Identifier
- St001885: Binary words ⟶ ℤ
Values
=>
0=>2
1=>2
00=>1
01=>2
10=>2
11=>1
000=>1
001=>4
010=>1
011=>4
100=>4
101=>1
110=>4
111=>1
0000=>1
0001=>6
0010=>3
0011=>6
0100=>3
0101=>1
0110=>3
0111=>6
1000=>6
1001=>3
1010=>1
1011=>3
1100=>6
1101=>3
1110=>6
1111=>1
00000=>1
00001=>12
00010=>5
00011=>12
00100=>1
00101=>12
00110=>5
00111=>12
01000=>5
01001=>2
01010=>1
01011=>12
01100=>5
01101=>2
01110=>5
01111=>12
10000=>12
10001=>5
10010=>2
10011=>5
10100=>12
10101=>1
10110=>2
10111=>5
11000=>12
11001=>5
11010=>12
11011=>1
11100=>12
11101=>5
11110=>12
11111=>1
000000=>1
000001=>20
000010=>11
000011=>20
000100=>3
000101=>20
000110=>11
000111=>20
001000=>3
001001=>1
001010=>11
001011=>20
001100=>3
001101=>20
001110=>11
001111=>20
010000=>11
010001=>3
010010=>1
010011=>20
010100=>11
010101=>1
010110=>11
010111=>20
011000=>11
011001=>3
011010=>11
011011=>1
011100=>11
011101=>3
011110=>11
011111=>20
100000=>20
100001=>11
100010=>3
100011=>11
100100=>1
100101=>11
100110=>3
100111=>11
101000=>20
101001=>11
101010=>1
101011=>11
101100=>20
101101=>1
101110=>3
101111=>11
110000=>20
110001=>11
110010=>20
110011=>3
110100=>20
110101=>11
110110=>1
110111=>3
111000=>20
111001=>11
111010=>20
111011=>3
111100=>20
111101=>11
111110=>20
111111=>1
0000000=>1
0000001=>40
0000010=>19
0000011=>40
0000100=>6
0000101=>40
0000110=>19
0000111=>40
0001000=>1
0001001=>40
0001010=>19
0001011=>40
0001100=>6
0001101=>40
0001110=>19
0001111=>40
0010000=>6
0010001=>2
0010010=>1
0010011=>40
0010100=>6
0010101=>40
0010110=>19
0010111=>40
0011000=>6
0011001=>2
0011010=>19
0011011=>40
0011100=>6
0011101=>40
0011110=>19
0011111=>40
0100000=>19
0100001=>8
0100010=>1
0100011=>40
0100100=>1
0100101=>8
0100110=>19
0100111=>40
0101000=>19
0101001=>8
0101010=>1
0101011=>40
0101100=>19
0101101=>8
0101110=>19
0101111=>40
0110000=>19
0110001=>8
0110010=>19
0110011=>2
0110100=>19
0110101=>8
0110110=>1
0110111=>40
0111000=>19
0111001=>8
0111010=>19
0111011=>2
0111100=>19
0111101=>8
0111110=>19
0111111=>40
1000000=>40
1000001=>19
1000010=>8
1000011=>19
1000100=>2
1000101=>19
1000110=>8
1000111=>19
1001000=>40
1001001=>1
1001010=>8
1001011=>19
1001100=>2
1001101=>19
1001110=>8
1001111=>19
1010000=>40
1010001=>19
1010010=>8
1010011=>19
1010100=>40
1010101=>1
1010110=>8
1010111=>19
1011000=>40
1011001=>19
1011010=>8
1011011=>1
1011100=>40
1011101=>1
1011110=>8
1011111=>19
1100000=>40
1100001=>19
1100010=>40
1100011=>6
1100100=>40
1100101=>19
1100110=>2
1100111=>6
1101000=>40
1101001=>19
1101010=>40
1101011=>6
1101100=>40
1101101=>1
1101110=>2
1101111=>6
1110000=>40
1110001=>19
1110010=>40
1110011=>6
1110100=>40
1110101=>19
1110110=>40
1110111=>1
1111000=>40
1111001=>19
1111010=>40
1111011=>6
1111100=>40
1111101=>19
1111110=>40
1111111=>1
00000000=>1
00000001=>74
00000010=>41
00000011=>74
00000100=>11
00000101=>74
00000110=>41
00000111=>74
00001000=>3
00001001=>74
00001010=>41
00001011=>74
00001100=>11
00001101=>74
00001110=>41
00001111=>74
00010000=>3
00010001=>1
00010010=>41
00010011=>74
00010100=>11
00010101=>74
00010110=>41
00010111=>74
00011000=>3
00011001=>74
00011010=>41
00011011=>74
00011100=>11
00011101=>74
00011110=>41
00011111=>74
00100000=>11
00100001=>4
00100010=>1
00100011=>74
00100100=>1
00100101=>74
00100110=>41
00100111=>74
00101000=>11
00101001=>4
00101010=>41
00101011=>74
00101100=>11
00101101=>74
00101110=>41
00101111=>74
00110000=>11
00110001=>4
00110010=>41
00110011=>1
00110100=>11
00110101=>74
00110110=>41
00110111=>74
00111000=>11
00111001=>4
00111010=>41
00111011=>74
00111100=>11
00111101=>74
00111110=>41
00111111=>74
01000000=>41
01000001=>13
01000010=>4
01000011=>74
01000100=>1
01000101=>13
01000110=>41
01000111=>74
01001000=>41
01001001=>1
01001010=>4
01001011=>74
01001100=>41
01001101=>13
01001110=>41
01001111=>74
01010000=>41
01010001=>13
01010010=>4
01010011=>74
01010100=>41
01010101=>1
01010110=>41
01010111=>74
01011000=>41
01011001=>13
01011010=>4
01011011=>74
01011100=>41
01011101=>13
01011110=>41
01011111=>74
01100000=>41
01100001=>13
01100010=>41
01100011=>4
01100100=>41
01100101=>13
01100110=>1
01100111=>74
01101000=>41
01101001=>13
01101010=>41
01101011=>4
01101100=>41
01101101=>1
01101110=>41
01101111=>74
01110000=>41
01110001=>13
01110010=>41
01110011=>4
01110100=>41
01110101=>13
01110110=>41
01110111=>1
01111000=>41
01111001=>13
01111010=>41
01111011=>4
01111100=>41
01111101=>13
01111110=>41
01111111=>74
10000000=>74
10000001=>41
10000010=>13
10000011=>41
10000100=>4
10000101=>41
10000110=>13
10000111=>41
10001000=>1
10001001=>41
10001010=>13
10001011=>41
10001100=>4
10001101=>41
10001110=>13
10001111=>41
10010000=>74
10010001=>41
10010010=>1
10010011=>41
10010100=>4
10010101=>41
10010110=>13
10010111=>41
10011000=>74
10011001=>1
10011010=>13
10011011=>41
10011100=>4
10011101=>41
10011110=>13
10011111=>41
10100000=>74
10100001=>41
10100010=>13
10100011=>41
10100100=>74
10100101=>4
10100110=>13
10100111=>41
10101000=>74
10101001=>41
10101010=>1
10101011=>41
10101100=>74
10101101=>4
10101110=>13
10101111=>41
10110000=>74
10110001=>41
10110010=>13
10110011=>41
10110100=>74
10110101=>4
10110110=>1
10110111=>41
10111000=>74
10111001=>41
10111010=>13
10111011=>1
10111100=>74
10111101=>4
10111110=>13
10111111=>41
11000000=>74
11000001=>41
11000010=>74
11000011=>11
11000100=>74
11000101=>41
11000110=>4
11000111=>11
11001000=>74
11001001=>41
11001010=>74
11001011=>11
11001100=>1
11001101=>41
11001110=>4
11001111=>11
11010000=>74
11010001=>41
11010010=>74
11010011=>11
11010100=>74
11010101=>41
11010110=>4
11010111=>11
11011000=>74
11011001=>41
11011010=>74
11011011=>1
11011100=>74
11011101=>1
11011110=>4
11011111=>11
11100000=>74
11100001=>41
11100010=>74
11100011=>11
11100100=>74
11100101=>41
11100110=>74
11100111=>3
11101000=>74
11101001=>41
11101010=>74
11101011=>11
11101100=>74
11101101=>41
11101110=>1
11101111=>3
11110000=>74
11110001=>41
11110010=>74
11110011=>11
11110100=>74
11110101=>41
11110110=>74
11110111=>3
11111000=>74
11111001=>41
11111010=>74
11111011=>11
11111100=>74
11111101=>41
11111110=>74
11111111=>1
000000000=>1
000000001=>148
000000010=>77
000000011=>148
000000100=>23
000000101=>148
000000110=>77
000000111=>148
000001000=>6
000001001=>148
000001010=>77
000001011=>148
000001100=>23
000001101=>148
000001110=>77
000001111=>148
000010000=>1
000010001=>148
000010010=>77
000010011=>148
000010100=>23
000010101=>148
000010110=>77
000010111=>148
000011000=>6
000011001=>148
000011010=>77
000011011=>148
000011100=>23
000011101=>148
000011110=>77
000011111=>148
000100000=>6
000100001=>2
000100010=>1
000100011=>148
000100100=>23
000100101=>148
000100110=>77
000100111=>148
000101000=>6
000101001=>148
000101010=>77
000101011=>148
000101100=>23
000101101=>148
000101110=>77
000101111=>148
000110000=>6
000110001=>2
000110010=>77
000110011=>148
000110100=>23
000110101=>148
000110110=>77
000110111=>148
000111000=>6
000111001=>148
000111010=>77
000111011=>148
000111100=>23
000111101=>148
000111110=>77
000111111=>148
001000000=>23
001000001=>7
001000010=>2
001000011=>148
001000100=>1
001000101=>148
001000110=>77
001000111=>148
001001000=>23
001001001=>1
001001010=>77
001001011=>148
001001100=>23
001001101=>148
001001110=>77
001001111=>148
001010000=>23
001010001=>7
001010010=>2
001010011=>148
001010100=>23
001010101=>148
001010110=>77
001010111=>148
001011000=>23
001011001=>7
001011010=>77
001011011=>148
001011100=>23
001011101=>148
001011110=>77
001011111=>148
001100000=>23
001100001=>7
001100010=>77
001100011=>2
001100100=>23
001100101=>148
001100110=>1
001100111=>148
001101000=>23
001101001=>7
001101010=>77
001101011=>148
001101100=>23
001101101=>148
001101110=>77
001101111=>148
001110000=>23
001110001=>7
001110010=>77
001110011=>2
001110100=>23
001110101=>148
001110110=>77
001110111=>148
001111000=>23
001111001=>7
001111010=>77
001111011=>148
001111100=>23
001111101=>148
001111110=>77
001111111=>148
010000000=>77
010000001=>30
010000010=>6
010000011=>148
010000100=>2
010000101=>30
010000110=>77
010000111=>148
010001000=>1
010001001=>30
010001010=>6
010001011=>148
010001100=>77
010001101=>30
010001110=>77
010001111=>148
010010000=>77
010010001=>30
010010010=>1
010010011=>148
010010100=>2
010010101=>30
010010110=>77
010010111=>148
010011000=>77
010011001=>30
010011010=>6
010011011=>148
010011100=>77
010011101=>30
010011110=>77
010011111=>148
010100000=>77
010100001=>30
010100010=>6
010100011=>148
010100100=>77
010100101=>2
010100110=>77
010100111=>148
010101000=>77
010101001=>30
010101010=>1
010101011=>148
010101100=>77
010101101=>30
010101110=>77
010101111=>148
010110000=>77
010110001=>30
010110010=>6
010110011=>148
010110100=>77
010110101=>2
010110110=>77
010110111=>148
010111000=>77
010111001=>30
010111010=>6
010111011=>148
010111100=>77
010111101=>30
010111110=>77
010111111=>148
011000000=>77
011000001=>30
011000010=>77
011000011=>7
011000100=>77
011000101=>30
011000110=>2
011000111=>148
011001000=>77
011001001=>30
011001010=>77
011001011=>7
011001100=>1
011001101=>30
011001110=>77
011001111=>148
011010000=>77
011010001=>30
011010010=>77
011010011=>7
011010100=>77
011010101=>30
011010110=>2
011010111=>148
011011000=>77
011011001=>30
011011010=>77
011011011=>1
011011100=>77
011011101=>30
011011110=>77
011011111=>148
011100000=>77
011100001=>30
011100010=>77
011100011=>7
011100100=>77
011100101=>30
011100110=>77
011100111=>2
011101000=>77
011101001=>30
011101010=>77
011101011=>7
011101100=>77
011101101=>30
011101110=>1
011101111=>148
011110000=>77
011110001=>30
011110010=>77
011110011=>7
011110100=>77
011110101=>30
011110110=>77
011110111=>2
011111000=>77
011111001=>30
011111010=>77
011111011=>7
011111100=>77
011111101=>30
011111110=>77
011111111=>148
100000000=>148
100000001=>77
100000010=>30
100000011=>77
100000100=>7
100000101=>77
100000110=>30
100000111=>77
100001000=>2
100001001=>77
100001010=>30
100001011=>77
100001100=>7
100001101=>77
100001110=>30
100001111=>77
100010000=>148
100010001=>1
100010010=>30
100010011=>77
100010100=>7
100010101=>77
100010110=>30
100010111=>77
100011000=>2
100011001=>77
100011010=>30
100011011=>77
100011100=>7
100011101=>77
100011110=>30
100011111=>77
100100000=>148
100100001=>77
100100010=>30
100100011=>77
100100100=>1
100100101=>77
100100110=>30
100100111=>77
100101000=>148
100101001=>2
100101010=>30
100101011=>77
100101100=>7
100101101=>77
100101110=>30
100101111=>77
100110000=>148
100110001=>77
100110010=>30
100110011=>1
100110100=>7
100110101=>77
100110110=>30
100110111=>77
100111000=>148
100111001=>2
100111010=>30
100111011=>77
100111100=>7
100111101=>77
100111110=>30
100111111=>77
101000000=>148
101000001=>77
101000010=>30
101000011=>77
101000100=>148
101000101=>6
101000110=>30
101000111=>77
101001000=>148
101001001=>77
101001010=>2
101001011=>77
101001100=>148
101001101=>6
101001110=>30
101001111=>77
101010000=>148
101010001=>77
101010010=>30
101010011=>77
101010100=>148
101010101=>1
101010110=>30
101010111=>77
101011000=>148
101011001=>77
101011010=>2
101011011=>77
101011100=>148
101011101=>6
101011110=>30
101011111=>77
101100000=>148
101100001=>77
101100010=>30
101100011=>77
101100100=>148
101100101=>6
101100110=>30
101100111=>77
101101000=>148
101101001=>77
101101010=>30
101101011=>2
101101100=>148
101101101=>1
101101110=>30
101101111=>77
101110000=>148
101110001=>77
101110010=>30
101110011=>77
101110100=>148
101110101=>6
101110110=>30
101110111=>1
101111000=>148
101111001=>77
101111010=>30
101111011=>2
101111100=>148
101111101=>6
101111110=>30
101111111=>77
110000000=>148
110000001=>77
110000010=>148
110000011=>23
110000100=>148
110000101=>77
110000110=>7
110000111=>23
110001000=>148
110001001=>77
110001010=>148
110001011=>23
110001100=>2
110001101=>77
110001110=>7
110001111=>23
110010000=>148
110010001=>77
110010010=>148
110010011=>23
110010100=>148
110010101=>77
110010110=>7
110010111=>23
110011000=>148
110011001=>1
110011010=>148
110011011=>23
110011100=>2
110011101=>77
110011110=>7
110011111=>23
110100000=>148
110100001=>77
110100010=>148
110100011=>23
110100100=>148
110100101=>77
110100110=>7
110100111=>23
110101000=>148
110101001=>77
110101010=>148
110101011=>23
110101100=>148
110101101=>2
110101110=>7
110101111=>23
110110000=>148
110110001=>77
110110010=>148
110110011=>23
110110100=>148
110110101=>77
110110110=>1
110110111=>23
110111000=>148
110111001=>77
110111010=>148
110111011=>1
110111100=>148
110111101=>2
110111110=>7
110111111=>23
111000000=>148
111000001=>77
111000010=>148
111000011=>23
111000100=>148
111000101=>77
111000110=>148
111000111=>6
111001000=>148
111001001=>77
111001010=>148
111001011=>23
111001100=>148
111001101=>77
111001110=>2
111001111=>6
111010000=>148
111010001=>77
111010010=>148
111010011=>23
111010100=>148
111010101=>77
111010110=>148
111010111=>6
111011000=>148
111011001=>77
111011010=>148
111011011=>23
111011100=>148
111011101=>1
111011110=>2
111011111=>6
111100000=>148
111100001=>77
111100010=>148
111100011=>23
111100100=>148
111100101=>77
111100110=>148
111100111=>6
111101000=>148
111101001=>77
111101010=>148
111101011=>23
111101100=>148
111101101=>77
111101110=>148
111101111=>1
111110000=>148
111110001=>77
111110010=>148
111110011=>23
111110100=>148
111110101=>77
111110110=>148
111110111=>6
111111000=>148
111111001=>77
111111010=>148
111111011=>23
111111100=>148
111111101=>77
111111110=>148
111111111=>1
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Description
The number of binary words with the same proper border set.
The proper border set of a binary word $w$ is the set of proper prefixes which are also suffixes of $w$.
For example, $0010000010$, $0010100010$ and $0010110010$ are the words with proper border set $\{0, 0010\}$, whereas $0010010010$ has proper border set $\{0, 0010, 0010010\}$.
The proper border set of a binary word $w$ is the set of proper prefixes which are also suffixes of $w$.
For example, $0010000010$, $0010100010$ and $0010110010$ are the words with proper border set $\{0, 0010\}$, whereas $0010010010$ has proper border set $\{0, 0010, 0010010\}$.
Code
@cached_function def proper_border_set(w): B = [] for i in range(1, len(w)): if w[:i] == w[-i:]: B.append(w[:i]) return B def statistic(w): B = proper_border_set(w) return sum(1 for u in Words([0,1], len(w)) if B == proper_border_set(u))
Created
Dec 21, 2022 at 11:43 by Martin Rubey
Updated
Dec 21, 2022 at 11:43 by Martin Rubey
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