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-----------------------------------------------------------------------------
Statistic identifier: St000631
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
Description: The number of distinct palindromic decompositions of a binary word.
A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.
-----------------------------------------------------------------------------
References: [1] [[mo201205]]
-----------------------------------------------------------------------------
Code:
def statistic(w):
return len(paldecs(w))
def paldecs(w):
"""
Return all distinct palindromic decompositions of w.
sage: w = Word([1,1])
sage: paldecs(w)
[[word: 1, word: 1], [word: 11]]
sage: w = Word([1,0,1])
sage: paldecs(w)
[[word: 1, word: 0, word: 1], [word: 101]]
sage: w = Word([1,0,1,0,0,1])
sage: paldecs(w)
[[word: 1, word: 0, word: 1, word: 0, word: 0, word: 1],
[word: 1, word: 0, word: 1, word: 00, word: 1],
[word: 1, word: 0, word: 1001],
[word: 1, word: 010, word: 0, word: 1],
[word: 101, word: 0, word: 0, word: 1],
[word: 101, word: 00, word: 1]]
sage: w = Word("referee")
sage: paldecs(w)
[[word: r, word: e, word: f, word: e, word: r, word: e, word: e],
[word: r, word: e, word: f, word: e, word: r, word: ee],
[word: r, word: e, word: f, word: ere, word: e],
[word: r, word: efe, word: r, word: e, word: e],
[word: r, word: efe, word: r, word: ee],
[word: refer, word: e, word: e],
[word: refer, word: ee]]
"""
if len(w) == 0:
return [[]]
P1 = w.palindrome_prefixes()
result = []
for p1 in P1:
l = len(p1)
if l > 0:
P2 = paldecs(w[l:])
for p2 in P2:
result.append([p1] + p2)
return result
-----------------------------------------------------------------------------
Statistic values:
0 => 1
1 => 1
00 => 2
01 => 1
10 => 1
11 => 2
000 => 4
001 => 2
010 => 2
011 => 2
100 => 2
101 => 2
110 => 2
111 => 4
0000 => 8
0001 => 4
0010 => 3
0011 => 4
0100 => 3
0101 => 3
0110 => 3
0111 => 4
1000 => 4
1001 => 3
1010 => 3
1011 => 3
1100 => 4
1101 => 3
1110 => 4
1111 => 8
00000 => 16
00001 => 8
00010 => 6
00011 => 8
00100 => 6
00101 => 5
00110 => 5
00111 => 8
01000 => 6
01001 => 4
01010 => 5
01011 => 5
01100 => 5
01101 => 4
01110 => 5
01111 => 8
10000 => 8
10001 => 5
10010 => 4
10011 => 5
10100 => 5
10101 => 5
10110 => 4
10111 => 6
11000 => 8
11001 => 5
11010 => 5
11011 => 6
11100 => 8
11101 => 6
11110 => 8
11111 => 16
000000 => 32
000001 => 16
000010 => 12
000011 => 16
000100 => 11
000101 => 10
000110 => 10
000111 => 16
001000 => 11
001001 => 8
001010 => 8
001011 => 8
001100 => 10
001101 => 7
001110 => 9
001111 => 16
010000 => 12
010001 => 7
010010 => 7
010011 => 7
010100 => 8
010101 => 8
010110 => 6
010111 => 10
011000 => 10
011001 => 6
011010 => 6
011011 => 8
011100 => 9
011101 => 7
011110 => 9
011111 => 16
100000 => 16
100001 => 9
100010 => 7
100011 => 9
100100 => 8
100101 => 6
100110 => 6
100111 => 10
101000 => 10
101001 => 6
101010 => 8
101011 => 8
101100 => 7
101101 => 7
101110 => 7
101111 => 12
110000 => 16
110001 => 9
110010 => 7
110011 => 10
110100 => 8
110101 => 8
110110 => 8
110111 => 11
111000 => 16
111001 => 10
111010 => 10
111011 => 11
111100 => 16
111101 => 12
111110 => 16
111111 => 32
0000000 => 64
0000001 => 32
0000010 => 24
0000011 => 32
0000100 => 22
0000101 => 20
0000110 => 20
0000111 => 32
0001000 => 22
0001001 => 15
0001010 => 16
0001011 => 16
0001100 => 19
0001101 => 14
0001110 => 18
0001111 => 32
0010000 => 22
0010001 => 13
0010010 => 12
0010011 => 14
0010100 => 14
0010101 => 13
0010110 => 10
0010111 => 16
0011000 => 19
0011001 => 12
0011010 => 11
0011011 => 14
0011100 => 18
0011101 => 13
0011110 => 17
0011111 => 32
0100000 => 24
0100001 => 13
0100010 => 11
0100011 => 13
0100100 => 12
0100101 => 10
0100110 => 9
0100111 => 14
0101000 => 16
0101001 => 10
0101010 => 13
0101011 => 13
0101100 => 11
0101101 => 10
0101110 => 11
0101111 => 20
0110000 => 20
0110001 => 11
0110010 => 9
0110011 => 12
0110100 => 10
0110101 => 10
0110110 => 11
0110111 => 15
0111000 => 18
0111001 => 11
0111010 => 11
0111011 => 13
0111100 => 17
0111101 => 13
0111110 => 17
0111111 => 32
1000000 => 32
1000001 => 17
1000010 => 13
1000011 => 17
1000100 => 13
1000101 => 11
1000110 => 11
1000111 => 18
1001000 => 15
1001001 => 11
1001010 => 10
1001011 => 10
1001100 => 12
1001101 => 9
1001110 => 11
1001111 => 20
1010000 => 20
1010001 => 11
1010010 => 10
1010011 => 11
1010100 => 13
1010101 => 13
1010110 => 10
1010111 => 16
1011000 => 14
1011001 => 9
1011010 => 10
1011011 => 12
1011100 => 13
1011101 => 11
1011110 => 13
1011111 => 24
1100000 => 32
1100001 => 17
1100010 => 13
1100011 => 18
1100100 => 14
1100101 => 11
1100110 => 12
1100111 => 19
1101000 => 16
1101001 => 10
1101010 => 13
1101011 => 14
1101100 => 14
1101101 => 12
1101110 => 13
1101111 => 22
1110000 => 32
1110001 => 18
1110010 => 14
1110011 => 19
1110100 => 16
1110101 => 16
1110110 => 15
1110111 => 22
1111000 => 32
1111001 => 20
1111010 => 20
1111011 => 22
1111100 => 32
1111101 => 24
1111110 => 32
1111111 => 64
00000000 => 128
00000001 => 64
00000010 => 48
00000011 => 64
00000100 => 44
00000101 => 40
00000110 => 40
00000111 => 64
00001000 => 43
00001001 => 30
00001010 => 32
00001011 => 32
00001100 => 38
00001101 => 28
00001110 => 36
00001111 => 64
00010000 => 43
00010001 => 26
00010010 => 23
00010011 => 26
00010100 => 27
00010101 => 26
00010110 => 20
00010111 => 32
00011000 => 38
00011001 => 23
00011010 => 22
00011011 => 28
00011100 => 35
00011101 => 26
00011110 => 34
00011111 => 64
00100000 => 44
00100001 => 24
00100010 => 20
00100011 => 24
00100100 => 23
00100101 => 18
00100110 => 17
00100111 => 28
00101000 => 27
00101001 => 17
00101010 => 21
00101011 => 21
00101100 => 18
00101101 => 17
00101110 => 18
00101111 => 32
00110000 => 38
00110001 => 21
00110010 => 17
00110011 => 24
00110100 => 18
00110101 => 18
00110110 => 19
00110111 => 26
00111000 => 35
00111001 => 22
00111010 => 21
00111011 => 24
00111100 => 34
00111101 => 25
00111110 => 33
00111111 => 64
01000000 => 48
01000001 => 25
01000010 => 20
01000011 => 25
01000100 => 20
01000101 => 17
01000110 => 16
01000111 => 26
01001000 => 23
01001001 => 16
01001010 => 16
01001011 => 17
01001100 => 17
01001101 => 13
01001110 => 16
01001111 => 28
01010000 => 32
01010001 => 18
01010010 => 16
01010011 => 18
01010100 => 21
01010101 => 21
01010110 => 16
01010111 => 26
01011000 => 22
01011001 => 14
01011010 => 16
01011011 => 18
01011100 => 21
01011101 => 17
01011110 => 21
01011111 => 40
01100000 => 40
01100001 => 21
01100010 => 16
01100011 => 22
01100100 => 17
01100101 => 14
01100110 => 16
01100111 => 23
01101000 => 20
01101001 => 13
01101010 => 16
01101011 => 17
01101100 => 19
01101101 => 16
01101110 => 17
01101111 => 30
01110000 => 36
01110001 => 20
01110010 => 16
01110011 => 21
01110100 => 18
01110101 => 18
01110110 => 17
01110111 => 26
01111000 => 34
01111001 => 21
01111010 => 21
01111011 => 24
01111100 => 33
01111101 => 25
01111110 => 33
01111111 => 64
10000000 => 64
10000001 => 33
10000010 => 25
10000011 => 33
10000100 => 24
10000101 => 21
10000110 => 21
10000111 => 34
10001000 => 26
10001001 => 17
10001010 => 18
10001011 => 18
10001100 => 21
10001101 => 16
10001110 => 20
10001111 => 36
10010000 => 30
10010001 => 17
10010010 => 16
10010011 => 19
10010100 => 17
10010101 => 16
10010110 => 13
10010111 => 20
10011000 => 23
10011001 => 16
10011010 => 14
10011011 => 17
10011100 => 22
10011101 => 16
10011110 => 21
10011111 => 40
10100000 => 40
10100001 => 21
10100010 => 17
10100011 => 21
10100100 => 18
10100101 => 16
10100110 => 14
10100111 => 22
10101000 => 26
10101001 => 16
10101010 => 21
10101011 => 21
10101100 => 18
10101101 => 16
10101110 => 18
10101111 => 32
10110000 => 28
10110001 => 16
10110010 => 13
10110011 => 17
10110100 => 17
10110101 => 16
10110110 => 16
10110111 => 23
10111000 => 26
10111001 => 16
10111010 => 17
10111011 => 20
10111100 => 25
10111101 => 20
10111110 => 25
10111111 => 48
11000000 => 64
11000001 => 33
11000010 => 25
11000011 => 34
11000100 => 24
11000101 => 21
11000110 => 22
11000111 => 35
11001000 => 26
11001001 => 19
11001010 => 18
11001011 => 18
11001100 => 24
11001101 => 17
11001110 => 21
11001111 => 38
11010000 => 32
11010001 => 18
11010010 => 17
11010011 => 18
11010100 => 21
11010101 => 21
11010110 => 17
11010111 => 27
11011000 => 28
11011001 => 17
11011010 => 18
11011011 => 23
11011100 => 24
11011101 => 20
11011110 => 24
11011111 => 44
11100000 => 64
11100001 => 34
11100010 => 26
11100011 => 35
11100100 => 28
11100101 => 22
11100110 => 23
11100111 => 38
11101000 => 32
11101001 => 20
11101010 => 26
11101011 => 27
11101100 => 26
11101101 => 23
11101110 => 26
11101111 => 43
11110000 => 64
11110001 => 36
11110010 => 28
11110011 => 38
11110100 => 32
11110101 => 32
11110110 => 30
11110111 => 43
11111000 => 64
11111001 => 40
11111010 => 40
11111011 => 44
11111100 => 64
11111101 => 48
11111110 => 64
11111111 => 128
000000000 => 256
000000001 => 128
000000010 => 96
000000011 => 128
000000100 => 88
000000101 => 80
000000110 => 80
000000111 => 128
000001000 => 86
000001001 => 60
000001010 => 64
000001011 => 64
000001100 => 76
000001101 => 56
000001110 => 72
000001111 => 128
000010000 => 86
000010001 => 51
000010010 => 46
000010011 => 52
000010100 => 54
000010101 => 52
000010110 => 40
000010111 => 64
000011000 => 75
000011001 => 46
000011010 => 44
000011011 => 56
000011100 => 70
000011101 => 52
000011110 => 68
000011111 => 128
000100000 => 86
000100001 => 47
000100010 => 39
000100011 => 48
000100100 => 43
000100101 => 34
000100110 => 32
000100111 => 52
000101000 => 54
000101001 => 33
000101010 => 42
000101011 => 42
000101100 => 36
000101101 => 34
000101110 => 36
000101111 => 64
000110000 => 75
000110001 => 42
000110010 => 33
000110011 => 46
000110100 => 36
000110101 => 36
000110110 => 38
000110111 => 52
000111000 => 70
000111001 => 43
000111010 => 42
000111011 => 48
000111100 => 67
000111101 => 50
000111110 => 66
000111111 => 128
001000000 => 88
001000001 => 46
001000010 => 36
001000011 => 46
001000100 => 37
001000101 => 31
001000110 => 30
001000111 => 48
001001000 => 43
001001001 => 31
001001010 => 29
001001011 => 30
001001100 => 33
001001101 => 25
001001110 => 31
001001111 => 56
001010000 => 54
001010001 => 30
001010010 => 27
001010011 => 31
001010100 => 35
001010101 => 34
001010110 => 26
001010111 => 42
001011000 => 36
001011001 => 23
001011010 => 26
001011011 => 30
001011100 => 34
001011101 => 28
001011110 => 34
001011111 => 64
001100000 => 76
001100001 => 40
001100010 => 31
001100011 => 42
001100100 => 33
001100101 => 27
001100110 => 30
001100111 => 46
001101000 => 36
001101001 => 23
001101010 => 29
001101011 => 31
001101100 => 34
001101101 => 28
001101110 => 30
001101111 => 52
001110000 => 70
001110001 => 39
001110010 => 31
001110011 => 42
001110100 => 34
001110101 => 34
001110110 => 32
001110111 => 48
001111000 => 67
001111001 => 42
001111010 => 41
001111011 => 46
001111100 => 66
001111101 => 49
001111110 => 65
001111111 => 128
010000000 => 96
010000001 => 49
010000010 => 38
010000011 => 49
010000100 => 36
010000101 => 32
010000110 => 31
010000111 => 50
010001000 => 39
010001001 => 26
010001010 => 27
010001011 => 28
010001100 => 31
010001101 => 23
010001110 => 29
010001111 => 52
010010000 => 46
010010001 => 26
010010010 => 26
010010011 => 28
010010100 => 27
010010101 => 26
010010110 => 21
010010111 => 34
010011000 => 33
010011001 => 22
010011010 => 20
010011011 => 25
010011100 => 31
010011101 => 23
010011110 => 30
010011111 => 56
010100000 => 64
010100001 => 34
010100010 => 27
010100011 => 34
010100100 => 29
010100101 => 25
010100110 => 23
010100111 => 36
010101000 => 42
010101001 => 26
010101010 => 34
010101011 => 34
010101100 => 29
010101101 => 26
010101110 => 29
010101111 => 52
010110000 => 44
010110001 => 25
010110010 => 20
010110011 => 27
010110100 => 26
010110101 => 25
010110110 => 24
010110111 => 34
010111000 => 42
010111001 => 26
010111010 => 28
010111011 => 31
010111100 => 41
010111101 => 32
010111110 => 41
010111111 => 80
011000000 => 80
011000001 => 41
011000010 => 31
011000011 => 42
011000100 => 30
011000101 => 26
011000110 => 28
011000111 => 43
011001000 => 32
011001001 => 23
011001010 => 23
011001011 => 23
011001100 => 30
011001101 => 22
011001110 => 26
011001111 => 46
011010000 => 40
011010001 => 23
011010010 => 21
011010011 => 23
011010100 => 26
011010101 => 26
011010110 => 22
011010111 => 33
011011000 => 38
011011001 => 23
011011010 => 24
011011011 => 31
011011100 => 32
011011101 => 26
011011110 => 32
011011111 => 60
011100000 => 72
011100001 => 38
011100010 => 29
011100011 => 39
011100100 => 31
011100101 => 25
011100110 => 26
011100111 => 42
011101000 => 36
011101001 => 23
011101010 => 29
011101011 => 30
011101100 => 30
011101101 => 26
011101110 => 31
011101111 => 51
011110000 => 68
011110001 => 38
011110010 => 30
011110011 => 40
011110100 => 34
011110101 => 34
011110110 => 32
011110111 => 47
011111000 => 66
011111001 => 41
011111010 => 41
011111011 => 46
011111100 => 65
011111101 => 49
011111110 => 65
011111111 => 128
100000000 => 128
100000001 => 65
100000010 => 49
100000011 => 65
100000100 => 46
100000101 => 41
100000110 => 41
100000111 => 66
100001000 => 47
100001001 => 32
100001010 => 34
100001011 => 34
100001100 => 40
100001101 => 30
100001110 => 38
100001111 => 68
100010000 => 51
100010001 => 31
100010010 => 26
100010011 => 30
100010100 => 30
100010101 => 29
100010110 => 23
100010111 => 36
100011000 => 42
100011001 => 26
100011010 => 25
100011011 => 31
100011100 => 39
100011101 => 29
100011110 => 38
100011111 => 72
100100000 => 60
100100001 => 32
100100010 => 26
100100011 => 32
100100100 => 31
100100101 => 24
100100110 => 23
100100111 => 38
100101000 => 33
100101001 => 22
100101010 => 26
100101011 => 26
100101100 => 23
100101101 => 21
100101110 => 23
100101111 => 40
100110000 => 46
100110001 => 26
100110010 => 22
100110011 => 30
100110100 => 23
100110101 => 23
100110110 => 23
100110111 => 32
100111000 => 43
100111001 => 28
100111010 => 26
100111011 => 30
100111100 => 42
100111101 => 31
100111110 => 41
100111111 => 80
101000000 => 80
101000001 => 41
101000010 => 32
101000011 => 41
101000100 => 31
101000101 => 28
101000110 => 26
101000111 => 42
101001000 => 34
101001001 => 24
101001010 => 25
101001011 => 26
101001100 => 27
101001101 => 20
101001110 => 25
101001111 => 44
101010000 => 52
101010001 => 29
101010010 => 26
101010011 => 29
101010100 => 34
101010101 => 34
101010110 => 26
101010111 => 42
101011000 => 36
101011001 => 23
101011010 => 25
101011011 => 29
101011100 => 34
101011101 => 27
101011110 => 34
101011111 => 64
101100000 => 56
101100001 => 30
101100010 => 23
101100011 => 31
101100100 => 25
101100101 => 20
101100110 => 22
101100111 => 33
101101000 => 34
101101001 => 21
101101010 => 26
101101011 => 27
101101100 => 28
101101101 => 26
101101110 => 26
101101111 => 46
101110000 => 52
101110001 => 29
101110010 => 23
101110011 => 31
101110100 => 28
101110101 => 27
101110110 => 26
101110111 => 39
101111000 => 50
101111001 => 31
101111010 => 32
101111011 => 36
101111100 => 49
101111101 => 38
101111110 => 49
101111111 => 96
110000000 => 128
110000001 => 65
110000010 => 49
110000011 => 66
110000100 => 46
110000101 => 41
110000110 => 42
110000111 => 67
110001000 => 48
110001001 => 32
110001010 => 34
110001011 => 34
110001100 => 42
110001101 => 31
110001110 => 39
110001111 => 70
110010000 => 52
110010001 => 30
110010010 => 28
110010011 => 34
110010100 => 31
110010101 => 29
110010110 => 23
110010111 => 36
110011000 => 46
110011001 => 30
110011010 => 27
110011011 => 33
110011100 => 42
110011101 => 31
110011110 => 40
110011111 => 76
110100000 => 64
110100001 => 34
110100010 => 28
110100011 => 34
110100100 => 30
110100101 => 26
110100110 => 23
110100111 => 36
110101000 => 42
110101001 => 26
110101010 => 34
110101011 => 35
110101100 => 31
110101101 => 27
110101110 => 30
110101111 => 54
110110000 => 56
110110001 => 31
110110010 => 25
110110011 => 33
110110100 => 30
110110101 => 29
110110110 => 31
110110111 => 43
110111000 => 48
110111001 => 30
110111010 => 31
110111011 => 37
110111100 => 46
110111101 => 36
110111110 => 46
110111111 => 88
111000000 => 128
111000001 => 66
111000010 => 50
111000011 => 67
111000100 => 48
111000101 => 42
111000110 => 43
111000111 => 70
111001000 => 52
111001001 => 38
111001010 => 36
111001011 => 36
111001100 => 46
111001101 => 33
111001110 => 42
111001111 => 75
111010000 => 64
111010001 => 36
111010010 => 34
111010011 => 36
111010100 => 42
111010101 => 42
111010110 => 33
111010111 => 54
111011000 => 52
111011001 => 32
111011010 => 34
111011011 => 43
111011100 => 48
111011101 => 39
111011110 => 47
111011111 => 86
111100000 => 128
111100001 => 68
111100010 => 52
111100011 => 70
111100100 => 56
111100101 => 44
111100110 => 46
111100111 => 75
111101000 => 64
111101001 => 40
111101010 => 52
111101011 => 54
111101100 => 52
111101101 => 46
111101110 => 51
111101111 => 86
111110000 => 128
111110001 => 72
111110010 => 56
111110011 => 76
111110100 => 64
111110101 => 64
111110110 => 60
111110111 => 86
111111000 => 128
111111001 => 80
111111010 => 80
111111011 => 88
111111100 => 128
111111101 => 96
111111110 => 128
111111111 => 256
0000000001 => 256
0010000001 => 90
0010101101 => 42
0010101011 => 55
0010100111 => 62
0010011101 => 45
0010011011 => 48
0010010111 => 60
0010001111 => 96
0001110101 => 68
0001110011 => 82
0001101101 => 56
0001101011 => 62
0001100111 => 88
0001011101 => 56
0001011011 => 60
0001010111 => 84
0001001111 => 104
0000111101 => 100
0000111011 => 96
0000110111 => 104
0000101111 => 128
0000011111 => 256
0000000000 => 512
0000000110 => 160
0000011000 => 150
0000011110 => 136
0000010010 => 92
0001100000 => 150
0001100110 => 58
0001111000 => 134
0001111110 => 130
0001110010 => 61
0001001000 => 82
0001001110 => 58
0001000010 => 71
0001011010 => 52
0010101010 => 55
0010101000 => 69
0010101100 => 47
0010100100 => 49
0010010100 => 49
0001010100 => 69
0010101110 => 47
0010100000 => 108
0010000100 => 67
0000010100 => 108
0000000010 => 192
0000001110 => 144
0000111110 => 132
0000000100 => 176
0000001010 => 128
0000010110 => 80
0000101110 => 72
0001011110 => 68
0001101110 => 60
0001110110 => 64
0001111010 => 82
0001111100 => 131
0000000101 => 160
0000001101 => 112
0000011101 => 104
0001111101 => 98
0000001001 => 120
0000010101 => 104
0000101101 => 68
0010110001 => 41
0001100001 => 79
0001000001 => 90
0000111111 => 256
0001110100 => 68
0001101100 => 67
0010011100 => 61
0001011100 => 68
0000111100 => 134
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0000011001 => 92
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0000000011 => 256
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0010000010 => 69
0001000011 => 90
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0001010000 => 107
0001000100 => 72
0001010101 => 68
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0010000000 => 176
0000000111 => 256
0000001111 => 256
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0001011111 => 128
0001111001 => 83
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0001000111 => 96
0000100111 => 104
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0001001001 => 58
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0010110000 => 72
0000011100 => 140
0000001000 => 172
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0010101111 => 84
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0001001100 => 62
0001000110 => 59
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0000110010 => 66
0010010000 => 86
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0001000000 => 172
0001011001 => 46
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0001110001 => 78
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0010101001 => 43
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0000111001 => 86
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0001001101 => 47
0001010001 => 60
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0001111011 => 92
0010011000 => 64
0001011000 => 72
0000100000 => 171
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0000110000 => 150
0000100001 => 94
0010100011 => 57
0010100001 => 57
0000111000 => 139
0001101000 => 72
0000010000 => 171
0000101001 => 66
0010100010 => 46
0010001001 => 48
0010000011 => 90
0010001010 => 50
0001110000 => 139
-----------------------------------------------------------------------------
Created: Oct 16, 2016 at 21:12 by Martin Rubey
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Last Updated: Oct 16, 2016 at 21:12 by Martin Rubey