# 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 instance
• universe - (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 universe
• immutable - (default: True) whether or not this sequence is immutable
• cr - (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 calling str() on this sequence.
• use_sage_types – (default: False) if True, coerce the
built-in 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 built-in 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):
...


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             # 32-bit
-4181190870548101704   # 64-bit


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 instance
• universe - (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 universe
• immutable - (default: True) whether or not this sequence is immutable
• cr - (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
built-in 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 built-in 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):
...


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             # 32-bit
-4181190870548101704   # 64-bit


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):
...
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):
...
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):
...

sort(key=None, reverse=False)

Sort this list IN PLACE.

INPUT:

• key - see Python list sort
• reverse - see Python list 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 instance
• universe - (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 universe
• immutable - (default: True) whether or not this sequence is immutable
• cr - (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 calling str() on this sequence.
• use_sage_types – (default: False) if True, coerce the
built-in 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 built-in 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):
...


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             # 32-bit
-4181190870548101704   # 64-bit


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