Miscellaneous

class sage.combinat.misc.DoublyLinkedList(l)

Bases: object

A doubly linked list class that provides constant time hiding and unhiding of entries.

Note that this list’s indexing is 1-based.

EXAMPLES:

Sage

sage: dll = sage.combinat.misc.DoublyLinkedList([1,2,3]); dll
Doubly linked list of [1, 2, 3]: [1, 2, 3]
sage: dll.hide(1); dll
Doubly linked list of [1, 2, 3]: [2, 3]
sage: dll.unhide(1); dll
Doubly linked list of [1, 2, 3]: [1, 2, 3]
sage: dll.hide(2); dll
Doubly linked list of [1, 2, 3]: [1, 3]
sage: dll.unhide(2); dll
Doubly linked list of [1, 2, 3]: [1, 2, 3]

Python

>>> from sage.all import *
>>> dll = sage.combinat.misc.DoublyLinkedList([Integer(1),Integer(2),Integer(3)]); dll
Doubly linked list of [1, 2, 3]: [1, 2, 3]
>>> dll.hide(Integer(1)); dll
Doubly linked list of [1, 2, 3]: [2, 3]
>>> dll.unhide(Integer(1)); dll
Doubly linked list of [1, 2, 3]: [1, 2, 3]
>>> dll.hide(Integer(2)); dll
Doubly linked list of [1, 2, 3]: [1, 3]
>>> dll.unhide(Integer(2)); dll
Doubly linked list of [1, 2, 3]: [1, 2, 3]

head()

hide(i)

next(j)

prev(j)

unhide(i)

class sage.combinat.misc.IterableFunctionCall(f, *args, **kwargs)

Bases: object

This class wraps functions with a yield statement (generators) by an object that can be iterated over. For example,

EXAMPLES:

Sage

sage: def f(): yield 'a'; yield 'b'

Python

>>> from sage.all import *
>>> def f(): yield 'a'; yield 'b'

This does not work:

Sage

sage: for z in f: print(z)
Traceback (most recent call last):
...
TypeError: 'function' object is not iterable

Python

>>> from sage.all import *
>>> for z in f: print(z)
Traceback (most recent call last):
...
TypeError: 'function' object is not iterable

Use IterableFunctionCall if you want something like the above to work:

Sage

sage: from sage.combinat.misc import IterableFunctionCall
sage: g = IterableFunctionCall(f)
sage: for z in g: print(z)
a
b

Python

>>> from sage.all import *
>>> from sage.combinat.misc import IterableFunctionCall
>>> g = IterableFunctionCall(f)
>>> for z in g: print(z)
a
b

If your function takes arguments, just put them after the function name. You needn’t enclose them in a tuple or anything, just put them there:

Sage

sage: def f(n, m): yield 'a' * n; yield 'b' * m; yield 'foo'
sage: g = IterableFunctionCall(f, 2, 3)
sage: for z in g: print(z)
aa
bbb
foo

Python

>>> from sage.all import *
>>> def f(n, m): yield 'a' * n; yield 'b' * m; yield 'foo'
>>> g = IterableFunctionCall(f, Integer(2), Integer(3))
>>> for z in g: print(z)
aa
bbb
foo

sage.combinat.misc.check_integer_list_constraints(l, **kwargs)

EXAMPLES:

Sage

sage: from sage.combinat.misc import check_integer_list_constraints
sage: cilc = check_integer_list_constraints
sage: l = [[2,1,3],[1,2],[3,3],[4,1,1]]
sage: cilc(l, min_part=2)
[[3, 3]]
sage: cilc(l, max_part=2)
[[1, 2]]
sage: cilc(l, length=2)
[[1, 2], [3, 3]]
sage: cilc(l, max_length=2)
[[1, 2], [3, 3]]
sage: cilc(l, min_length=3)
[[2, 1, 3], [4, 1, 1]]
sage: cilc(l, max_slope=0)
[[3, 3], [4, 1, 1]]
sage: cilc(l, min_slope=1)
[[1, 2]]
sage: cilc(l, outer=[2,2])
[[1, 2]]
sage: cilc(l, inner=[2,2])
[[3, 3]]

Python

>>> from sage.all import *
>>> from sage.combinat.misc import check_integer_list_constraints
>>> cilc = check_integer_list_constraints
>>> l = [[Integer(2),Integer(1),Integer(3)],[Integer(1),Integer(2)],[Integer(3),Integer(3)],[Integer(4),Integer(1),Integer(1)]]
>>> cilc(l, min_part=Integer(2))
[[3, 3]]
>>> cilc(l, max_part=Integer(2))
[[1, 2]]
>>> cilc(l, length=Integer(2))
[[1, 2], [3, 3]]
>>> cilc(l, max_length=Integer(2))
[[1, 2], [3, 3]]
>>> cilc(l, min_length=Integer(3))
[[2, 1, 3], [4, 1, 1]]
>>> cilc(l, max_slope=Integer(0))
[[3, 3], [4, 1, 1]]
>>> cilc(l, min_slope=Integer(1))
[[1, 2]]
>>> cilc(l, outer=[Integer(2),Integer(2)])
[[1, 2]]
>>> cilc(l, inner=[Integer(2),Integer(2)])
[[3, 3]]

Sage

sage: cilc([1,2,3], length=3, singleton=True)
[1, 2, 3]
sage: cilc([1,2,3], length=2, singleton=True) is None
True

Python

>>> from sage.all import *
>>> cilc([Integer(1),Integer(2),Integer(3)], length=Integer(3), singleton=True)
[1, 2, 3]
>>> cilc([Integer(1),Integer(2),Integer(3)], length=Integer(2), singleton=True) is None
True

sage.combinat.misc.umbral_operation(poly)

Returns the umbral operation \(\downarrow\) applied to poly.

The umbral operation replaces each instance of \(x_i^{a_i}\) with \(x_i*(x_i - 1)*\cdots*(x_i - a_i + 1)\).

EXAMPLES:

Sage

sage: P = PolynomialRing(QQ, 2, 'x')
sage: x = P.gens()
sage: from sage.combinat.misc import umbral_operation
sage: umbral_operation(x[0]^3) == x[0]*(x[0]-1)*(x[0]-2)
True
sage: umbral_operation(x[0]*x[1])
x0*x1
sage: umbral_operation(x[0]+x[1])
x0 + x1
sage: umbral_operation(x[0]^2*x[1]^2) == x[0]*(x[0]-1)*x[1]*(x[1]-1)
True

Python

>>> from sage.all import *
>>> P = PolynomialRing(QQ, Integer(2), 'x')
>>> x = P.gens()
>>> from sage.combinat.misc import umbral_operation
>>> umbral_operation(x[Integer(0)]**Integer(3)) == x[Integer(0)]*(x[Integer(0)]-Integer(1))*(x[Integer(0)]-Integer(2))
True
>>> umbral_operation(x[Integer(0)]*x[Integer(1)])
x0*x1
>>> umbral_operation(x[Integer(0)]+x[Integer(1)])
x0 + x1
>>> umbral_operation(x[Integer(0)]**Integer(2)*x[Integer(1)]**Integer(2)) == x[Integer(0)]*(x[Integer(0)]-Integer(1))*x[Integer(1)]*(x[Integer(1)]-Integer(1))
True