在Python中将分区设置为k组(包括NULL集)

Question

有一些像这样的帖子我有一个数字列表，如何生成所有唯一的k分区?

但是我想知道是否有一些新的有效的库来解决这个问题(itertools ?sagemath ?)

我有一个数字列表，如何生成所有唯一有序的k分区?例如，如果我有[1,2,3,4,5]和k=3

[[1,2],[3],[4,5]]就是这样一个分区但是[[4,5],[3],[1,2]]也是这样一个分区

我还想包括NULL集作为k子集中的一个可能集，例如

[[2,3],[],[1,4,5]]

顺序关系到

[[1,2],[3],[4,5]]

和[[4,5],[3],[1,2]]

但[[2,1],[3],[5 4]]被认为是一样的[[1,2]、[3]、[4,5]]如果你跟我来…

据我所知，OrderedSetPartitions(5,3)从Sagemath将不会提供我的问题的答案，因为它排除了NULL集

编辑:这里是一个(未优化的)尝试使用SAGEMATH

天真地解决这个问题。

def OrderedSetPartitions_0(A,k):
    cols={i for i in range(k)}
    # returns the list of k-OrderedSetPartitions of A, allowing for the empty set
    s=Subsets(cols).list()
    res=[]
    count=0
    P=[OrderedSetPartitions(A,i) for i in range(k+1)]
    for sub in s:
           print("sub=")
           print(sub)
           tmp=[ {} for i in range(k)]
           c=sub.cardinality()
           for part in P[c]:
               print("part=")
               print(part)
               for i in range(c):
                   tmp[sub[i]]=part[i]
               print("tmp=")
               print(tmp)
               res=res.append([tmp])
               # res = res.append(tmp) # tried this too
               print("res=")
               print(res)
               count=count+1
    return(res)
    # print(count)
A=range(3)
k=2
A
P=[OrderedSetPartitions(A,i) for i in range(k+1)]
# note that P[2].list is a list of list !
P[2].list()

[[{0, 1}, {2}],
 [{0, 2}, {1}],
 [{1, 2}, {0}],
 [{0}, {1, 2}],
 [{1}, {0, 2}],
 [{2}, {0, 1}]]

myset=OrderedSetPartitions_0(A,k)

我得到这个错误信息，我承认我根本没有得到它，因为它在编码时看起来很好，但不知何故res似乎是"None"而不是[]

sub=
{}
sub=
{0}
part=
[{0, 1, 2}]
tmp=
[{0, 1, 2}, {}]
res=
None
sub=
{1}
part=
[{0, 1, 2}]
tmp=
[{}, {0, 1, 2}]
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_21.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("#

-- coding: utf-8 --n" + support.preparse_worksheet_cell(base64.b64decode("bXlzZXQ9T3JkZXJlZFNldFBhcnRpdGlvbnNfMChBLGsp")，globals())+"n");execfile (os.path.abspath("代码 . py"))文件"，第1行，在

  File "/private/var/folders/gm/z065gk616xg6g0xgn4c7_bvc0000gn/T/tmpryfYOj/___code___.py", line 2, in <module>
    exec compile(u'myset=OrderedSetPartitions_0(A,k)
  File "", line 1, in <module>
  File "/private/var/folders/gm/z065gk616xg6g0xgn4c7_bvc0000gn/T/tmpSH_9LF/___code___.py", line 27, in OrderedSetPartitions_0
    res=res.append([tmp])
AttributeError: 'NoneType' object has no attribute 'append'

问题是关于将列表聚合到res.如果我在所有涉及res的行上加一个尖号，我可以正确地枚举输出

编辑:谢谢你的回答

实际上我把res=res.append(tmp)改成了res.append(tmp)当做print(tmp)

时，我得到了正确的枚举

[{0, 1, 2}, {}, {}] [{}, {0, 1, 2}, {}] [{}, {}, {0, 1, 2}] [{0, 1}, {2}, {}] [{0, 2}, {1}, {}] [{1, 2}, {0}, {}] [{0}, {1, 2}, {}] [{1}, {0, 2}, {}] [{2}, {0, 1}, {}] [{0, 1}, {}, {2}] [{0, 2}, {}, {1}] [{1, 2}, {}, {0}] [{0}, {}, {1, 2}] [{1}, {}, {0, 2}] [{2}, {}, {0, 1}] [{}, {0, 1}, {2}] [{}, {0, 2}, {1}] [{}, {1, 2}, {0}] [{}, {0}, {1, 2}] [{}, {1}, {0, 2}] [{}, {2}, {0, 1}] [{0}, {1}, {2}] [{0}, {2}, {1}] [{1}, {0}, {2}] [{2}, {0}, {1}] [{1}, {2}, {0}] [{2}, {1}, {0}]

但奇怪的是res是错误的，一定有一些副作用

[[{0, 1, 2}, {}, {}],
 [{}, {0, 1, 2}, {}],
 [{}, {}, {0, 1, 2}],
 [{2}, {0, 1}, {}],
 [{2}, {0, 1}, {}],
 [{2}, {0, 1}, {}],
 [{2}, {0, 1}, {}],
 [{2}, {0, 1}, {}],
 [{2}, {0, 1}, {}],
 [{2}, {}, {0, 1}],
 [{2}, {}, {0, 1}],
 [{2}, {}, {0, 1}],
 [{2}, {}, {0, 1}],
 [{2}, {}, {0, 1}],
 [{2}, {}, {0, 1}],
 [{}, {2}, {0, 1}],
 [{}, {2}, {0, 1}],
 [{}, {2}, {0, 1}],
 [{}, {2}, {0, 1}],
 [{}, {2}, {0, 1}],
 [{}, {2}, {0, 1}],
 [{2}, {1}, {0}],
 [{2}, {1}, {0}],
 [{2}, {1}, {0}],
 [{2}, {1}, {0}],
 [{2}, {1}, {0}],
 [{2}, {1}, {0}]]

前3行是正确的，然后它开始与我从print(tmp)得到的不同。这对我来说很奇怪，因为print(tmp)和res.append(tmp)之间没有指令!!!!!

Answer 1

下面是Sagemath中使用NumPy数组和itertools的解决方案。其思想与代码中的相同:创建OrderedSetPartitions并使用空集加强它们。为了在没有太多循环的情况下做到这一点，使用NumPy数组:关键部分是partitions[:, s] = P，其中2D数组partitions的某些列最初由空集填充，被来自OrderedSetPartitions的非空集取代。

import numpy as np
from itertools import combinations
A = Set([1, 2, 3, 4, 5])        # Sage set, not Python set
k = 3                           # number of elements in partition
all_partitions = np.array(OrderedSetPartitions(A, k).list())
for i in range(k-1, 0, -1):
    P = np.array(OrderedSetPartitions(A, i).list()) if i > 1 else [[A]]
    for s in combinations(range(k), i):
        partitions = np.empty((len(P), k), dtype=object)
        partitions[:, :] = [[Set()]]
        partitions[:, s] = P
        all_partitions = np.vstack((all_partitions, partitions))
print all_partitions

输出是一个double NumPy数组。如果需要Python列表，可以返回all_partitions.tolist()。

技术

Sage集合(用Set([1,2,3])创建)和Python集合(用set([1,2,3])或{1,2,3,4,5}创建)是不同类的对象。在Sagemath中，Sage集的输出看起来更好:它们显示为{1,2,3}，而Python集显示为set([1,2,3])。因此，在Sagemath中首选Sage集。同样，OrderedSetPartitions返回Sage集。

但是让NumPy与Sage集合一起玩需要更多的努力:特别是，我无法让np.full接受空的Sage集合Set()作为填充对象。这就是使用np.empty然后填充它的原因。

一个类似的问题导致i == 1被不同地对待:NumPy试图将[[Set([1,2,3,4,5])]]转换为一个三维数字数组，而不是一个包含一个Sage set对象的二维数组。