r"""
Combinatorial Functions
AUTHORS:
- David Joyner (2006-07): initial implementation.
- William Stein (2006-07): editing of docs and code; many
optimizations, refinements, and bug fixes in corner cases
- David Joyner (2006-09): bug fix for combinations, added
permutations_iterator, combinations_iterator from Python Cookbook,
edited docs.
- David Joyner (2007-11): changed permutations, added hadamard_matrix
- Florent Hivert (2009-02): combinatorial class cleanup
- Fredrik Johansson (2010-07): fast implementation of ``stirling_number2``
This module implements some combinatorial functions, as listed
below. For a more detailed description, see the relevant
docstrings.
Sequences:
- Bell numbers, ``bell_number``
- Bernoulli numbers, ``bernoulli_number`` (though
PARI's bernoulli is better)
- Catalan numbers, ``catalan_number`` (not to be
confused with the Catalan constant)
- Eulerian/Euler numbers, ``euler_number`` (Maxima)
- Fibonacci numbers, ``fibonacci`` (PARI) and
``fibonacci_number`` (GAP) The PARI version is
better.
- Lucas numbers, ``lucas_number1``,
``lucas_number2``.
- Stirling numbers, ``stirling_number1``,
``stirling_number2``.
Set-theoretic constructions:
- Combinations of a multiset, ``combinations``,
``combinations_iterator``, and
``number_of_combinations``. These are unordered
selections without repetition of k objects from a multiset S.
- Arrangements of a multiset, ``arrangements`` and
``number_of_arrangements`` These are ordered
selections without repetition of k objects from a multiset S.
- Derangements of a multiset, ``derangements`` and
``number_of_derangements``.
- Tuples of a multiset, ``tuples`` and
``number_of_tuples``. An ordered tuple of length k of
set S is a ordered selection with repetitions of S and is
represented by a sorted list of length k containing elements from
S.
- Unordered tuples of a set, ``unordered_tuple`` and
``number_of_unordered_tuples``. An unordered tuple
of length k of set S is an unordered selection with repetitions of S
and is represented by a sorted list of length k containing elements
from S.
- Permutations of a multiset, ``permutations``,
``permutations_iterator``,
``number_of_permutations``. A permutation is a list
that contains exactly the same elements but possibly in different
order.
Partitions:
- Partitions of a set, ``partitions_set``,
``number_of_partitions_set``. An unordered partition
of set S is a set of pairwise disjoint nonempty sets with union S
and is represented by a sorted list of such sets.
- Partitions of an integer, ``partitions_list``,
``number_of_partitions_list``. An unordered
partition of n is an unordered sum
`n = p_1+p_2 +\ldots+ p_k` of positive integers and is
represented by the list `p = [p_1,p_2,\ldots,p_k]`, in
nonincreasing order, i.e., `p1\geq p_2 ...\geq p_k`.
- Ordered partitions of an integer,
``ordered_partitions``,
``number_of_ordered_partitions``. An ordered
partition of n is an ordered sum
`n = p_1+p_2 +\ldots+ p_k` of positive integers and is
represented by the list `p = [p_1,p_2,\ldots,p_k]`, in
nonincreasing order, i.e., `p1\geq p_2 ...\geq p_k`.
- Restricted partitions of an integer,
``partitions_restricted``,
``number_of_partitions_restricted``. An unordered
restricted partition of n is an unordered sum
`n = p_1+p_2 +\ldots+ p_k` of positive integers
`p_i` belonging to a given set `S`, and is
represented by the list `p = [p_1,p_2,\ldots,p_k]`, in
nonincreasing order, i.e., `p1\geq p_2 ...\geq p_k`.
- ``partitions_greatest`` implements a special type
of restricted partition.
- ``partitions_greatest_eq`` is another type of
restricted partition.
- Tuples of partitions, ``partition_tuples``,
``number_of_partition_tuples``. A `k`-tuple
of partitions is represented by a list of all `k`-tuples of
partitions which together form a partition of `n`.
- Powers of a partition, ``partition_power(pi, k)``.
The power of a partition corresponds to the `k`-th power of
a permutation with cycle structure `\pi`.
- Sign of a partition, ``partition_sign( pi )`` This
means the sign of a permutation with cycle structure given by the
partition pi.
- Associated partition, ``partition_associated( pi
)`` The "associated" (also called "conjugate" in the
literature) partition of the partition pi which is obtained by
transposing the corresponding Ferrers diagram.
- Ferrers diagram, ``ferrers_diagram``. Analogous to
the Young diagram of an irreducible representation of
`S_n`.
Related functions:
- Bernoulli polynomials, ``bernoulli_polynomial``
Implemented in other modules (listed for completeness):
The ``sage.rings.arith`` module contains the following
combinatorial functions:
- binomial the binomial coefficient (wrapped from PARI)
- factorial (wrapped from PARI)
- partition (from the Python Cookbook) Generator of the list of
all the partitions of the integer `n`.
- ``number_of_partitions`` (wrapped from PARI) the
*number* of partitions:
- ``falling_factorial`` Definition: for integer
`a \ge 0` we have `x(x-1) \cdots (x-a+1)`. In all
other cases we use the GAMMA-function:
`\frac {\Gamma(x+1)} {\Gamma(x-a+1)}`.
- ``rising_factorial`` Definition: for integer
`a \ge 0` we have `x(x+1) \cdots (x+a-1)`. In all
other cases we use the GAMMA-function:
`\frac {\Gamma(x+a)} {\Gamma(x)}`.
- gaussian_binomial the gaussian binomial
.. math::
\binom{n}{k}_q = \frac{(1-q^m)(1-q^{m-1})\cdots (1-q^{m-r+1})} {(1-q)(1-q^2)\cdots (1-q^r)}.
The ``sage.groups.perm_gps.permgroup_elements``
contains the following combinatorial functions:
- matrix method of PermutationGroupElement yielding the
permutation matrix of the group element.
.. TODO::
GUAVA commands:
* MOLS returns a list of n Mutually Orthogonal Latin Squares (MOLS).
* VandermondeMat
* GrayMat returns a list of all different vectors of length n over
the field F, using Gray ordering.
Not in GAP:
* Rencontres numbers
http://en.wikipedia.org/wiki/Rencontres_number
REFERENCES:
- http://en.wikipedia.org/wiki/Twelvefold_way (general reference)
"""
from sage.interfaces.all import gap, maxima
from sage.rings.all import QQ, ZZ, Integer
from sage.rings.arith import bernoulli, binomial
from sage.rings.polynomial.polynomial_element import Polynomial
from sage.misc.sage_eval import sage_eval
from sage.libs.all import pari
from sage.misc.prandom import randint
from sage.misc.misc import prod
from sage.structure.sage_object import SageObject
from sage.structure.parent import Parent
from sage.misc.lazy_attribute import lazy_attribute
from sage.misc.misc import deprecation
from combinat_cython import _stirling_number2
def bell_number(n):
r"""
Returns the n-th Bell number (the number of ways to partition a set
of n elements into pairwise disjoint nonempty subsets).
If `n \leq 0`, returns `1`.
Wraps GAP's Bell.
EXAMPLES::
sage: bell_number(10)
115975
sage: bell_number(2)
2
sage: bell_number(-10)
1
sage: bell_number(1)
1
sage: bell_number(1/3)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
"""
ans=gap.eval("Bell(%s)"%ZZ(n))
return ZZ(ans)
def catalan_number(n):
r"""
Returns the n-th Catalan number
Catalan numbers: The `n`-th Catalan number is given
directly in terms of binomial coefficients by
.. math::
C_n = \frac{1}{n+1}{2n\choose n} = \frac{(2n)!}{(n+1)!\,n!} \qquad\mbox{ for }\quad n\ge 0.
Consider the set `S = \{ 1, ..., n \}`. A noncrossing
partition of `S` is a partition in which no two blocks
"cross" each other, i.e., if a and b belong to one block and x and
y to another, they are not arranged in the order axby.
`C_n` is the number of noncrossing partitions of the set
`S`. There are many other interpretations (see
REFERENCES).
When `n=-1`, this function raises a ZeroDivisionError; for
other `n<0` it returns `0`.
INPUT:
- ``n`` - integer
OUTPUT: integer
EXAMPLES::
sage: [catalan_number(i) for i in range(7)]
[1, 1, 2, 5, 14, 42, 132]
sage: taylor((-1/2)*sqrt(1 - 4*x^2), x, 0, 15)
132*x^14 + 42*x^12 + 14*x^10 + 5*x^8 + 2*x^6 + x^4 + x^2 - 1/2
sage: [catalan_number(i) for i in range(-7,7) if i != -1]
[0, 0, 0, 0, 0, 0, 1, 1, 2, 5, 14, 42, 132]
sage: catalan_number(-1)
Traceback (most recent call last):
...
ZeroDivisionError
sage: [catalan_number(n).mod(2) for n in range(16)]
[1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1]
REFERENCES:
- http://en.wikipedia.org/wiki/Catalan_number
- http://www-history.mcs.st-andrews.ac.uk/~history/Miscellaneous/CatalanNumbers/catalan.html
"""
from sage.rings.arith import binomial
n = ZZ(n)
return binomial(2*n,n).divide_knowing_divisible_by(n+1)
def euler_number(n):
"""
Returns the n-th Euler number
IMPLEMENTATION: Wraps Maxima's euler.
EXAMPLES::
sage: [euler_number(i) for i in range(10)]
[1, 0, -1, 0, 5, 0, -61, 0, 1385, 0]
sage: maxima.eval("taylor (2/(exp(x)+exp(-x)), x, 0, 10)")
'1-x^2/2+5*x^4/24-61*x^6/720+277*x^8/8064-50521*x^10/3628800'
sage: [euler_number(i)/factorial(i) for i in range(11)]
[1, 0, -1/2, 0, 5/24, 0, -61/720, 0, 277/8064, 0, -50521/3628800]
sage: euler_number(-1)
Traceback (most recent call last):
...
ValueError: n (=-1) must be a nonnegative integer
REFERENCES:
- http://en.wikipedia.org/wiki/Euler_number
"""
n = ZZ(n)
if n < 0:
raise ValueError, "n (=%s) must be a nonnegative integer"%n
return ZZ(maxima.eval("euler(%s)"%n))
def fibonacci(n, algorithm="pari"):
"""
Returns the n-th Fibonacci number. The Fibonacci sequence
`F_n` is defined by the initial conditions
`F_1=F_2=1` and the recurrence relation
`F_{n+2} = F_{n+1} + F_n`. For negative `n` we
define `F_n = (-1)^{n+1}F_{-n}`, which is consistent with
the recurrence relation.
INPUT:
- ``algorithm`` - string:
- ``"pari"`` - (default) - use the PARI C library's
fibo function.
- ``"gap"`` - use GAP's Fibonacci function
.. note::
PARI is tens to hundreds of times faster than GAP here;
moreover, PARI works for every large input whereas GAP doesn't.
EXAMPLES::
sage: fibonacci(10)
55
sage: fibonacci(10, algorithm='gap')
55
::
sage: fibonacci(-100)
-354224848179261915075
sage: fibonacci(100)
354224848179261915075
::
sage: fibonacci(0)
0
sage: fibonacci(1/2)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
"""
n = ZZ(n)
if algorithm == 'pari':
return ZZ(pari(n).fibonacci())
elif algorithm == 'gap':
return ZZ(gap.eval("Fibonacci(%s)"%n))
else:
raise ValueError, "no algorithm %s"%algorithm
def lucas_number1(n,P,Q):
"""
Returns the n-th Lucas number "of the first kind" (this is not
standard terminology). The Lucas sequence `L^{(1)}_n` is
defined by the initial conditions `L^{(1)}_1=0`,
`L^{(1)}_2=1` and the recurrence relation
`L^{(1)}_{n+2} = P*L^{(1)}_{n+1} - Q*L^{(1)}_n`.
Wraps GAP's Lucas(...)[1].
P=1, Q=-1 gives the Fibonacci sequence.
INPUT:
- ``n`` - integer
- ``P, Q`` - integer or rational numbers
OUTPUT: integer or rational number
EXAMPLES::
sage: lucas_number1(5,1,-1)
5
sage: lucas_number1(6,1,-1)
8
sage: lucas_number1(7,1,-1)
13
sage: lucas_number1(7,1,-2)
43
::
sage: lucas_number1(5,2,3/5)
229/25
sage: lucas_number1(5,2,1.5)
Traceback (most recent call last):
...
TypeError: no canonical coercion from Real Field with 53 bits of precision to Rational Field
There was a conjecture that the sequence `L_n` defined by
`L_{n+2} = L_{n+1} + L_n`, `L_1=1`,
`L_2=3`, has the property that `n` prime implies
that `L_n` is prime.
::
sage: lucas = lambda n : Integer((5/2)*lucas_number1(n,1,-1)+(1/2)*lucas_number2(n,1,-1))
sage: [[lucas(n),is_prime(lucas(n)),n+1,is_prime(n+1)] for n in range(15)]
[[1, False, 1, False],
[3, True, 2, True],
[4, False, 3, True],
[7, True, 4, False],
[11, True, 5, True],
[18, False, 6, False],
[29, True, 7, True],
[47, True, 8, False],
[76, False, 9, False],
[123, False, 10, False],
[199, True, 11, True],
[322, False, 12, False],
[521, True, 13, True],
[843, False, 14, False],
[1364, False, 15, False]]
Can you use Sage to find a counterexample to the conjecture?
"""
ans=gap.eval("Lucas(%s,%s,%s)[1]"%(QQ._coerce_(P),QQ._coerce_(Q),ZZ(n)))
return sage_eval(ans)
def lucas_number2(n,P,Q):
r"""
Returns the n-th Lucas number "of the second kind" (this is not
standard terminology). The Lucas sequence `L^{(2)}_n` is
defined by the initial conditions `L^{(2)}_1=2`,
`L^{(2)}_2=P` and the recurrence relation
`L^{(2)}_{n+2} = P*L^{(2)}_{n+1} - Q*L^{(2)}_n`.
Wraps GAP's Lucas(...)[2].
INPUT:
- ``n`` - integer
- ``P, Q`` - integer or rational numbers
OUTPUT: integer or rational number
EXAMPLES::
sage: [lucas_number2(i,1,-1) for i in range(10)]
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76]
sage: [fibonacci(i-1)+fibonacci(i+1) for i in range(10)]
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76]
::
sage: n = lucas_number2(5,2,3); n
2
sage: type(n)
<type 'sage.rings.integer.Integer'>
sage: n = lucas_number2(5,2,-3/9); n
418/9
sage: type(n)
<type 'sage.rings.rational.Rational'>
The case P=1, Q=-1 is the Lucas sequence in Brualdi's Introductory
Combinatorics, 4th ed., Prentice-Hall, 2004::
sage: [lucas_number2(n,1,-1) for n in range(10)]
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76]
"""
ans=gap.eval("Lucas(%s,%s,%s)[2]"%(QQ._coerce_(P),QQ._coerce_(Q),ZZ(n)))
return sage_eval(ans)
def stirling_number1(n,k):
"""
Returns the n-th Stilling number `S_1(n,k)` of the first
kind (the number of permutations of n points with k cycles). Wraps
GAP's Stirling1.
EXAMPLES::
sage: stirling_number1(3,2)
3
sage: stirling_number1(5,2)
50
sage: 9*stirling_number1(9,5)+stirling_number1(9,4)
269325
sage: stirling_number1(10,5)
269325
Indeed, `S_1(n,k) = S_1(n-1,k-1) + (n-1)S_1(n-1,k)`.
"""
return ZZ(gap.eval("Stirling1(%s,%s)"%(ZZ(n),ZZ(k))))
def stirling_number2(n, k, algorithm=None):
"""
Returns the n-th Stirling number `S_2(n,k)` of the second
kind (the number of ways to partition a set of n elements into k
pairwise disjoint nonempty subsets). (The n-th Bell number is the
sum of the `S_2(n,k)`'s, `k=0,...,n`.)
INPUT:
* ``n`` - nonnegative machine-size integer
* ``k`` - nonnegative machine-size integer
* ``algorithm``:
* None (default) - use native implementation
* ``"maxima"`` - use Maxima's stirling2 function
* ``"gap"`` - use GAP's Stirling2 function
EXAMPLES:
Print a table of the first several Stirling numbers of the second kind::
sage: for n in range(10):
... for k in range(10):
... print str(stirling_number2(n,k)).rjust(k and 6),
... print
...
1 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
0 1 1 0 0 0 0 0 0 0
0 1 3 1 0 0 0 0 0 0
0 1 7 6 1 0 0 0 0 0
0 1 15 25 10 1 0 0 0 0
0 1 31 90 65 15 1 0 0 0
0 1 63 301 350 140 21 1 0 0
0 1 127 966 1701 1050 266 28 1 0
0 1 255 3025 7770 6951 2646 462 36 1
Stirling numbers satisfy `S_2(n,k) = S_2(n-1,k-1) + kS_2(n-1,k)`::
sage: 5*stirling_number2(9,5) + stirling_number2(9,4)
42525
sage: stirling_number2(10,5)
42525
TESTS::
sage: stirling_number2(500,501)
0
sage: stirling_number2(500,500)
1
sage: stirling_number2(500,499)
124750
sage: stirling_number2(500,498)
7739801875
sage: stirling_number2(500,497)
318420320812125
sage: stirling_number2(500,0)
0
sage: stirling_number2(500,1)
1
sage: stirling_number2(500,2)
1636695303948070935006594848413799576108321023021532394741645684048066898202337277441635046162952078575443342063780035504608628272942696526664263794687
sage: stirling_number2(500,3)
6060048632644989473730877846590553186337230837666937173391005972096766698597315914033083073801260849147094943827552228825899880265145822824770663507076289563105426204030498939974727520682393424986701281896187487826395121635163301632473646
sage: stirling_number2(500,30)
13707767141249454929449108424328432845001327479099713037876832759323918134840537229737624018908470350134593241314462032607787062188356702932169472820344473069479621239187226765307960899083230982112046605340713218483809366970996051181537181362810003701997334445181840924364501502386001705718466534614548056445414149016614254231944272872440803657763210998284198037504154374028831561296154209804833852506425742041757849726214683321363035774104866182331315066421119788248419742922490386531970053376982090046434022248364782970506521655684518998083846899028416459701847828711541840099891244700173707021989771147674432503879702222276268661726508226951587152781439224383339847027542755222936463527771486827849728880
sage: stirling_number2(500,31)
5832088795102666690960147007601603328246123996896731854823915012140005028360632199516298102446004084519955789799364757997824296415814582277055514048635928623579397278336292312275467402957402880590492241647229295113001728653772550743446401631832152281610081188041624848850056657889275564834450136561842528589000245319433225808712628826136700651842562516991245851618481622296716433577650218003181535097954294609857923077238362717189185577756446945178490324413383417876364657995818830270448350765700419876347023578011403646501685001538551891100379932684279287699677429566813471166558163301352211170677774072447414719380996777162087158124939742564291760392354506347716119002497998082844612434332155632097581510486912
sage: n = stirling_number2(20,11)
sage: n
1900842429486
sage: type(n)
<type 'sage.rings.integer.Integer'>
sage: n = stirling_number2(20,11,algorithm='gap')
sage: n
1900842429486
sage: type(n)
<type 'sage.rings.integer.Integer'>
sage: n = stirling_number2(20,11,algorithm='maxima')
sage: n
1900842429486
sage: type(n)
<type 'sage.rings.integer.Integer'>
Sage's implementation splitting the computation of the Stirling
numbers of the second kind in two cases according to `n`, let us
check the result it gives agree with both maxima and gap.
For `n<200`::
sage: for n in Subsets(range(100,200), 5).random_element():
... for k in Subsets(range(n), 5).random_element():
... s_sage = stirling_number2(n,k)
... s_maxima = stirling_number2(n,k, algorithm = "maxima")
... s_gap = stirling_number2(n,k, algorithm = "gap")
... if not (s_sage == s_maxima and s_sage == s_gap):
... print "Error with n<200"
For `n\geq 200`::
sage: for n in Subsets(range(200,300), 5).random_element():
... for k in Subsets(range(n), 5).random_element():
... s_sage = stirling_number2(n,k)
... s_maxima = stirling_number2(n,k, algorithm = "maxima")
... s_gap = stirling_number2(n,k, algorithm = "gap")
... if not (s_sage == s_maxima and s_sage == s_gap):
... print "Error with n<200"
TESTS:
Checking an exception is raised whenever a wrong value is given
for ``algorithm``::
sage: s_sage = stirling_number2(50,3, algorithm = "CloudReading")
Traceback (most recent call last):
...
ValueError: unknown algorithm: CloudReading
"""
if algorithm is None:
return _stirling_number2(n, k)
elif algorithm == 'gap':
return ZZ(gap.eval("Stirling2(%s,%s)"%(ZZ(n),ZZ(k))))
elif algorithm == 'maxima':
return ZZ(maxima.eval("stirling2(%s,%s)"%(ZZ(n),ZZ(k))))
else:
raise ValueError("unknown algorithm: %s" % algorithm)
class CombinatorialObject(SageObject):
def __init__(self, l):
"""
CombinatorialObject provides a thin wrapper around a list. The main
differences are that __setitem__ is disabled so that
CombinatorialObjects are shallowly immutable, and the intention is
that they are semantically immutable.
Because of this, CombinatorialObjects provide a __hash__
function which computes the hash of the string representation of a
list and the hash of its parent's class. Thus, each
CombinatorialObject should have a unique string representation.
INPUT:
- ``l`` - a list or any object that can be convert to a list by
``list``
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: c == loads(dumps(c))
True
sage: c._list
[1, 2, 3]
sage: c._hash is None
True
"""
if isinstance(l, list):
self._list = l
else:
self._list = list(l)
self._hash = None
def __str__(self):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: str(c)
'[1, 2, 3]'
"""
return str(self._list)
def _repr_(self):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: c.__repr__()
'[1, 2, 3]'
"""
return self._list.__repr__()
def __eq__(self, other):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: d = CombinatorialObject([2,3,4])
sage: c == [1,2,3]
True
sage: c == [2,3,4]
False
sage: c == d
False
"""
if isinstance(other, CombinatorialObject):
return self._list.__eq__(other._list)
else:
return self._list.__eq__(other)
def __lt__(self, other):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: d = CombinatorialObject([2,3,4])
sage: c < d
True
sage: c < [2,3,4]
True
"""
if isinstance(other, CombinatorialObject):
return self._list.__lt__(other._list)
else:
return self._list.__lt__(other)
def __le__(self, other):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: d = CombinatorialObject([2,3,4])
sage: c <= c
True
sage: c <= d
True
sage: c <= [1,2,3]
True
"""
if isinstance(other, CombinatorialObject):
return self._list.__le__(other._list)
else:
return self._list.__le__(other)
def __gt__(self, other):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: d = CombinatorialObject([2,3,4])
sage: c > c
False
sage: c > d
False
sage: c > [1,2,3]
False
"""
if isinstance(other, CombinatorialObject):
return self._list.__gt__(other._list)
else:
return self._list.__gt__(other)
def __ge__(self, other):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: d = CombinatorialObject([2,3,4])
sage: c >= c
True
sage: c >= d
False
sage: c >= [1,2,3]
True
"""
if isinstance(other, CombinatorialObject):
return self._list.__ge__(other._list)
else:
return self._list.__ge__(other)
def __ne__(self, other):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: d = CombinatorialObject([2,3,4])
sage: c != c
False
sage: c != d
True
sage: c != [1,2,3]
False
"""
if isinstance(other, CombinatorialObject):
return self._list.__ne__(other._list)
else:
return self._list.__ne__(other)
def __add__(self, other):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: c + [4]
[1, 2, 3, 4]
sage: type(_)
<type 'list'>
"""
return self._list + other
def __hash__(self):
"""
Computes the hash of self by computing the hash of the string
representation of self._list. The hash is cached and stored in
self._hash.
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: c._hash is None
True
sage: hash(c) #random
1335416675971793195
sage: c._hash #random
1335416675971793195
"""
if self._hash is None:
self._hash = str(self._list).__hash__()
return self._hash
def __len__(self):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: len(c)
3
sage: c.__len__()
3
"""
return self._list.__len__()
def __getitem__(self, key):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: c[0]
1
sage: c[1:]
[2, 3]
sage: type(_)
<type 'list'>
"""
return self._list.__getitem__(key)
def __iter__(self):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: list(iter(c))
[1, 2, 3]
"""
return self._list.__iter__()
def __contains__(self, item):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: 1 in c
True
sage: 5 in c
False
"""
return self._list.__contains__(item)
def index(self, key):
"""
EXAMPLES::
sage: c = CombinatorialObject([1,2,3])
sage: c.index(1)
0
sage: c.index(3)
2
"""
return self._list.index(key)
from sage.misc.classcall_metaclass import ClasscallMetaclass
from sage.categories.enumerated_sets import EnumeratedSets
class CombinatorialClass(Parent):
"""
This class is deprecated, and will disappear as soon as all derived
classes in Sage's library will have been fixed. Please derive
directly from Parent and use the category :class:`EnumeratedSets`,
:class:`FiniteEnumeratedSets`, or :class:`InfiniteEnumeratedSets`, as
appropriate.
For examples, see::
sage: FiniteEnumeratedSets().example()
An example of a finite enumerated set: {1,2,3}
sage: InfiniteEnumeratedSets().example()
An example of an infinite enumerated set: the non negative integers
"""
__metaclass__ = ClasscallMetaclass
def __init__(self, category = None, *keys, **opts):
"""
TESTS::
sage: C = sage.combinat.combinat.CombinatorialClass()
sage: C.category()
Category of enumerated sets
sage: C.__class__
<class 'sage.combinat.combinat.CombinatorialClass_with_category'>
sage: isinstance(C, Parent)
True
sage: C = sage.combinat.combinat.CombinatorialClass(category = FiniteEnumeratedSets())
sage: C.category()
Category of finite enumerated sets
"""
Parent.__init__(self, category = EnumeratedSets().or_subcategory(category))
def __len__(self):
"""
__len__ has been removed ! to get the number of element in a
combinatorial class, use .cardinality instead.
TEST::
sage: len(Partitions(5))
Traceback (most recent call last):
...
AttributeError: __len__ has been removed; use .cardinality() instead
"""
raise AttributeError, "__len__ has been removed; use .cardinality() instead"
def count(self):
"""
Deprecated ! Please use ``.cardinality`` instead.
TEST::
sage: class C(CombinatorialClass):
... def __iter__(self):
... return iter([1,2,3])
...
sage: C().count() #indirect doctest
doctest:1: DeprecationWarning: The usage of count for combinatorial classes is deprecated. Please use cardinality
3
"""
deprecation("The usage of count for combinatorial classes is deprecated. Please use cardinality")
return self.cardinality()
def is_finite(self):
"""
Returns whether self is finite or not.
EXAMPLES::
sage: Partitions(5).is_finite()
True
sage: Permutations().is_finite()
False
"""
from sage.rings.all import infinity
return self.cardinality() != infinity
def __getitem__(self, i):
"""
Returns the combinatorial object of rank i.
EXAMPLES::
sage: p5 = Partitions(5)
sage: p5[0]
[5]
sage: p5[6]
[1, 1, 1, 1, 1]
sage: p5[7]
Traceback (most recent call last):
...
ValueError: the value must be between 0 and 6 inclusive
"""
return self.unrank(i)
def __str__(self):
"""
Returns a string representation of self.
EXAMPLES::
sage: str(Partitions(5))
'Partitions of the integer 5'
"""
return self.__repr__()
def _repr_(self):
"""
EXAMPLES::
sage: repr(Partitions(5)) # indirect doctest
'Partitions of the integer 5'
"""
if hasattr(self, '_name') and self._name:
return self._name
else:
return "Combinatorial Class -- REDEFINE ME!"
def __contains__(self, x):
"""
Tests whether or not the combinatorial class contains the object x.
This raises a NotImplementedError as a default since _all_
subclasses of CombinatorialClass should override this.
Note that we could replace this with a default implementation that
just iterates through the elements of the combinatorial class and
checks for equality. However, since we use __contains__ for
type checking, this operation should be cheap and should be
implemented manually for each combinatorial class.
EXAMPLES::
sage: C = CombinatorialClass()
sage: x in C
Traceback (most recent call last):
...
NotImplementedError
"""
raise NotImplementedError
def __cmp__(self, x):
"""
Compares two different combinatorial classes. For now, the
comparison is done just on their repr's.
EXAMPLES::
sage: p5 = Partitions(5)
sage: p6 = Partitions(6)
sage: repr(p5) == repr(p6)
False
sage: p5 == p6
False
"""
return cmp(repr(self), repr(x))
def __cardinality_from_iterator(self):
"""
Default implementation of cardinality which just goes through the iterator
of the combinatorial class to count the number of objects.
EXAMPLES::
sage: class C(CombinatorialClass):
... def __iter__(self):
... return iter([1,2,3])
...
sage: C().cardinality() #indirect doctest
3
"""
c = Integer(0)
one = Integer(1)
for _ in self:
c += one
return c
cardinality = __cardinality_from_iterator
def __call__(self, x):
"""
Returns x as an element of the combinatorial class's object class.
EXAMPLES::
sage: p5 = Partitions(5)
sage: a = [2,2,1]
sage: type(a)
<type 'list'>
sage: a = p5(a)
sage: type(a)
<class 'sage.combinat.partition.Partition_class'>
sage: p5([2,1])
Traceback (most recent call last):
...
ValueError: [2, 1] not in Partitions of the integer 5
"""
if x in self:
return self._element_constructor_(x)
else:
raise ValueError, "%s not in %s"%(x, self)
Element = CombinatorialObject
@lazy_attribute
def element_class(self):
"""
This function is a temporary helper so that a CombinatorialClass
behaves as a parent for creating elements. This will disappear when
combinatorial classes will be turned into actual parents (in the
category EnumeratedSets).
TESTS::
sage: P5 = Partitions(5)
sage: P5.element_class
<class 'sage.combinat.partition.Partition_class'>
"""
if hasattr(self, "object_class"):
from sage.misc.misc import deprecation
deprecation("Using object_class for specifying the class of the elements of a combinatorial class is deprecated. Please use Element instead")
return self.Element
def _element_constructor_(self, x):
"""
This function is a temporary helper so that a CombinatorialClass
behaves as a parent for creating elements. This will disappear when
combinatorial classes will be turned into actual parents (in the
category EnumeratedSets).
TESTS::
sage: P5 = Partitions(5)
sage: p = P5([3,2]) # indirect doctest
sage: type(p)
<class 'sage.combinat.partition.Partition_class'>
"""
return self.element_class(x)
def __list_from_iterator(self):
"""
The default implementation of list which builds the list from the
iterator.
EXAMPLES::
sage: class C(CombinatorialClass):
... def __iter__(self):
... return iter([1,2,3])
...
sage: C().list() #indirect doctest
[1, 2, 3]
"""
return [x for x in self]
list = __list_from_iterator
Element = CombinatorialObject
def __iterator_from_next(self):
"""
An iterator to use when .first() and .next() are provided.
EXAMPLES::
sage: C = CombinatorialClass()
sage: C.first = lambda: 0
sage: C.next = lambda c: c+1
sage: it = iter(C) # indirect doctest
sage: [it.next() for _ in range(4)]
[0, 1, 2, 3]
"""
f = self.first()
yield f
while True:
try:
f = self.next(f)
except (TypeError, ValueError ):
break
if f is None or f is False :
break
else:
yield f
def __iterator_from_previous(self):
"""
An iterator to use when .last() and .previous() are provided. Note
that this requires the combinatorial class to be finite. It is not
recommended to implement combinatorial classes using last and
previous.
EXAMPLES::
sage: C = CombinatorialClass()
sage: C.last = lambda: 4
sage: def prev(c):
... if c <= 1:
... return None
... else:
... return c-1
...
sage: C.previous = prev
sage: it = iter(C) # indirect doctest
sage: [it.next() for _ in range(4)]
[1, 2, 3, 4]
"""
l = self.last()
li = [l]
while True:
try:
l = self.previous(l)
except (TypeError, ValueError):
break
if l == None:
break
else:
li.append(l)
return reversed(li)
def __iterator_from_unrank(self):
"""
An iterator to use when .unrank() is provided.
EXAMPLES::
sage: C = CombinatorialClass()
sage: l = [1,2,3]
sage: C.unrank = lambda c: l[c]
sage: list(C) # indirect doctest
[1, 2, 3]
"""
r = 0
u = self.unrank(r)
yield u
while True:
r += 1
try:
u = self.unrank(r)
except (TypeError, ValueError, IndexError):
break
if u == None:
break
else:
yield u
def __iterator_from_list(self):
"""
An iterator to use when .list() is provided()
EXAMPLES::
sage: C = CombinatorialClass()
sage: C.list = lambda: [1, 2, 3]
sage: list(C) # indirect doctest
[1, 2, 3]
"""
for x in self.list():
yield x
def iterator(self):
"""
Iterator is deprecated for combinatorial classes.
EXAMPLES::
sage: p5 = Partitions(3)
sage: it = p5.iterator()
doctest:1: DeprecationWarning: The usage of iterator for combinatorial classes is deprecated. Please use the class itself
sage: [i for i in it]
[[3], [2, 1], [1, 1, 1]]
sage: [i for i in p5]
[[3], [2, 1], [1, 1, 1]]
"""
from sage.misc.misc import deprecation
deprecation("The usage of iterator for combinatorial classes is deprecated. Please use the class itself")
return self.__iter__()
def __iter__(self):
"""
Allows the combinatorial class to be treated as an iterator. Default
implementation.
EXAMPLES::
sage: p5 = Partitions(5)
sage: [i for i in p5]
[[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1]]
sage: C = CombinatorialClass()
sage: iter(C)
Traceback (most recent call last):
...
NotImplementedError: iterator called but not implemented
"""
if ( self.first != self.__first_from_iterator and
self.next != self.__next_from_iterator ):
return self.__iterator_from_next()
elif ( self.last != self.__last_from_iterator and
self.previous != self.__previous_from_iterator):
return self.__iterator_from_previous()
elif self.unrank != self.__unrank_from_iterator:
return self.__iterator_from_unrank()
elif self.list != self.__list_from_iterator:
return self.__iterator_from_list()
else:
raise NotImplementedError, "iterator called but not implemented"
def __unrank_from_iterator(self, r):
"""
Default implementation of unrank which goes through the iterator.
EXAMPLES::
sage: C = CombinatorialClass()
sage: C.list = lambda: [1,2,3]
sage: C.unrank(1) # indirect doctest
2
"""
counter = 0
for u in self:
if counter == r:
return u
counter += 1
raise ValueError, "the value must be between %s and %s inclusive"%(0,counter-1)
unrank = __unrank_from_iterator
def __random_element_from_unrank(self):
"""
Default implementation of random which uses unrank.
EXAMPLES::
sage: C = CombinatorialClass()
sage: C.list = lambda: [1,2,3]
sage: C.random_element() # indirect doctest
1
"""
c = self.cardinality()
r = randint(0, c-1)
return self.unrank(r)
random_element = __random_element_from_unrank
def random(self):
"""
Deprecated. Use self.random_element() instead.
EXAMPLES::
sage: C = CombinatorialClass()
sage: C.random()
Traceback (most recent call last):
...
NotImplementedError: Deprecated: use random_element() instead
"""
raise NotImplementedError, "Deprecated: use random_element() instead"
def __rank_from_iterator(self, obj):
"""
Default implementation of rank which uses iterator.
EXAMPLES::
sage: C = CombinatorialClass()
sage: C.list = lambda: [1,2,3]
sage: C.rank(3) # indirect doctest
2
"""
r = 0
for i in self:
if i == obj:
return r
r += 1
raise ValueError
rank = __rank_from_iterator
def __first_from_iterator(self):
"""
Default implementation for first which uses iterator.
EXAMPLES::
sage: C = CombinatorialClass()
sage: C.list = lambda: [1,2,3]
sage: C.first() # indirect doctest
1
"""
for i in self:
return i
first = __first_from_iterator
def __last_from_iterator(self):
"""
Default implementation for first which uses iterator.
EXAMPLES::
sage: C = CombinatorialClass()
sage: C.list = lambda: [1,2,3]
sage: C.last() # indirect doctest
3
"""
for i in self:
pass
return i
last = __last_from_iterator
def __next_from_iterator(self, obj):
"""
Default implementation for next which uses iterator.
EXAMPLES::
sage: C = CombinatorialClass()
sage: C.list = lambda: [1,2,3]
sage: C.next(2) # indirect doctest
3
"""
found = False
for i in self:
if found:
return i
if i == obj:
found = True
return None
next = __next_from_iterator
def __previous_from_iterator(self, obj):
"""
Default implementation for next which uses iterator.
EXAMPLES::
sage: C = CombinatorialClass()
sage: C.list = lambda: [1,2,3]
sage: C.previous(2) # indirect doctest
1
"""
prev = None
for i in self:
if i == obj:
break
prev = i
return prev
previous = __previous_from_iterator
def filter(self, f, name=None):
"""
Returns the combinatorial subclass of f which consists of the
elements x of self such that f(x) is True.
EXAMPLES::
sage: P = Permutations(3).filter(lambda x: x.avoids([1,2]))
sage: P.list()
[[3, 2, 1]]
"""
return FilteredCombinatorialClass(self, f, name=name)
def union(self, right_cc, name=None):
"""
Returns the combinatorial class representing the union of self and
right_cc.
EXAMPLES::
sage: P = Permutations(2).union(Permutations(1))
sage: P.list()
[[1, 2], [2, 1], [1]]
"""
if not isinstance(right_cc, CombinatorialClass):
raise TypeError, "right_cc must be a CombinatorialClass"
return UnionCombinatorialClass(self, right_cc, name=name)
def map(self, f, name=None):
r"""
Returns the image `\{f(x) | x \in \text{self}\}` of this combinatorial
class by `f`, as a combinatorial class.
`f` is supposed to be injective.
EXAMPLES::
sage: R = Permutations(3).map(attrcall('reduced_word')); R
Image of Standard permutations of 3 by *.reduced_word()
sage: R.cardinality()
6
sage: R.list()
[[], [2], [1], [1, 2], [2, 1], [2, 1, 2]]
sage: [ r for r in R]
[[], [2], [1], [1, 2], [2, 1], [2, 1, 2]]
If the function is not injective, then there may be repeated elements:
sage: P = Partitions(4)
sage: P.list()
[[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]
sage: P.map(len).list()
[1, 2, 2, 3, 4]
TESTS::
sage: R = Permutations(3).map(attrcall('reduced_word'))
sage: R == loads(dumps(R))
True
"""
return MapCombinatorialClass(self, f, name)
class FilteredCombinatorialClass(CombinatorialClass):
def __init__(self, combinatorial_class, f, name=None):
"""
A filtered combinatorial class F is a subset of another
combinatorial class C specified by a function f that takes in an
element c of C and returns True if and only if c is in F.
TESTS::
sage: Permutations(3).filter(lambda x: x.avoids([1,2]))
Filtered sublass of Standard permutations of 3
"""
self.f = f
self.combinatorial_class = combinatorial_class
self._name = name
def __repr__(self):
"""
EXAMPLES::
sage: P = Permutations(3).filter(lambda x: x.avoids([1,2]))
sage: P.__repr__()
'Filtered sublass of Standard permutations of 3'
sage: P._name = 'Permutations avoiding [1, 2]'
sage: P.__repr__()
'Permutations avoiding [1, 2]'
"""
if self._name:
return self._name
else:
return "Filtered sublass of " + repr(self.combinatorial_class)
def __contains__(self, x):
"""
EXAMPLES::
sage: P = Permutations(3).filter(lambda x: x.avoids([1,2]))
sage: 'cat' in P
False
sage: [4,3,2,1] in P
False
sage: Permutation([1,2,3]) in P
False
sage: Permutation([3,2,1]) in P
True
"""
return x in self.combinatorial_class and self.f(x)
def cardinality(self):
"""
EXAMPLES::
sage: P = Permutations(3).filter(lambda x: x.avoids([1,2]))
sage: P.cardinality()
1
"""
c = 0
for _ in self:
c += 1
return c
def __iter__(self):
"""
EXAMPLES::
sage: P = Permutations(3).filter(lambda x: x.avoids([1,2]))
sage: list(P)
[[3, 2, 1]]
"""
for x in self.combinatorial_class:
if self.f(x):
yield x
class UnionCombinatorialClass(CombinatorialClass):
def __init__(self, left_cc, right_cc, name=None):
"""
A UnionCombinatorialClass is a union of two other combinatorial
classes.
TESTS::
sage: P = Permutations(3).union(Permutations(2))
sage: P == loads(dumps(P))
True
"""
self.left_cc = left_cc
self.right_cc = right_cc
self._name = name
def __repr__(self):
"""
TESTS::
sage: print repr(Permutations(3).union(Permutations(2)))
Union combinatorial class of
Standard permutations of 3
and
Standard permutations of 2
"""
if self._name:
return self._name
else:
return "Union combinatorial class of \n %s\nand\n %s"%(self.left_cc, self.right_cc)
def __contains__(self, x):
"""
EXAMPLES::
sage: P = Permutations(3).union(Permutations(2))
sage: [1,2] in P
True
sage: [3,2,1] in P
True
sage: [1,2,3,4] in P
False
"""
return x in self.left_cc or x in self.right_cc
def cardinality(self):
"""
EXAMPLES::
sage: P = Permutations(3).union(Permutations(2))
sage: P.cardinality()
8
"""
return self.left_cc.cardinality() + self.right_cc.cardinality()
def list(self):
"""
EXAMPLES::
sage: P = Permutations(3).union(Permutations(2))
sage: P.list()
[[1, 2, 3],
[1, 3, 2],
[2, 1, 3],
[2, 3, 1],
[3, 1, 2],
[3, 2, 1],
[1, 2],
[2, 1]]
"""
return self.left_cc.list() + self.right_cc.list()
def __iter__(self):
"""
EXAMPLES::
sage: P = Permutations(3).union(Permutations(2))
sage: list(P)
[[1, 2, 3],
[1, 3, 2],
[2, 1, 3],
[2, 3, 1],
[3, 1, 2],
[3, 2, 1],
[1, 2],
[2, 1]]
"""
for x in self.left_cc:
yield x
for x in self.right_cc:
yield x
def first(self):
"""
EXAMPLES::
sage: P = Permutations(3).union(Permutations(2))
sage: P.first()
[1, 2, 3]
"""
return self.left_cc.first()
def last(self):
"""
EXAMPLES::
sage: P = Permutations(3).union(Permutations(2))
sage: P.last()
[2, 1]
"""
return self.right_cc.last()
def rank(self, x):
"""
EXAMPLES::
sage: P = Permutations(3).union(Permutations(2))
sage: P.rank(Permutation([2,1]))
7
sage: P.rank(Permutation([1,2,3]))
0
"""
try:
return self.left_cc.rank(x)
except (TypeError, ValueError):
return self.left_cc.cardinality() + self.right_cc.rank(x)
def unrank(self, x):
"""
EXAMPLES::
sage: P = Permutations(3).union(Permutations(2))
sage: P.unrank(7)
[2, 1]
sage: P.unrank(0)
[1, 2, 3]
"""
try:
return self.left_cc.unrank(x)
except (TypeError, ValueError):
return self.right_cc.unrank(x - self.left_cc.cardinality())
class MapCombinatorialClass(CombinatorialClass):
r"""
A MapCombinatorialClass models the image of a combinatorial
class through a function which is assumed to be injective
See CombinatorialClass.map for examples
"""
def __init__(self, cc, f, name=None):
"""
TESTS::
sage: Partitions(3).map(attrcall('conjugate'))
Image of Partitions of the integer 3 by *.conjugate()
"""
self.cc = cc
self.f = f
self._name = name
def __repr__(self):
"""
TESTS::
sage: Partitions(3).map(attrcall('conjugate'))
Image of Partitions of the integer 3 by *.conjugate()
"""
if self._name:
return self._name
else:
return "Image of %s by %s"%(self.cc, self.f)
def cardinality(self):
"""
Returns the cardinality of this combinatorial class
EXAMPLES::
sage: R = Permutations(10).map(attrcall('reduced_word'))
sage: R.cardinality()
3628800
"""
return self.cc.cardinality()
def __iter__(self):
"""
Returns an iterator over the elements of this combinatorial class
EXAMPLES::
sage: R = Permutations(10).map(attrcall('reduced_word'))
sage: R.cardinality()
3628800
"""
for x in self.cc:
yield self.f(x)
def an_element(self):
"""
Returns an element of this combinatorial class
EXAMPLES::
sage: R = SymmetricGroup(10).map(attrcall('reduced_word'))
sage: R.an_element()
[9, 8, 7, 6, 5, 4, 3, 2, 1]
"""
return self.f(self.cc.an_element())
from sage.rings.all import infinity
class InfiniteAbstractCombinatorialClass(CombinatorialClass):
r"""
This is an internal class that should not be used directly. A class which
inherits from InfiniteAbstractCombinatorialClass inherits the standard
methods list and count.
If self._infinite_cclass_slice exists then self.__iter__ returns an
iterator for self, otherwise raise NotImplementedError. The method
self._infinite_cclass_slice is supposed to accept any integer as an
argument and return something which is iterable.
"""
def cardinality(self):
"""
Counts the elements of the combinatorial class.
EXAMPLES:
sage: R = InfiniteAbstractCombinatorialClass()
sage: R.cardinality()
+Infinity
"""
return infinity
def list(self):
"""
Returns an error since self is an infinite combinatorial class.
EXAMPLES:
sage: R = InfiniteAbstractCombinatorialClass()
sage: R.list()
Traceback (most recent call last):
...
NotImplementedError: infinite list
"""
raise NotImplementedError, "infinite list"
def __iter__(self):
"""
Returns an iterator for the infinite combinatorial class self if
possible or raise a NotImplementedError.
EXAMPLES:
sage: R = InfiniteAbstractCombinatorialClass()
sage: iter(R).next()
Traceback (most recent call last):
...
NotImplementedError
sage: c = iter(Compositions()) # indirect doctest
sage: c.next(), c.next(), c.next(), c.next(), c.next(), c.next()
([], [1], [1, 1], [2], [1, 1, 1], [1, 2])
sage: c.next(), c.next(), c.next(), c.next(), c.next(), c.next()
([2, 1], [3], [1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [1, 3])
"""
try:
finite = self._infinite_cclass_slice
except AttributeError:
raise NotImplementedError
i = 0
while True:
for c in finite(i):
yield c
i+=1
def hurwitz_zeta(s,x,N):
"""
Returns the value of the `\zeta(s,x)` to `N`
decimals, where s and x are real.
The Hurwitz zeta function is one of the many zeta functions. It
defined as
.. math::
\zeta(s,x) = \sum_{k=0}^\infty (k+x)^{-s}.
When `x = 1`, this coincides with Riemann's zeta function.
The Dirichlet L-functions may be expressed as a linear combination
of Hurwitz zeta functions.
Note that if you use floating point inputs, then the results may be
slightly off.
EXAMPLES::
sage: hurwitz_zeta(3,1/2,6)
8.41439000000000
sage: hurwitz_zeta(11/10,1/2,6)
12.1041000000000
sage: hurwitz_zeta(11/10,1/2,50)
12.10381349568375510570907741296668061903364861809
REFERENCES:
- http://en.wikipedia.org/wiki/Hurwitz_zeta_function
"""
maxima.eval('load ("bffac")')
s = maxima.eval("bfhzeta (%s,%s,%s)"%(s,x,N))
i = s.rfind('b')
if i == -1:
return sage_eval(s)
else:
if s[i+1:] == '0':
return sage_eval(s[:i])
else:
return sage_eval(s[:i])*10**sage_eval(s[i+1:])
return s
def combinations(mset,k):
r"""
A combination of a multiset (a list of objects which may contain
the same object several times) mset is an unordered selection
without repetitions and is represented by a sorted sublist of mset.
Returns the set of all combinations of the multiset mset with k
elements.
.. warning::
Wraps GAP's Combinations. Hence mset must be a list of objects
that have string representations that can be interpreted by the
GAP interpreter. If mset consists of at all complicated Sage
objects, this function does *not* do what you expect. A proper
function should be written! (TODO!)
EXAMPLES::
sage: mset = [1,1,2,3,4,4,5]
sage: combinations(mset,2)
[[1, 1],
[1, 2],
[1, 3],
[1, 4],
[1, 5],
[2, 3],
[2, 4],
[2, 5],
[3, 4],
[3, 5],
[4, 4],
[4, 5]]
sage: mset = ["d","a","v","i","d"]
sage: combinations(mset,3)
['add', 'adi', 'adv', 'aiv', 'ddi', 'ddv', 'div']
.. note::
For large lists, this raises a string error.
"""
ans=gap.eval("Combinations(%s,%s)"%(mset,ZZ(k))).replace("\n","")
return eval(ans)
def combinations_iterator(mset,n=None):
"""
Posted by Raymond Hettinger, 2006/03/23, to the Python Cookbook:
http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/474124
Much faster than combinations.
EXAMPLES::
sage: X = combinations_iterator([1,2,3,4,5],3)
sage: [x for x in X]
[[1, 2, 3],
[1, 2, 4],
[1, 2, 5],
[1, 3, 4],
[1, 3, 5],
[1, 4, 5],
[2, 3, 4],
[2, 3, 5],
[2, 4, 5],
[3, 4, 5]]
"""
items = mset
if n is None:
n = len(items)
for i in range(len(items)):
v = items[i:i+1]
if n == 1:
yield v
else:
rest = items[i+1:]
for c in combinations_iterator(rest, n-1):
yield v + c
def number_of_combinations(mset,k):
"""
Returns the size of combinations(mset,k). IMPLEMENTATION: Wraps
GAP's NrCombinations.
.. note::
``mset`` must be a list of integers or strings (i.e., this is
very restricted).
EXAMPLES::
sage: mset = [1,1,2,3,4,4,5]
sage: number_of_combinations(mset,2)
12
"""
return ZZ(gap.eval("NrCombinations(%s,%s)"%(mset,ZZ(k))))
def arrangements(mset,k):
r"""
An arrangement of mset is an ordered selection without repetitions
and is represented by a list that contains only elements from mset,
but maybe in a different order.
``arrangements`` returns the set of arrangements of the
multiset mset that contain k elements.
IMPLEMENTATION: Wraps GAP's Arrangements.
.. warning::
Wraps GAP - hence mset must be a list of objects that have
string representations that can be interpreted by the GAP
interpreter. If mset consists of at all complicated Sage
objects, this function does *not* do what you expect. A proper
function should be written! (TODO!)
EXAMPLES::
sage: mset = [1,1,2,3,4,4,5]
sage: arrangements(mset,2)
[[1, 1],
[1, 2],
[1, 3],
[1, 4],
[1, 5],
[2, 1],
[2, 3],
[2, 4],
[2, 5],
[3, 1],
[3, 2],
[3, 4],
[3, 5],
[4, 1],
[4, 2],
[4, 3],
[4, 4],
[4, 5],
[5, 1],
[5, 2],
[5, 3],
[5, 4]]
sage: arrangements( ["c","a","t"], 2 )
['ac', 'at', 'ca', 'ct', 'ta', 'tc']
sage: arrangements( ["c","a","t"], 3 )
['act', 'atc', 'cat', 'cta', 'tac', 'tca']
"""
ans=gap.eval("Arrangements(%s,%s)"%(mset,k))
return eval(ans)
def number_of_arrangements(mset,k):
"""
Returns the size of arrangements(mset,k). Wraps GAP's
NrArrangements.
EXAMPLES::
sage: mset = [1,1,2,3,4,4,5]
sage: number_of_arrangements(mset,2)
22
"""
return ZZ(gap.eval("NrArrangements(%s,%s)"%(mset,ZZ(k))))
def derangements(mset):
"""
A derangement is a fixed point free permutation of list and is
represented by a list that contains exactly the same elements as
mset, but possibly in different order. Derangements returns the set
of all derangements of a multiset.
Wraps GAP's Derangements.
.. warning::
Wraps GAP - hence mset must be a list of objects that have
string representations that can be interpreted by the GAP
interpreter. If mset consists of at all complicated Sage
objects, this function does *not* do what you expect. A proper
function should be written! (TODO!)
EXAMPLES::
sage: mset = [1,2,3,4]
sage: derangements(mset)
[[2, 1, 4, 3],
[2, 3, 4, 1],
[2, 4, 1, 3],
[3, 1, 4, 2],
[3, 4, 1, 2],
[3, 4, 2, 1],
[4, 1, 2, 3],
[4, 3, 1, 2],
[4, 3, 2, 1]]
sage: derangements(["c","a","t"])
['atc', 'tca']
"""
ans=gap.eval("Derangements(%s)"%mset)
return eval(ans)
def number_of_derangements(mset):
"""
Returns the size of derangements(mset). Wraps GAP's
NrDerangements.
EXAMPLES::
sage: mset = [1,2,3,4]
sage: number_of_derangements(mset)
9
"""
ans=gap.eval("NrDerangements(%s)"%mset)
return ZZ(ans)
def tuples(S,k):
"""
An ordered tuple of length k of set is an ordered selection with
repetition and is represented by a list of length k containing
elements of set. tuples returns the set of all ordered tuples of
length k of the set.
EXAMPLES::
sage: S = [1,2]
sage: tuples(S,3)
[[1, 1, 1], [2, 1, 1], [1, 2, 1], [2, 2, 1], [1, 1, 2], [2, 1, 2], [1, 2, 2], [2, 2, 2]]
sage: mset = ["s","t","e","i","n"]
sage: tuples(mset,2)
[['s', 's'], ['t', 's'], ['e', 's'], ['i', 's'], ['n', 's'], ['s', 't'], ['t', 't'],
['e', 't'], ['i', 't'], ['n', 't'], ['s', 'e'], ['t', 'e'], ['e', 'e'], ['i', 'e'],
['n', 'e'], ['s', 'i'], ['t', 'i'], ['e', 'i'], ['i', 'i'], ['n', 'i'], ['s', 'n'],
['t', 'n'], ['e', 'n'], ['i', 'n'], ['n', 'n']]
The Set(...) comparisons are necessary because finite fields are
not enumerated in a standard order.
::
sage: K.<a> = GF(4, 'a')
sage: mset = [x for x in K if x!=0]
sage: tuples(mset,2)
[[a, a], [a + 1, a], [1, a], [a, a + 1], [a + 1, a + 1], [1, a + 1], [a, 1], [a + 1, 1], [1, 1]]
AUTHORS:
- Jon Hanke (2006-08)
"""
import copy
if k<=0:
return [[]]
if k==1:
return [[x] for x in S]
ans = []
for s in S:
for x in tuples(S,k-1):
y = copy.copy(x)
y.append(s)
ans.append(y)
return ans
def number_of_tuples(S,k):
"""
Returns the size of tuples(S,k). Wraps GAP's NrTuples.
EXAMPLES::
sage: S = [1,2,3,4,5]
sage: number_of_tuples(S,2)
25
sage: S = [1,1,2,3,4,5]
sage: number_of_tuples(S,2)
25
"""
ans=gap.eval("NrTuples(%s,%s)"%(S,ZZ(k)))
return ZZ(ans)
def unordered_tuples(S,k):
"""
An unordered tuple of length k of set is a unordered selection with
repetitions of set and is represented by a sorted list of length k
containing elements from set.
unordered_tuples returns the set of all unordered tuples of length
k of the set. Wraps GAP's UnorderedTuples.
.. warning::
Wraps GAP - hence mset must be a list of objects that have
string representations that can be interpreted by the GAP
interpreter. If mset consists of at all complicated Sage
objects, this function does *not* do what you expect. A proper
function should be written! (TODO!)
EXAMPLES::
sage: S = [1,2]
sage: unordered_tuples(S,3)
[[1, 1, 1], [1, 1, 2], [1, 2, 2], [2, 2, 2]]
sage: unordered_tuples(["a","b","c"],2)
['aa', 'ab', 'ac', 'bb', 'bc', 'cc']
"""
ans=gap.eval("UnorderedTuples(%s,%s)"%(S,ZZ(k)))
return eval(ans)
def number_of_unordered_tuples(S,k):
"""
Returns the size of unordered_tuples(S,k). Wraps GAP's
NrUnorderedTuples.
EXAMPLES::
sage: S = [1,2,3,4,5]
sage: number_of_unordered_tuples(S,2)
15
"""
ans=gap.eval("NrUnorderedTuples(%s,%s)"%(S,ZZ(k)))
return ZZ(ans)
def permutations(mset):
"""
A permutation is represented by a list that contains exactly the
same elements as mset, but possibly in different order. If mset is
a proper set there are `|mset| !` such permutations.
Otherwise if the first elements appears `k_1` times, the
second element appears `k_2` times and so on, the number
of permutations is `|mset|! / (k_1! k_2! \ldots)`, which
is sometimes called a multinomial coefficient.
permutations returns the set of all permutations of a multiset.
Calls a function written by Mike Hansen, not GAP.
EXAMPLES::
sage: mset = [1,1,2,2,2]
sage: permutations(mset)
[[1, 1, 2, 2, 2],
[1, 2, 1, 2, 2],
[1, 2, 2, 1, 2],
[1, 2, 2, 2, 1],
[2, 1, 1, 2, 2],
[2, 1, 2, 1, 2],
[2, 1, 2, 2, 1],
[2, 2, 1, 1, 2],
[2, 2, 1, 2, 1],
[2, 2, 2, 1, 1]]
sage: MS = MatrixSpace(GF(2),2,2)
sage: A = MS([1,0,1,1])
sage: permutations(A.rows())
[[(1, 0), (1, 1)], [(1, 1), (1, 0)]]
"""
from sage.combinat.permutation import Permutations
ans = Permutations(mset)
return ans.list()
def permutations_iterator(mset,n=None):
"""
Do not use this function. It will be deprecated in future version
of Sage and eventually removed. Use Permutations instead; instead
of
for p in permutations_iterator(range(1, m+1), n)
use
for p in Permutations(m, n).
Note that Permutations, unlike this function, treats repeated
elements as identical.
If you insist on using this now:
Returns an iterator (http://docs.python.org/lib/typeiter.html)
which can be used in place of permutations(mset) if all you need it
for is a 'for' loop.
Posted by Raymond Hettinger, 2006/03/23, to the Python Cookbook:
http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/474124
Note- This function considers repeated elements as different
entries, so for example::
sage: from sage.combinat.combinat import permutations, permutations_iterator
sage: mset = [1,2,2]
sage: permutations(mset)
[[1, 2, 2], [2, 1, 2], [2, 2, 1]]
sage: for p in permutations_iterator(mset): print p
[1, 2, 2]
[1, 2, 2]
[2, 1, 2]
[2, 2, 1]
[2, 1, 2]
[2, 2, 1]
EXAMPLES::
sage: X = permutations_iterator(range(3),2)
sage: [x for x in X]
[[0, 1], [0, 2], [1, 0], [1, 2], [2, 0], [2, 1]]
"""
items = mset
if n is None:
n = len(items)
for i in range(len(items)):
v = items[i:i+1]
if n == 1:
yield v
else:
rest = items[:i] + items[i+1:]
for p in permutations_iterator(rest, n-1):
yield v + p
def number_of_permutations(mset):
"""
Do not use this function. It will be deprecated in future version
of Sage and eventually removed. Use Permutations instead; instead
of
number_of_permutations(mset)
use
Permutations(mset).cardinality().
If you insist on using this now:
Returns the size of permutations(mset).
AUTHORS:
- Robert L. Miller
EXAMPLES::
sage: mset = [1,1,2,2,2]
sage: number_of_permutations(mset)
10
"""
from sage.rings.arith import factorial
m = len(mset)
n = []
seen = []
for element in mset:
try:
n[seen.index(element)] += 1
except (IndexError, ValueError):
n.append(1)
seen.append(element)
return factorial(m)/prod([factorial(k) for k in n])
def cyclic_permutations(mset):
"""
Returns a list of all cyclic permutations of mset. Treats mset as a
list, not a set, i.e. entries with the same value are distinct.
AUTHORS:
- Emily Kirkman
EXAMPLES::
sage: from sage.combinat.combinat import cyclic_permutations, cyclic_permutations_iterator
sage: cyclic_permutations(range(4))
[[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 1, 3], [0, 2, 3, 1], [0, 3, 1, 2], [0, 3, 2, 1]]
sage: for cycle in cyclic_permutations(['a', 'b', 'c']):
... print cycle
['a', 'b', 'c']
['a', 'c', 'b']
Note that lists with repeats are not handled intuitively::
sage: cyclic_permutations([1,1,1])
[[1, 1, 1], [1, 1, 1]]
"""
return list(cyclic_permutations_iterator(mset))
def cyclic_permutations_iterator(mset):
"""
Iterates over all cyclic permutations of mset in cycle notation.
Treats mset as a list, not a set, i.e. entries with the same value
are distinct.
AUTHORS:
- Emily Kirkman
EXAMPLES::
sage: from sage.combinat.combinat import cyclic_permutations, cyclic_permutations_iterator
sage: cyclic_permutations(range(4))
[[0, 1, 2, 3], [0, 1, 3, 2], [0, 2, 1, 3], [0, 2, 3, 1], [0, 3, 1, 2], [0, 3, 2, 1]]
sage: for cycle in cyclic_permutations(['a', 'b', 'c']):
... print cycle
['a', 'b', 'c']
['a', 'c', 'b']
Note that lists with repeats are not handled intuitively::
sage: cyclic_permutations([1,1,1])
[[1, 1, 1], [1, 1, 1]]
"""
if len(mset) > 2:
for perm in permutations_iterator(mset[1:]):
yield [mset[0]] + perm
else:
yield mset
def bell_polynomial(n, k):
r"""
This function returns the Bell Polynomial
.. math::
B_{n,k}(x_1, x_2, \ldots, x_{n-k+1}) = \sum_{\sum{j_i}=k, \sum{i j_i}
=n} \frac{n!}{j_1!j_2!\ldots} \frac{x_1}{1!}^j_1 \frac{x_2}{2!}^j_2
\ldots
INPUT:
- ``n`` - integer
- ``k`` - integer
OUTPUT:
- polynomial expression (SymbolicArithmetic)
EXAMPLES::
sage: bell_polynomial(6,2)
10*x_3^2 + 15*x_2*x_4 + 6*x_1*x_5
sage: bell_polynomial(6,3)
15*x_2^3 + 60*x_1*x_2*x_3 + 15*x_1^2*x_4
REFERENCES:
- E.T. Bell, "Partition Polynomials"
AUTHORS:
- Blair Sutton (2009-01-26)
"""
from sage.combinat.partition import Partitions
from sage.rings.arith import factorial
vars = ZZ[tuple(['x_'+str(i) for i in range(1, n-k+2)])].gens()
result = 0
for p in Partitions(n, length=k):
factorial_product = 1
power_factorial_product = 1
for part, count in p.to_exp_dict().iteritems():
factorial_product *= factorial(count)
power_factorial_product *= factorial(part)**count
coefficient = factorial(n) / (factorial_product * power_factorial_product)
result += coefficient * prod([vars[i-1] for i in p])
return result
def fibonacci_sequence(start, stop=None, algorithm=None):
r"""
Returns an iterator over the Fibonacci sequence, for all fibonacci
numbers `f_n` from ``n = start`` up to (but
not including) ``n = stop``
INPUT:
- ``start`` - starting value
- ``stop`` - stopping value
- ``algorithm`` - default (None) - passed on to
fibonacci function (or not passed on if None, i.e., use the
default).
EXAMPLES::
sage: fibs = [i for i in fibonacci_sequence(10, 20)]
sage: fibs
[55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181]
::
sage: sum([i for i in fibonacci_sequence(100, 110)])
69919376923075308730013
.. seealso::
:func:`fibonacci_xrange`
AUTHORS:
- Bobby Moretti
"""
if stop is None:
stop = ZZ(start)
start = ZZ(0)
else:
start = ZZ(start)
stop = ZZ(stop)
if algorithm:
for n in xrange(start, stop):
yield fibonacci(n, algorithm=algorithm)
else:
for n in xrange(start, stop):
yield fibonacci(n)
def fibonacci_xrange(start, stop=None, algorithm='pari'):
r"""
Returns an iterator over all of the Fibonacci numbers in the given
range, including ``f_n = start`` up to, but not
including, ``f_n = stop``.
EXAMPLES::
sage: fibs_in_some_range = [i for i in fibonacci_xrange(10^7, 10^8)]
sage: len(fibs_in_some_range)
4
sage: fibs_in_some_range
[14930352, 24157817, 39088169, 63245986]
::
sage: fibs = [i for i in fibonacci_xrange(10, 100)]
sage: fibs
[13, 21, 34, 55, 89]
::
sage: list(fibonacci_xrange(13, 34))
[13, 21]
A solution to the second Project Euler problem::
sage: sum([i for i in fibonacci_xrange(10^6) if is_even(i)])
1089154
.. seealso::
:func:`fibonacci_sequence`
AUTHORS:
- Bobby Moretti
"""
if stop is None:
stop = ZZ(start)
start = ZZ(0)
else:
start = ZZ(start)
stop = ZZ(stop)
fn = 0
n = 0
while fn < start:
n += 1
fn = fibonacci(n)
while True:
fn = fibonacci(n)
n += 1
if fn < stop:
yield fn
else:
return
def bernoulli_polynomial(x, n):
r"""
Return the nth Bernoulli polynomial evaluated at x.
The generating function for the Bernoulli polynomials is
.. math::
\frac{t e^{xt}}{e^t-1}= \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!},
and they are given directly by
.. math::
B_n(x) = \sum_{i=0}^n \binom{n}{i}B_{n-i}x^i.
One has `B_n(x) = - n\zeta(1 - n,x)`, where
`\zeta(s,x)` is the Hurwitz zeta function. Thus, in a
certain sense, the Hurwitz zeta function generalizes the
Bernoulli polynomials to non-integer values of n.
EXAMPLES::
sage: y = QQ['y'].0
sage: bernoulli_polynomial(y, 5)
y^5 - 5/2*y^4 + 5/3*y^3 - 1/6*y
sage: bernoulli_polynomial(y, 5)(12)
199870
sage: bernoulli_polynomial(12, 5)
199870
sage: bernoulli_polynomial(y^2 + 1, 5)
y^10 + 5/2*y^8 + 5/3*y^6 - 1/6*y^2
sage: P.<t> = ZZ[]
sage: p = bernoulli_polynomial(t, 6)
sage: p.parent()
Univariate Polynomial Ring in t over Rational Field
We verify an instance of the formula which is the origin of
the Bernoulli polynomials (and numbers)::
sage: power_sum = sum(k^4 for k in range(10))
sage: 5*power_sum == bernoulli_polynomial(10, 5) - bernoulli(5)
True
REFERENCES:
- http://en.wikipedia.org/wiki/Bernoulli_polynomials
"""
try:
n = ZZ(n)
if n < 0:
raise TypeError
except TypeError:
raise ValueError, "The second argument must be a non-negative integer"
if n == 0:
return ZZ(1)
if n == 1:
return x - ZZ(1)/2
k = n.mod(2)
coeffs = [0]*k + sum(([binomial(n, i)*bernoulli(n-i), 0]
for i in range(k, n+1, 2)), [])
coeffs[-3] = -n/2
if isinstance(x, Polynomial):
try:
return x.parent()(coeffs)(x)
except TypeError:
pass
x2 = x*x
xi = x**k
s = 0
for i in range(k, n-1, 2):
s += coeffs[i]*xi
t = xi
xi *= x2
s += xi - t*x*n/2
return s
