Finite Homogenous Sequences¶
A mutable sequence of elements with a common guaranteed category, which can be set immutable.
Sequence derives from list, so has all the functionality of lists and
can be used wherever lists are used. When a sequence is created
without explicitly given the common universe of the elements, the
constructor coerces the first and second element to some
canonical common parent, if possible, then the second and
third, etc. If this is possible, it then coerces everything into the
canonical parent at the end. (Note that canonical coercion is very
restrictive.) The sequence then has a function universe()
which returns either the common canonical parent (if the coercion
succeeded), or the category of all objects (Objects()). So if you
have a list \(v\) and type:
sage: v = [1, 2/3, 5]
sage: w = Sequence(v)
sage: w.universe()
Rational Field
then since w.universe()
is \(\QQ\), you’re guaranteed that all
elements of \(w\) are rationals:
sage: v[0].parent()
Integer Ring
sage: w[0].parent()
Rational Field
If you do assignment to \(w\) this property of being rationals is guaranteed to be preserved:
sage: w[0] = 2
sage: w[0].parent()
Rational Field
sage: w[0] = 'hi'
Traceback (most recent call last):
...
TypeError: unable to convert 'hi' to a rational
However, if you do w = Sequence(v)
and the resulting universe
is Objects()
, the elements are not guaranteed to have any
special parent. This is what should happen, e.g., with finite field
elements of different characteristics:
sage: v = Sequence([GF(3)(1), GF(7)(1)])
sage: v.universe()
Category of objects
You can make a list immutable with v.freeze()
. Assignment is
never again allowed on an immutable list.
Creation of a sequence involves making a copy of the input list, and substantial coercions. It can be greatly sped up by explicitly specifying the universe of the sequence:
sage: v = Sequence(range(10000), universe=ZZ)

sage.structure.sequence.
Sequence
(x, universe=None, check=True, immutable=False, cr=False, cr_str=None, use_sage_types=False)¶ A mutable list of elements with a common guaranteed universe, which can be set immutable.
A universe is either an object that supports coercion (e.g., a parent), or a category.
INPUT:
x
 a list or tuple instanceuniverse
 (default: None) the universe of elements; if None determined using canonical coercions and the entire list of elements. If list is empty, is category Objects() of all objects.check
– (default: True) whether to coerce the elements of x into the universeimmutable
 (default: True) whether or not this sequence is immutablecr
 (default: False) if True, then print a carriage return after each comma when printing this sequence.cr_str
 (default: False) if True, then print a carriage return after each comma when callingstr()
on this sequence.use_sage_types
– (default: False) if True, coerce the builtin Python numerical types int, float, complex to the corresponding Sage types (this makes functions like vector() more flexible)
OUTPUT:
 a sequence
EXAMPLES:
sage: v = Sequence(range(10)) sage: v.universe() <type 'int'> sage: v [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
We can request that the builtin Python numerical types be coerced to Sage objects:
sage: v = Sequence(range(10), use_sage_types=True) sage: v.universe() Integer Ring sage: v [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
You can also use seq for “Sequence”, which is identical to using Sequence:
sage: v = seq([1,2,1/1]); v [1, 2, 1] sage: v.universe() Rational Field
Note that assignment coerces if possible,:
sage: v = Sequence(range(10), ZZ) sage: a = QQ(5) sage: v[3] = a sage: parent(v[3]) Integer Ring sage: parent(a) Rational Field sage: v[3] = 2/3 Traceback (most recent call last): ... TypeError: no conversion of this rational to integer
Sequences can be used absolutely anywhere lists or tuples can be used:
sage: isinstance(v, list) True
Sequence can be immutable, so entries can’t be changed:
sage: v = Sequence([1,2,3], immutable=True) sage: v.is_immutable() True sage: v[0] = 5 Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead.
Only immutable sequences are hashable (unlike Python lists), though the hashing is potentially slow, since it first involves conversion of the sequence to a tuple, and returning the hash of that.:
sage: v = Sequence(range(10), ZZ, immutable=True) sage: hash(v) 1591723448 # 32bit 4181190870548101704 # 64bit
If you really know what you are doing, you can circumvent the type checking (for an efficiency gain):
sage: list.__setitem__(v, int(1), 2/3) # bad circumvention sage: v [0, 2/3, 2, 3, 4, 5, 6, 7, 8, 9] sage: list.__setitem__(v, int(1), int(2)) # not so bad circumvention
You can make a sequence with a new universe from an old sequence.:
sage: w = Sequence(v, QQ) sage: w [0, 2, 2, 3, 4, 5, 6, 7, 8, 9] sage: w.universe() Rational Field sage: w[1] = 2/3 sage: w [0, 2/3, 2, 3, 4, 5, 6, 7, 8, 9]
The default universe for any sequence, if no compatible parent structure can be found, is the universe of all Sage objects.
This example illustrates how every element of a list is taken into account when constructing a sequence.:
sage: v = Sequence([1,7,6,GF(5)(3)]); v [1, 2, 1, 3] sage: v.universe() Finite Field of size 5

class
sage.structure.sequence.
Sequence_generic
(x, universe=None, check=True, immutable=False, cr=False, cr_str=None, use_sage_types=False)¶ Bases:
sage.structure.sage_object.SageObject
,list
A mutable list of elements with a common guaranteed universe, which can be set immutable.
A universe is either an object that supports coercion (e.g., a parent), or a category.
INPUT:
x
 a list or tuple instanceuniverse
 (default: None) the universe of elements; if None determined using canonical coercions and the entire list of elements. If list is empty, is category Objects() of all objects.check
– (default: True) whether to coerce the elements of x into the universeimmutable
 (default: True) whether or not this sequence is immutablecr
 (default: False) if True, then print a carriage return after each comma when printing this sequence.use_sage_types
– (default: False) if True, coerce the builtin Python numerical types int, float, complex to the corresponding Sage types (this makes functions like vector() more flexible)
OUTPUT:
 a sequence
EXAMPLES:
sage: v = Sequence(range(10)) sage: v.universe() <type 'int'> sage: v [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
We can request that the builtin Python numerical types be coerced to Sage objects:
sage: v = Sequence(range(10), use_sage_types=True) sage: v.universe() Integer Ring sage: v [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
You can also use seq for “Sequence”, which is identical to using Sequence:
sage: v = seq([1,2,1/1]); v [1, 2, 1] sage: v.universe() Rational Field
Note that assignment coerces if possible,
sage: v = Sequence(range(10), ZZ) sage: a = QQ(5) sage: v[3] = a sage: parent(v[3]) Integer Ring sage: parent(a) Rational Field sage: v[3] = 2/3 Traceback (most recent call last): ... TypeError: no conversion of this rational to integer
Sequences can be used absolutely anywhere lists or tuples can be used:
sage: isinstance(v, list) True
Sequence can be immutable, so entries can’t be changed:
sage: v = Sequence([1,2,3], immutable=True) sage: v.is_immutable() True sage: v[0] = 5 Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead.
Only immutable sequences are hashable (unlike Python lists), though the hashing is potentially slow, since it first involves conversion of the sequence to a tuple, and returning the hash of that.
sage: v = Sequence(range(10), ZZ, immutable=True) sage: hash(v) 1591723448 # 32bit 4181190870548101704 # 64bit
If you really know what you are doing, you can circumvent the type checking (for an efficiency gain):
sage: list.__setitem__(v, int(1), 2/3) # bad circumvention sage: v [0, 2/3, 2, 3, 4, 5, 6, 7, 8, 9] sage: list.__setitem__(v, int(1), int(2)) # not so bad circumvention
You can make a sequence with a new universe from an old sequence.
sage: w = Sequence(v, QQ) sage: w [0, 2, 2, 3, 4, 5, 6, 7, 8, 9] sage: w.universe() Rational Field sage: w[1] = 2/3 sage: w [0, 2/3, 2, 3, 4, 5, 6, 7, 8, 9]
The default universe for any sequence, if no compatible parent structure can be found, is the universe of all Sage objects.
This example illustrates how every element of a list is taken into account when constructing a sequence.
sage: v = Sequence([1,7,6,GF(5)(3)]); v [1, 2, 1, 3] sage: v.universe() Finite Field of size 5

append
(x)¶ EXAMPLES:
sage: v = Sequence([1,2,3,4], immutable=True) sage: v.append(34) Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead. sage: v = Sequence([1/3,2,3,4]) sage: v.append(4) sage: type(v[4]) <type 'sage.rings.rational.Rational'>

extend
(iterable)¶ Extend list by appending elements from the iterable.
EXAMPLES:
sage: B = Sequence([1,2,3]) sage: B.extend(range(4)) sage: B [1, 2, 3, 0, 1, 2, 3]

insert
(index, object)¶ Insert object before index.
EXAMPLES:
sage: B = Sequence([1,2,3]) sage: B.insert(10, 5) sage: B [1, 2, 3, 5]

is_immutable
()¶ Return True if this object is immutable (can not be changed) and False if it is not.
To make this object immutable use
set_immutable()
.EXAMPLES:
sage: v = Sequence([1,2,3,4/5]) sage: v[0] = 5 sage: v [5, 2, 3, 4/5] sage: v.is_immutable() False sage: v.set_immutable() sage: v.is_immutable() True

is_mutable
()¶ EXAMPLES:
sage: a = Sequence([1,2/3,2/5]) sage: a.is_mutable() True sage: a[0] = 100 sage: type(a[0]) <type 'sage.rings.rational.Rational'> sage: a.set_immutable() sage: a[0] = 50 Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead. sage: a.is_mutable() False

pop
(index=1)¶ Remove and return item at index (default last)
EXAMPLES:
sage: B = Sequence([1,2,3]) sage: B.pop(1) 2 sage: B [1, 3]

remove
(value)¶ Remove first occurrence of value
EXAMPLES:
sage: B = Sequence([1,2,3]) sage: B.remove(2) sage: B [1, 3]

reverse
()¶ Reverse the elements of self, in place.
EXAMPLES:
sage: B = Sequence([1,2,3]) sage: B.reverse(); B [3, 2, 1]

set_immutable
()¶ Make this object immutable, so it can never again be changed.
EXAMPLES:
sage: v = Sequence([1,2,3,4/5]) sage: v[0] = 5 sage: v [5, 2, 3, 4/5] sage: v.set_immutable() sage: v[3] = 7 Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead.

sort
(key=None, reverse=False)¶ Sort this list IN PLACE.
INPUT:
key
 see Pythonlist sort
reverse
 see Pythonlist sort
EXAMPLES:
sage: B = Sequence([3,2,1/5]) sage: B.sort() sage: B [1/5, 2, 3] sage: B.sort(reverse=True); B [3, 2, 1/5]

universe
()¶ EXAMPLES:
sage: Sequence([1,2/3,2/5]).universe() Rational Field sage: Sequence([1,2/3,'2/5']).universe() Category of objects

sage.structure.sequence.
seq
(x, universe=None, check=True, immutable=False, cr=False, cr_str=None, use_sage_types=False)¶ A mutable list of elements with a common guaranteed universe, which can be set immutable.
A universe is either an object that supports coercion (e.g., a parent), or a category.
INPUT:
x
 a list or tuple instanceuniverse
 (default: None) the universe of elements; if None determined using canonical coercions and the entire list of elements. If list is empty, is category Objects() of all objects.check
– (default: True) whether to coerce the elements of x into the universeimmutable
 (default: True) whether or not this sequence is immutablecr
 (default: False) if True, then print a carriage return after each comma when printing this sequence.cr_str
 (default: False) if True, then print a carriage return after each comma when callingstr()
on this sequence.use_sage_types
– (default: False) if True, coerce the builtin Python numerical types int, float, complex to the corresponding Sage types (this makes functions like vector() more flexible)
OUTPUT:
 a sequence
EXAMPLES:
sage: v = Sequence(range(10)) sage: v.universe() <type 'int'> sage: v [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
We can request that the builtin Python numerical types be coerced to Sage objects:
sage: v = Sequence(range(10), use_sage_types=True) sage: v.universe() Integer Ring sage: v [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
You can also use seq for “Sequence”, which is identical to using Sequence:
sage: v = seq([1,2,1/1]); v [1, 2, 1] sage: v.universe() Rational Field
Note that assignment coerces if possible,:
sage: v = Sequence(range(10), ZZ) sage: a = QQ(5) sage: v[3] = a sage: parent(v[3]) Integer Ring sage: parent(a) Rational Field sage: v[3] = 2/3 Traceback (most recent call last): ... TypeError: no conversion of this rational to integer
Sequences can be used absolutely anywhere lists or tuples can be used:
sage: isinstance(v, list) True
Sequence can be immutable, so entries can’t be changed:
sage: v = Sequence([1,2,3], immutable=True) sage: v.is_immutable() True sage: v[0] = 5 Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead.
Only immutable sequences are hashable (unlike Python lists), though the hashing is potentially slow, since it first involves conversion of the sequence to a tuple, and returning the hash of that.:
sage: v = Sequence(range(10), ZZ, immutable=True) sage: hash(v) 1591723448 # 32bit 4181190870548101704 # 64bit
If you really know what you are doing, you can circumvent the type checking (for an efficiency gain):
sage: list.__setitem__(v, int(1), 2/3) # bad circumvention sage: v [0, 2/3, 2, 3, 4, 5, 6, 7, 8, 9] sage: list.__setitem__(v, int(1), int(2)) # not so bad circumvention
You can make a sequence with a new universe from an old sequence.:
sage: w = Sequence(v, QQ) sage: w [0, 2, 2, 3, 4, 5, 6, 7, 8, 9] sage: w.universe() Rational Field sage: w[1] = 2/3 sage: w [0, 2/3, 2, 3, 4, 5, 6, 7, 8, 9]
The default universe for any sequence, if no compatible parent structure can be found, is the universe of all Sage objects.
This example illustrates how every element of a list is taken into account when constructing a sequence.:
sage: v = Sequence([1,7,6,GF(5)(3)]); v [1, 2, 1, 3] sage: v.universe() Finite Field of size 5