Inverse permutation giving negative results in Python

# The inverse ``~p`` of a SymPy permutation ``p`` gives the expected result when accessed like ``i^~p``, but ``~p(i) gives a negative result. This script converts these weird results into normal permutations, and shows the resulting pattern.
# https://en.wikiversity.org/wiki/Permutation_notation#SymPy

from itertools import permutations
from sympy.combinatorics import Permutation
from bidict import bidict
from math import factorial

for n in range(2, 6):
    print('\n============================== ', n, ' ==============================\n')
    fact_n = factorial(n)

    perms_iter = permutations([i for i in range(n)])
    perms = bidict()

    for i, perm in enumerate(perms_iter):
        perms[fact_n-1-i] = Permutation(perm[::-1])  # create reverse colexicographic order

    converted_list = []

    for index1 in range(fact_n):
        perm1 = perms[index1]
        for index2 in range(fact_n):
            perm2 = perms[index2]
            if ~perm2 == perm1:
                weird_perm = [~perm2(i) for i in range(n)]  # the result with negative numbers
                converted_perm = [n+weird_perm[i] for i in range(n)]
                converted_perm_object = Permutation(converted_perm)
                for index3 in range(fact_n):
                    perm3 = perms[index3]
                    if perm3 == converted_perm_object:
                        converted_list.append(index3)
                        print(index1, converted_perm, index3)

    print('\n', converted_list)


"""
==============================  2  ==============================

0 [1, 0] 1
1 [0, 1] 0

 [1, 0]

==============================  3  ==============================

0 [2, 1, 0] 5
1 [1, 2, 0] 4
2 [2, 0, 1] 3
3 [1, 0, 2] 1
4 [0, 2, 1] 2
5 [0, 1, 2] 0

 [5, 4, 3, 1, 2, 0]

==============================  4  ==============================

0 [3, 2, 1, 0] 23
1 [2, 3, 1, 0] 22
2 [3, 1, 2, 0] 21
3 [2, 1, 3, 0] 19
4 [1, 3, 2, 0] 20
5 [1, 2, 3, 0] 18
6 [3, 2, 0, 1] 17
7 [2, 3, 0, 1] 16
8 [3, 1, 0, 2] 11
9 [2, 1, 0, 3] 5
10 [1, 3, 0, 2] 10
11 [1, 2, 0, 3] 4
12 [3, 0, 2, 1] 15
13 [2, 0, 3, 1] 13
14 [3, 0, 1, 2] 9
15 [2, 0, 1, 3] 3
16 [1, 0, 3, 2] 7
17 [1, 0, 2, 3] 1
18 [0, 3, 2, 1] 14
19 [0, 2, 3, 1] 12
20 [0, 3, 1, 2] 8
21 [0, 2, 1, 3] 2
22 [0, 1, 3, 2] 6
23 [0, 1, 2, 3] 0

 [23, 22, 21, 19, 20, 18, 17, 16, 11, 5, 10, 4, 15, 13, 9, 3, 7, 1, 14, 12, 8, 2, 6, 0]

==============================  5  ==============================

0 [4, 3, 2, 1, 0] 119
1 [3, 4, 2, 1, 0] 118
2 [4, 2, 3, 1, 0] 117
3 [3, 2, 4, 1, 0] 115
4 [2, 4, 3, 1, 0] 116
5 [2, 3, 4, 1, 0] 114
6 [4, 3, 1, 2, 0] 113
7 [3, 4, 1, 2, 0] 112
8 [4, 2, 1, 3, 0] 107
9 [3, 2, 1, 4, 0] 101
10 [2, 4, 1, 3, 0] 106
11 [2, 3, 1, 4, 0] 100
12 [4, 1, 3, 2, 0] 111
13 [3, 1, 4, 2, 0] 109
14 [4, 1, 2, 3, 0] 105
15 [3, 1, 2, 4, 0] 99
16 [2, 1, 4, 3, 0] 103
17 [2, 1, 3, 4, 0] 97
18 [1, 4, 3, 2, 0] 110
19 [1, 3, 4, 2, 0] 108
20 [1, 4, 2, 3, 0] 104
21 [1, 3, 2, 4, 0] 98
22 [1, 2, 4, 3, 0] 102
23 [1, 2, 3, 4, 0] 96
24 [4, 3, 2, 0, 1] 95
25 [3, 4, 2, 0, 1] 94
26 [4, 2, 3, 0, 1] 93
27 [3, 2, 4, 0, 1] 91
28 [2, 4, 3, 0, 1] 92
29 [2, 3, 4, 0, 1] 90
30 [4, 3, 1, 0, 2] 71
31 [3, 4, 1, 0, 2] 70
32 [4, 2, 1, 0, 3] 47
33 [3, 2, 1, 0, 4] 23
34 [2, 4, 1, 0, 3] 46
35 [2, 3, 1, 0, 4] 22
36 [4, 1, 3, 0, 2] 69
37 [3, 1, 4, 0, 2] 67
38 [4, 1, 2, 0, 3] 45
39 [3, 1, 2, 0, 4] 21
40 [2, 1, 4, 0, 3] 43
41 [2, 1, 3, 0, 4] 19
42 [1, 4, 3, 0, 2] 68
43 [1, 3, 4, 0, 2] 66
44 [1, 4, 2, 0, 3] 44
45 [1, 3, 2, 0, 4] 20
46 [1, 2, 4, 0, 3] 42
47 [1, 2, 3, 0, 4] 18
48 [4, 3, 0, 2, 1] 89
49 [3, 4, 0, 2, 1] 88
50 [4, 2, 0, 3, 1] 83
51 [3, 2, 0, 4, 1] 77
52 [2, 4, 0, 3, 1] 82
53 [2, 3, 0, 4, 1] 76
54 [4, 3, 0, 1, 2] 65
55 [3, 4, 0, 1, 2] 64
56 [4, 2, 0, 1, 3] 41
57 [3, 2, 0, 1, 4] 17
58 [2, 4, 0, 1, 3] 40
59 [2, 3, 0, 1, 4] 16
60 [4, 1, 0, 3, 2] 59
61 [3, 1, 0, 4, 2] 53
62 [4, 1, 0, 2, 3] 35
63 [3, 1, 0, 2, 4] 11
64 [2, 1, 0, 4, 3] 29
65 [2, 1, 0, 3, 4] 5
66 [1, 4, 0, 3, 2] 58
67 [1, 3, 0, 4, 2] 52
68 [1, 4, 0, 2, 3] 34
69 [1, 3, 0, 2, 4] 10
70 [1, 2, 0, 4, 3] 28
71 [1, 2, 0, 3, 4] 4
72 [4, 0, 3, 2, 1] 87
73 [3, 0, 4, 2, 1] 85
74 [4, 0, 2, 3, 1] 81
75 [3, 0, 2, 4, 1] 75
76 [2, 0, 4, 3, 1] 79
77 [2, 0, 3, 4, 1] 73
78 [4, 0, 3, 1, 2] 63
79 [3, 0, 4, 1, 2] 61
80 [4, 0, 2, 1, 3] 39
81 [3, 0, 2, 1, 4] 15
82 [2, 0, 4, 1, 3] 37
83 [2, 0, 3, 1, 4] 13
84 [4, 0, 1, 3, 2] 57
85 [3, 0, 1, 4, 2] 51
86 [4, 0, 1, 2, 3] 33
87 [3, 0, 1, 2, 4] 9
88 [2, 0, 1, 4, 3] 27
89 [2, 0, 1, 3, 4] 3
90 [1, 0, 4, 3, 2] 55
91 [1, 0, 3, 4, 2] 49
92 [1, 0, 4, 2, 3] 31
93 [1, 0, 3, 2, 4] 7
94 [1, 0, 2, 4, 3] 25
95 [1, 0, 2, 3, 4] 1
96 [0, 4, 3, 2, 1] 86
97 [0, 3, 4, 2, 1] 84
98 [0, 4, 2, 3, 1] 80
99 [0, 3, 2, 4, 1] 74
100 [0, 2, 4, 3, 1] 78
101 [0, 2, 3, 4, 1] 72
102 [0, 4, 3, 1, 2] 62
103 [0, 3, 4, 1, 2] 60
104 [0, 4, 2, 1, 3] 38
105 [0, 3, 2, 1, 4] 14
106 [0, 2, 4, 1, 3] 36
107 [0, 2, 3, 1, 4] 12
108 [0, 4, 1, 3, 2] 56
109 [0, 3, 1, 4, 2] 50
110 [0, 4, 1, 2, 3] 32
111 [0, 3, 1, 2, 4] 8
112 [0, 2, 1, 4, 3] 26
113 [0, 2, 1, 3, 4] 2
114 [0, 1, 4, 3, 2] 54
115 [0, 1, 3, 4, 2] 48
116 [0, 1, 4, 2, 3] 30
117 [0, 1, 3, 2, 4] 6
118 [0, 1, 2, 4, 3] 24
119 [0, 1, 2, 3, 4] 0

 [119, 118, 117, 115, 116, 114, 113, 112, 107, 101, 106, 100, 111, 109, 105, 99, 103, 97, 110, 108, 104, 98, 102, 96, 95, 94, 93, 91, 92, 90, 71, 70, 47, 23, 46, 22, 69, 67, 45, 21, 43, 19, 68, 66, 44, 20, 42, 18, 89, 88, 83, 77, 82, 76, 65, 64, 41, 17, 40, 16, 59, 53, 35, 11, 29, 5, 58, 52, 34, 10, 28, 4, 87, 85, 81, 75, 79, 73, 63, 61, 39, 15, 37, 13, 57, 51, 33, 9, 27, 3, 55, 49, 31, 7, 25, 1, 86, 84, 80, 74, 78, 72, 62, 60, 38, 14, 36, 12, 56, 50, 32, 8, 26, 2, 54, 48, 30, 6, 24, 0]
"""