5)

sage: C2 = Compositions(int(5))

sage: C3 = Compositions(ZZ(5))

sage: C is C2

True

sage: C is C3

True

"""

return super(Compositions_n, cls).__classcall__(cls, Integer(n))

def __init__(self, n):

"""

TESTS::

sage: C = Compositions(3)

sage: C == loads(dumps(C))

True

sage: TestSuite(C).run()

"""

self.n = n

Compositions.__init__(self, False)

def _repr_(self):

"""

Return a string representation of ``self``.

TESTS::

sage: repr(Compositions(3))

'Compositions of 3'

"""

return "Compositions of %s"%self.n

def __contains__(self, x):

"""

TESTS::

sage: [2,1,3] in Compositions(6)

True

sage: [2,1,2] in Compositions(6)

False

sage: [] in Compositions(0)

True

sage: [0] in Compositions(0)

True

"""

return x in Compositions() and sum(x) == self.n

def cardinality(self):

"""

Return the number of compositions of `n`.

TESTS::

sage: Compositions(3).cardinality()

sage: Compositions(0).cardinality()

"""

if self.n >= 1:

return ZZ(2) ** (self.n-1)

elif self.n == 0:

return ZZ(1)

else:

return ZZ(0)

def random_element(self):

r"""

Return a random ``Composition`` with uniform probability.

This method generates a random binary word starting with a 1

and then uses the bijection between compositions and their code.

EXAMPLES::

sage: Compositions(5).random_element() # random

[2, 1, 1, 1]

sage: Compositions(0).random_element()

[]

sage: Compositions(1).random_element()

[1]

TESTS::

sage: all(Compositions(10).random_element() in Compositions(10) for i in range(20))

True

"""

from sage.misc.prandom import choice

if self.n == 0:

return Compositions()([])

return Compositions().from_code([1] + [choice([0,1]) for _ in range(self.n - 1)])

def __iter__(self):

"""

Iterate over the compositions of `n`.

TESTS::

sage: Compositions(4).list()

[[1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [1, 3], [2, 1, 1], [2, 2], [3, 1], [4]]

sage: Compositions(0).list()

[[]]

"""

for c in composition_iterator_fast(self.n):

yield self.element_class(self, c)

def composition_iterator_fast(n):

"""

Iterator over compositions of ``n`` yielded as lists.

TESTS::

sage: from sage.combinat.composition import composition_iterator_fast

sage: L = list(composition_iterator_fast(4)); L

[[1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [1, 3], [2, 1, 1], [2, 2], [3, 1], [4]]

sage: type(L[0])

<... 'list'>

"""

# Special cases

if n < 0:

return

if n == 0:

yield []

return

s = Integer(0) # Current sum

cur = [Integer(0)]

while cur:

cur[-1] += 1

s += 1

# Note that because we are adding 1 every time,

# we will never have s > n

if s == n:

yield list(cur)

s -= cur.pop()

else:

cur.append(Integer(0))

from sage.structure.sage_object import register_unpickle_override

register_unpickle_override('sage.combinat.composition', 'Composition_class', Composition)

Coverage for local/lib/python2.7/site-packages/sage/combinat/composition.py : 78%

404 statements 315 run 89 missing 0 excluded