Python permutations.Permutation类代码示例

def test_unrank_lex():
    assert Permutation.unrank_lex(5, 10).rank == 10
    assert Permutation.unrank_lex(15, 225).rank == 225
    assert Permutation.unrank_lex(10, 0).is_Identity
    p = Permutation.unrank_lex(4, 23)
    assert p.rank == 23
    assert p.array_form == [3, 2, 1, 0]

def test_coset_factor():
    a = Permutation([0, 2, 1])
    G = PermutationGroup([a])
    c = Permutation([2, 1, 0])
    assert not G.coset_factor(c)
    assert G.coset_rank(c) is None

    a = Permutation([2, 0, 1, 3, 4, 5])
    b = Permutation([2, 1, 3, 4, 5, 0])
    g = PermutationGroup([a, b])
    assert g.order() == 360
    d = Permutation([1, 0, 2, 3, 4, 5])
    assert not g.coset_factor(d.array_form)
    assert not g.contains(d)
    assert Permutation(2) in G
    c = Permutation([1, 0, 2, 3, 5, 4])
    v = g.coset_factor(c, True)
    tr = g.basic_transversals
    p = Permutation.rmul(*[tr[i][v[i]] for i in range(len(g.base))])
    assert p == c
    v = g.coset_factor(c)
    p = Permutation.rmul(*v)
    assert p == c
    assert g.contains(c)
    G = PermutationGroup([Permutation([2, 1, 0])])
    p = Permutation([1, 0, 2])
    assert G.coset_factor(p) == []

def test_muln():
    n = 6
    m = 8
    a = [Permutation.unrank_nonlex(n, i).array_form for i in range(m)]
    h = range(n)
    for i in range(m):
        h = perm_af_mul(h, a[i])
        h2 = perm_af_muln(*a[:i+1])
        assert h == h2

def test_mul():
    a, b = [0, 2, 1, 3], [0, 1, 3, 2]
    assert _af_rmul(a, b) == [0, 2, 3, 1]
    assert _af_rmuln(a, b, range(4)) == [0, 2, 3, 1]
    assert rmul(Permutation(a), Permutation(b)).array_form == [0, 2, 3, 1]

    a = Permutation([0, 2, 1, 3])
    b = (0, 1, 3, 2)
    c = (3, 1, 2, 0)
    assert Permutation.rmul(a, b, c) == Permutation([1, 2, 3, 0])
    assert Permutation.rmul(a, c) == Permutation([3, 2, 1, 0])
    raises(TypeError, lambda: Permutation.rmul(b, c))

    n = 6
    m = 8
    a = [Permutation.unrank_nonlex(n, i).array_form for i in range(m)]
    h = range(n)
    for i in range(m):
        h = _af_rmul(h, a[i])
        h2 = _af_rmuln(*a[:i + 1])
        assert h == h2

def test_Permutation():
    # don't auto fill 0
    raises(ValueError, lambda: Permutation([1]))
    p = Permutation([0, 1, 2, 3])
    # call as bijective
    assert [p(i) for i in range(p.size)] == list(p)
    # call as operator
    assert p(range(p.size)) == list(p)
    # call as function
    assert list(p(1, 2)) == [0, 2, 1, 3]
    # conversion to list
    assert list(p) == range(4)
    # cycle form with size
    assert Permutation([[1, 2]], size=4) == Permutation([[1, 2], [0], [3]])
    # random generation
    assert Permutation.random(2) in (Permutation([1, 0]), Permutation([0, 1]))

    p = Permutation([2, 5, 1, 6, 3, 0, 4])
    q = Permutation([[1], [0, 3, 5, 6, 2, 4]])
    assert len(set([p, p])) == 1
    r = Permutation([1, 3, 2, 0, 4, 6, 5])
    ans = Permutation(_af_rmuln(*[w.array_form for w in (p, q, r)])).array_form
    assert rmul(p, q, r).array_form == ans
    # make sure no other permutation of p, q, r could have given
    # that answer
    for a, b, c in permutations((p, q, r)):
        if (a, b, c) == (p, q, r):
            continue
        assert rmul(a, b, c).array_form != ans

    assert p.support() == range(7)
    assert q.support() == [0, 2, 3, 4, 5, 6]
    assert Permutation(p.cyclic_form).array_form == p.array_form
    assert p.cardinality == 5040
    assert q.cardinality == 5040
    assert q.cycles == 2
    assert rmul(q, p) == Permutation([4, 6, 1, 2, 5, 3, 0])
    assert rmul(p, q) == Permutation([6, 5, 3, 0, 2, 4, 1])
    assert _af_rmul(p.array_form, q.array_form) == \
        [6, 5, 3, 0, 2, 4, 1]

    assert rmul(Permutation([[1, 2, 3], [0, 4]]),
                Permutation([[1, 2, 4], [0], [3]])).cyclic_form == \
        [[0, 4, 2], [1, 3]]
    assert q.array_form == [3, 1, 4, 5, 0, 6, 2]
    assert q.cyclic_form == [[0, 3, 5, 6, 2, 4]]
    assert q.full_cyclic_form == [[0, 3, 5, 6, 2, 4], [1]]
    assert p.cyclic_form == [[0, 2, 1, 5], [3, 6, 4]]
    t = p.transpositions()
    assert t == [(0, 5), (0, 1), (0, 2), (3, 4), (3, 6)]
    assert Permutation.rmul(*[Permutation(Cycle(*ti)) for ti in (t)])
    assert Permutation([1, 0]).transpositions() == [(0, 1)]

    assert p**13 == p
    assert q**0 == Permutation(range(q.size))
    assert q**-2 == ~q**2
    assert q**2 == Permutation([5, 1, 0, 6, 3, 2, 4])
    assert q**3 == q**2*q
    assert q**4 == q**2*q**2

    a = Permutation(1, 3)
    b = Permutation(2, 0, 3)
    I = Permutation(3)
    assert ~a == a**-1
    assert a*~a == I
    assert a*b**-1 == a*~b

    ans = Permutation(0, 5, 3, 1, 6)(2, 4)
    assert (p + q.rank()).rank() == ans.rank()
    assert (p + q.rank())._rank == ans.rank()
    assert (q + p.rank()).rank() == ans.rank()
    raises(TypeError, lambda: p + Permutation(range(10)))

    assert (p - q.rank()).rank() == Permutation(0, 6, 3, 1, 2, 5, 4).rank()
    assert p.rank() - q.rank() < 0  # for coverage: make sure mod is used
    assert (q - p.rank()).rank() == Permutation(1, 4, 6, 2)(3, 5).rank()

    assert p*q == Permutation(_af_rmuln(*[list(w) for w in (q, p)]))
    assert p*Permutation([]) == p
    assert Permutation([])*p == p
    assert p*Permutation([[0, 1]]) == Permutation([2, 5, 0, 6, 3, 1, 4])
    assert Permutation([[0, 1]])*p == Permutation([5, 2, 1, 6, 3, 0, 4])

    pq = p^q
    assert pq == Permutation([5, 6, 0, 4, 1, 2, 3])
    assert pq == rmul(q, p, ~q)
    qp = q^p
    assert qp == Permutation([4, 3, 6, 2, 1, 5, 0])
    assert qp == rmul(p, q, ~p)
    raises(ValueError, lambda: p^Permutation([]))

    assert p.commutator(q) == Permutation(0, 1, 3, 4, 6, 5, 2)
    assert q.commutator(p) == Permutation(0, 2, 5, 6, 4, 3, 1)
    assert p.commutator(q) == ~q.commutator(p)
    raises(ValueError, lambda: p.commutator(Permutation([])))

    assert len(p.atoms()) == 7
    assert q.atoms() == set([0, 1, 2, 3, 4, 5, 6])

    assert p.inversion_vector() == [2, 4, 1, 3, 1, 0]
#.........这里部分代码省略.........

def test_from_sequence():
    assert Permutation.from_sequence('SymPy') == Permutation(4)(0, 1, 3)
    assert Permutation.from_sequence('SymPy', key=lambda x: x.lower()) == \
        Permutation(4)(0, 2)(1, 3)

def test_ranking():
    assert Permutation.unrank_lex(5, 10).rank() == 10
    p = Permutation.unrank_lex(15, 225)
    assert p.rank() == 225
    p1 = p.next_lex()
    assert p1.rank() == 226
    assert Permutation.unrank_lex(15, 225).rank() == 225
    assert Permutation.unrank_lex(10, 0).is_Identity
    p = Permutation.unrank_lex(4, 23)
    assert p.rank() == 23
    assert p.array_form == [3, 2, 1, 0]
    assert p.next_lex() is None

    p = Permutation([1, 5, 2, 0, 3, 6, 4])
    q = Permutation([[1, 2, 3, 5, 6], [0, 4]])
    a = [Permutation.unrank_trotterjohnson(4, i).array_form for i in range(5)]
    assert a == [[0, 1, 2, 3], [0, 1, 3, 2], [0, 3, 1, 2], [3, 0, 1,
        2], [3, 0, 2, 1] ]
    assert [Permutation(pa).rank_trotterjohnson() for pa in a] == range(5)
    assert Permutation([0, 1, 2, 3]).next_trotterjohnson() == \
        Permutation([0, 1, 3, 2])

    assert q.rank_trotterjohnson() == 2283
    assert p.rank_trotterjohnson() == 3389
    assert Permutation([1, 0]).rank_trotterjohnson() == 1
    a = Permutation(range(3))
    b = a
    l = []
    tj = []
    for i in range(6):
        l.append(a)
        tj.append(b)
        a = a.next_lex()
        b = b.next_trotterjohnson()
    assert a == b is None
    assert set([tuple(a) for a in l]) == set([tuple(a) for a in tj])

    p = Permutation([2, 5, 1, 6, 3, 0, 4])
    q = Permutation([[6], [5], [0, 1, 2, 3, 4]])
    assert p.rank() == 1964
    assert q.rank() == 870
    assert Permutation([]).rank_nonlex() == 0
    prank = p.rank_nonlex()
    assert prank == 1600
    assert Permutation.unrank_nonlex(7, 1600) == p
    qrank = q.rank_nonlex()
    assert qrank == 41
    assert Permutation.unrank_nonlex(7, 41) == Permutation(q.array_form)

    a = [Permutation.unrank_nonlex(4, i).array_form for i in range(24)]
    assert a == [
        [1, 2, 3, 0], [3, 2, 0, 1], [1, 3, 0, 2], [1, 2, 0, 3], [2, 3, 1, 0],
        [2, 0, 3, 1], [3, 0, 1, 2], [2, 0, 1, 3], [1, 3, 2, 0], [3, 0, 2, 1],
        [1, 0, 3, 2], [1, 0, 2, 3], [2, 1, 3, 0], [2, 3, 0, 1], [3, 1, 0, 2],
        [2, 1, 0, 3], [3, 2, 1, 0], [0, 2, 3, 1], [0, 3, 1, 2], [0, 2, 1, 3],
        [3, 1, 2, 0], [0, 3, 2, 1], [0, 1, 3, 2], [0, 1, 2, 3]]

    ok = []
    p = Permutation([1, 0])
    for i in range(3):
        ok.append(p.array_form)
        p = p.next_nonlex()
        if p is None:
            ok.append(None)
            break
    assert ok == [[1, 0], [0, 1], None]
    assert Permutation([3, 2, 0, 1]).next_nonlex() == Permutation([1, 3, 0, 2])
    assert [Permutation(pa).rank_nonlex() for pa in a] == range(24)

def test_josephus():
    assert Permutation.josephus(4, 6, 1) == Permutation([3, 1, 0, 2, 5, 4])
    assert Permutation.josephus(1, 5, 1).is_Identity

def test_Permutation():
    p = Permutation([2, 5, 1, 6, 3, 0, 4])
    q = Permutation([[1], [0, 3, 5, 6, 2, 4]])

    assert Permutation(p.cyclic_form).array_form == p.array_form
    assert p.cardinality == 5040
    assert q.cardinality == 5040
    assert q.cycles == 2
    assert q*p == Permutation([4, 6, 1, 2, 5, 3, 0])
    assert p*q == Permutation([6, 5, 3, 0, 2, 4, 1])
    assert perm_af_mul([2, 5, 1, 6, 3, 0, 4], [3, 1, 4, 5, 0, 6, 2]) == \
        [6, 5, 3, 0, 2, 4, 1]

    assert cyclic([(2,3,5)], 5) == [[1, 2, 4], [0], [3]]
    assert (Permutation([[1,2,3],[0,4]])*Permutation([[1,2,4],[0],[3]])).cyclic_form == \
        [[1, 3], [0, 4, 2]]
    assert q.array_form == [3, 1, 4, 5, 0, 6, 2]
    assert p.cyclic_form == [[3, 6, 4], [0, 2, 1, 5]]
    assert p.transpositions() == [(3, 4), (3, 6), (0, 5), (0, 1), (0, 2)]

    assert p**13 == p
    assert q**2 == Permutation([5, 1, 0, 6, 3, 2, 4])

    assert p+q == Permutation([5, 6, 3, 1, 2, 4, 0])
    assert q+p == p+q

    assert p-q == Permutation([6, 3, 5, 1, 2, 4, 0])
    assert q-p == Permutation([1, 4, 2, 6, 5, 3, 0])

    a = p-q
    b = q-p
    assert (a+b).is_Identity

    assert p.conjugate(q) == Permutation([5, 3, 0, 4, 6, 2, 1])
    assert p.conjugate(q) == ~q*p*q == p**q
    assert q.conjugate(p) == Permutation([6, 3, 2, 0, 1, 4, 5])
    assert q.conjugate(p) == ~p*q*p == q**p

    assert p.commutator(q) == Permutation([1, 4, 5, 6, 3, 0, 2])
    assert q.commutator(p) == Permutation([5, 0, 6, 4, 1, 2, 3])
    assert p.commutator(q) == ~ q.commutator(p)

    assert len(p.atoms()) == 7
    assert q.atoms() == set([0, 1, 2, 3, 4, 5, 6])

    assert p.inversion_vector() == [2, 4, 1, 3, 1, 0]
    assert q.inversion_vector() == [3, 1, 2, 2, 0, 1]

    assert Permutation.from_inversion_vector(p.inversion_vector()) == p
    assert Permutation.from_inversion_vector(q.inversion_vector()).array_form\
           == q.array_form
    assert Permutation([i for i in range(500,-1,-1)]).inversions() == 125250

    assert Permutation([0, 4, 1, 3, 2]).parity() == 0
    assert Permutation([0, 1, 4, 3, 2]).parity() == 1
    assert perm_af_parity([0, 4, 1, 3, 2]) == 0
    assert perm_af_parity([0, 1, 4, 3, 2]) == 1

    s = Permutation([0])

    assert s.is_Singleton

    r = Permutation([3, 2, 1, 0])
    assert (r**2).is_Identity

    assert (p*(~p)).is_Identity
    assert (~p)**13 == Permutation([5, 2, 0, 4, 6, 1, 3])
    assert ~(r**2).is_Identity
    assert p.max() == 6
    assert p.min() == 0

    q = Permutation([[6], [5], [0, 1, 2, 3, 4]])

    assert q.max() == 4
    assert q.min() == 0

    p = Permutation([1, 5, 2, 0, 3, 6, 4])
    q = Permutation([[1, 2, 3, 5, 6], [0, 4]])

    assert p.ascents() == [0, 3, 4]
    assert q.ascents() == [1, 2, 4]
    assert r.ascents() == []

    assert p.descents() == [1, 2, 5]
    assert q.descents() == [0, 3, 5]
    assert Permutation(r.descents()).is_Identity

    assert p.inversions() == 7
    assert p.signature() == -1
    assert q.inversions() == 11
    assert q.signature() == -1
    assert (p*(~p)).inversions() == 0
    assert (p*(~p)).signature() == 1

    assert p.order() == 6
    assert q.order() == 10
    assert (p**(p.order())).is_Identity

    assert p.length() == 6
    assert q.length() == 7
#.........这里部分代码省略.........

def test_ranking():
    assert Permutation.unrank_lex(5, 10).rank() == 10
    p = Permutation.unrank_lex(15, 225)
    assert p.rank() == 225
    p1 = p.next_lex()
    assert p1.rank() == 226
    assert Permutation.unrank_lex(15, 225).rank() == 225
    assert Permutation.unrank_lex(10, 0).is_Identity
    p = Permutation.unrank_lex(4, 23)
    assert p.rank() == 23
    assert p.array_form == [3, 2, 1, 0]
    assert p.next_lex() == None

    p = Permutation([1, 5, 2, 0, 3, 6, 4])
    q = Permutation([[1, 2, 3, 5, 6], [0, 4]])
    a = [Permutation.unrank_trotterjohnson(4, i).array_form for i in range(5)]
    assert a == [[0,1,2,3], [0,1,3,2], [0,3,1,2], [3,0,1,2], [3,0,2,1] ]
    assert [Permutation(pa).rank_trotterjohnson() for pa in a] == range(5)
    assert Permutation([0,1,2,3]).next_trotterjohnson() == \
        Permutation([0,1,3,2])

    assert q.rank_trotterjohnson() == 2283
    assert p.rank_trotterjohnson() == 3389

    p = Permutation([2, 5, 1, 6, 3, 0, 4])
    q = Permutation([[6], [5], [0, 1, 2, 3, 4]])
    assert p.rank() == 1964
    assert q.rank() == 870
    assert Permutation([]).rank_nonlex() == 0
    prank = p.rank_nonlex()
    assert prank == 1600
    assert Permutation.unrank_nonlex(7, 1600) == p
    qrank = q.rank_nonlex()
    assert qrank == 41
    assert Permutation.unrank_nonlex(7, 41) == Permutation(q.array_form)

    a = [Permutation.unrank_nonlex(4, i).array_form for i in range(24)]
    assert a == \
    [[1, 2, 3, 0], [3, 2, 0, 1], [1, 3, 0, 2], [1, 2, 0, 3], [2, 3, 1, 0], \
     [2, 0, 3, 1], [3, 0, 1, 2], [2, 0, 1, 3], [1, 3, 2, 0], [3, 0, 2, 1], \
     [1, 0, 3, 2], [1, 0, 2, 3], [2, 1, 3, 0], [2, 3, 0, 1], [3, 1, 0, 2], \
     [2, 1, 0, 3], [3, 2, 1, 0], [0, 2, 3, 1], [0, 3, 1, 2], [0, 2, 1, 3], \
     [3, 1, 2, 0], [0, 3, 2, 1], [0, 1, 3, 2], [0, 1, 2, 3]]

    assert Permutation([3, 2, 0, 1]).next_nonlex() == Permutation([1, 3, 0, 2])
    assert [Permutation(pa).rank_nonlex() for pa in a] == range(24)

def test_Permutation():
    p = Permutation([2,5,1,6,3,0,4])
    q = Permutation([[1,4,5],[2,0,6],[3]])

    assert q.cycles == 3
    assert p*q == Permutation([4, 6, 1, 2, 5, 3, 0])
    assert q*p == Permutation([6, 5, 3, 0, 2, 4, 1])

    assert q.array_form == [3, 1, 4, 5, 0, 6, 2]
    assert p.cyclic_form == [[3, 6, 4], [0, 2, 1, 5]]

    assert p**13 == p
    assert q**2 == Permutation([5, 1, 0, 6, 3, 2, 4])

    assert p+q == Permutation([5, 6, 3, 1, 2, 4, 0])
    assert q+p == p+q

    assert p-q == Permutation([6, 3, 5, 1, 2, 4, 0])
    assert q-p == Permutation([1, 4, 2, 6, 5, 3, 0])

    a = p-q
    b = q-p
    assert (a+b).is_Identity

    assert len(p.atoms()) == 7
    assert q.atoms() == set([0, 1, 2, 3, 4, 5, 6])

    assert p.inversion_vector == [2, 4, 1, 3, 1, 0]
    assert q.inversion_vector == [3, 1, 2, 2, 0, 1]

    assert Permutation.from_inversion_vector(p.inversion_vector) == p
    assert Permutation.from_inversion_vector(q.inversion_vector).array_form\
           == q.array_form

    s = Permutation([0])

    assert s.is_Singleton

    r = Permutation([3,2,1,0])
    assert (r**2).is_Identity

    assert (p*(~p)).is_Identity
    assert (~p)**13 == Permutation([5, 2, 0, 4, 6, 1, 3])
    assert ~(r**2).is_Identity
    assert p.max == 6
    assert p.min == 0

    q = Permutation([[4,1,2,3],[0,5,6]])

    assert q.max == 4
    assert q.min == 0

    assert p.rank_nonlex() == 14830
    assert q.rank_nonlex() == 8441
    assert Permutation.unrank_nonlex(7, 41) == Permutation([4, 2, 3, 5, 1, 0, 6])

    assert q.rank == 870
    assert p.rank == 1964

    p = Permutation([1,5,2,0,3,6,4])
    q = Permutation([[2,3,5],[1,0,6],[4]])

    assert p.ascents == [0, 3, 4]
    assert q.ascents == [1, 2, 4]
    assert r.ascents == []

    assert p.descents == [1, 2, 5]
    assert q.descents == [0, 3, 5]
    assert Permutation(r.descents).is_Identity

    assert p.inversions == 7
    assert p.signature == -1
    assert q.inversions == 11
    assert q.signature == -1
    assert (p*(~p)).inversions == 0
    assert (p*(~p)).signature == 1

    assert p.order == 6
    assert q.order == 3
    assert (p**(p.order)).is_Identity

    assert p.length == 6
    assert q.length == 7
    assert r.length == 4

    assert not p.is_Positive
    assert p.is_Negative
    assert not q.is_Positive
    assert q.is_Negative
    assert r.is_Positive
    assert not r.is_Negative

    assert p.runs() == [[1, 5], [2], [0, 3, 6], [4]]
    assert q.runs() == [[4], [2, 3, 5], [0, 6], [1]]
    assert r.runs() == [[3], [2], [1], [0]]

    assert p.index == 8
    assert q.index == 8
    assert r.index == 3

#.........这里部分代码省略.........

def test_Permutation():
    p = Permutation([2, 5, 1, 6, 3, 0, 4])
    q = Permutation([[1], [0, 3, 5, 6, 2, 4]])

    assert Permutation(p.cyclic_form).array_form == p.array_form
    assert p.cardinality == 5040
    assert q.cardinality == 5040
    assert q.cycles == 2
    assert q*p == Permutation([4, 6, 1, 2, 5, 3, 0])
    assert p*q == Permutation([6, 5, 3, 0, 2, 4, 1])

    assert (Permutation([[1,2,3],[0,4]])*Permutation([[1,2,4],[0],[3]])).cyclic_form == \
        [[1, 3], [0, 4, 2]]
    assert q.array_form == [3, 1, 4, 5, 0, 6, 2]
    assert p.cyclic_form == [[3, 6, 4], [0, 2, 1, 5]]

    assert p**13 == p
    assert q**2 == Permutation([5, 1, 0, 6, 3, 2, 4])

    assert p+q == Permutation([5, 6, 3, 1, 2, 4, 0])
    assert q+p == p+q

    assert p-q == Permutation([6, 3, 5, 1, 2, 4, 0])
    assert q-p == Permutation([1, 4, 2, 6, 5, 3, 0])

    a = p-q
    b = q-p
    assert (a+b).is_Identity

    assert len(p.atoms()) == 7
    assert q.atoms() == set([0, 1, 2, 3, 4, 5, 6])

    assert p.inversion_vector() == [2, 4, 1, 3, 1, 0]
    assert q.inversion_vector() == [3, 1, 2, 2, 0, 1]

    assert Permutation.from_inversion_vector(p.inversion_vector()) == p
    assert Permutation.from_inversion_vector(q.inversion_vector()).array_form\
           == q.array_form

    assert Permutation([0, 4, 1, 3, 2]).parity() == 0
    assert Permutation([0, 1, 4, 3, 2]).parity() == 1
    s = Permutation([0])

    assert s.is_Singleton

    r = Permutation([3, 2, 1, 0])
    assert (r**2).is_Identity

    assert (p*(~p)).is_Identity
    assert (~p)**13 == Permutation([5, 2, 0, 4, 6, 1, 3])
    assert ~(r**2).is_Identity
    assert p.max() == 6
    assert p.min() == 0

    q = Permutation([[6], [5], [0, 1, 2, 3, 4]])

    assert q.max() == 4
    assert q.min() == 0

    assert Permutation([]).rank_nonlex() == 0
    prank = p.rank_nonlex()
    assert prank == 1600
    assert Permutation.unrank_nonlex(7, 1600) == p
    qrank = q.rank_nonlex()
    assert qrank == 41
    assert Permutation.unrank_nonlex(7, 41) == Permutation(q.array_form)

    a = [Permutation.unrank_nonlex(4, i).array_form for i in range(24)]
    assert a == \
    [[1, 2, 3, 0], [3, 2, 0, 1], [1, 3, 0, 2], [1, 2, 0, 3], [2, 3, 1, 0], \
     [2, 0, 3, 1], [3, 0, 1, 2], [2, 0, 1, 3], [1, 3, 2, 0], [3, 0, 2, 1], \
     [1, 0, 3, 2], [1, 0, 2, 3], [2, 1, 3, 0], [2, 3, 0, 1], [3, 1, 0, 2], \
     [2, 1, 0, 3], [3, 2, 1, 0], [0, 2, 3, 1], [0, 3, 1, 2], [0, 2, 1, 3], \
     [3, 1, 2, 0], [0, 3, 2, 1], [0, 1, 3, 2], [0, 1, 2, 3]]

    assert [Permutation(pa).rank_nonlex() for pa in a] == range(24)

    assert q.rank() == 870
    assert p.rank() == 1964

    p = Permutation([1, 5, 2, 0, 3, 6, 4])
    q = Permutation([[1, 2, 3, 5, 6], [0, 4]])

    assert p.ascents() == [0, 3, 4]
    assert q.ascents() == [1, 2, 4]
    assert r.ascents() == []

    assert p.descents() == [1, 2, 5]
    assert q.descents() == [0, 3, 5]
    assert Permutation(r.descents()).is_Identity

    assert p.inversions() == 7
    assert p.signature() == -1
    assert q.inversions() == 11
    assert q.signature() == -1
    assert (p*(~p)).inversions() == 0
    assert (p*(~p)).signature() == 1

    assert p.order() == 6
    assert q.order() == 10
#.........这里部分代码省略.........