Identifier
- St001414: Binary words ⟶ ℤ
Values
=>
0=>0
1=>0
00=>0
01=>0
10=>0
11=>0
000=>1
001=>0
010=>1
011=>0
100=>0
101=>1
110=>0
111=>1
0000=>1
0001=>1
0010=>0
0011=>0
0100=>1
0101=>1
0110=>0
0111=>0
1000=>0
1001=>0
1010=>1
1011=>1
1100=>0
1101=>0
1110=>1
1111=>1
00000=>2
00001=>1
00010=>1
00011=>1
00100=>2
00101=>0
00110=>0
00111=>0
01000=>1
01001=>1
01010=>2
01011=>1
01100=>0
01101=>0
01110=>2
01111=>0
10000=>0
10001=>2
10010=>0
10011=>0
10100=>1
10101=>2
10110=>1
10111=>1
11000=>0
11001=>0
11010=>0
11011=>2
11100=>1
11101=>1
11110=>1
11111=>2
000000=>2
000001=>2
000010=>1
000011=>1
000100=>1
000101=>1
000110=>1
000111=>1
001000=>2
001001=>2
001010=>0
001011=>0
001100=>0
001101=>0
001110=>0
001111=>0
010000=>1
010001=>1
010010=>1
010011=>1
010100=>2
010101=>2
010110=>1
010111=>1
011000=>0
011001=>0
011010=>0
011011=>0
011100=>2
011101=>2
011110=>0
011111=>0
100000=>0
100001=>0
100010=>2
100011=>2
100100=>0
100101=>0
100110=>0
100111=>0
101000=>1
101001=>1
101010=>2
101011=>2
101100=>1
101101=>1
101110=>1
101111=>1
110000=>0
110001=>0
110010=>0
110011=>0
110100=>0
110101=>0
110110=>2
110111=>2
111000=>1
111001=>1
111010=>1
111011=>1
111100=>1
111101=>1
111110=>2
111111=>2
0000000=>3
0000001=>2
0000010=>2
0000011=>2
0000100=>1
0000101=>1
0000110=>1
0000111=>1
0001000=>3
0001001=>1
0001010=>1
0001011=>1
0001100=>1
0001101=>1
0001110=>1
0001111=>1
0010000=>2
0010001=>2
0010010=>2
0010011=>2
0010100=>3
0010101=>0
0010110=>0
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0011000=>0
0011001=>0
0011010=>0
0011011=>0
0011100=>3
0011101=>0
0011110=>0
0011111=>0
0100000=>1
0100001=>1
0100010=>3
0100011=>1
0100100=>1
0100101=>1
0100110=>1
0100111=>1
0101000=>2
0101001=>2
0101010=>3
0101011=>2
0101100=>1
0101101=>1
0101110=>1
0101111=>1
0110000=>0
0110001=>0
0110010=>0
0110011=>0
0110100=>0
0110101=>0
0110110=>3
0110111=>0
0111000=>2
0111001=>2
0111010=>2
0111011=>2
0111100=>0
0111101=>0
0111110=>3
0111111=>0
1000000=>0
1000001=>3
1000010=>0
1000011=>0
1000100=>2
1000101=>2
1000110=>2
1000111=>2
1001000=>0
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1001010=>0
1001011=>0
1001100=>0
1001101=>0
1001110=>0
1001111=>0
1010000=>1
1010001=>1
1010010=>1
1010011=>1
1010100=>2
1010101=>3
1010110=>2
1010111=>2
1011000=>1
1011001=>1
1011010=>1
1011011=>1
1011100=>1
1011101=>3
1011110=>1
1011111=>1
1100000=>0
1100001=>0
1100010=>0
1100011=>3
1100100=>0
1100101=>0
1100110=>0
1100111=>0
1101000=>0
1101001=>0
1101010=>0
1101011=>3
1101100=>2
1101101=>2
1101110=>2
1101111=>2
1110000=>1
1110001=>1
1110010=>1
1110011=>1
1110100=>1
1110101=>1
1110110=>1
1110111=>3
1111000=>1
1111001=>1
1111010=>1
1111011=>1
1111100=>2
1111101=>2
1111110=>2
1111111=>3
00000000=>3
00000001=>3
00000010=>2
00000011=>2
00000100=>2
00000101=>2
00000110=>2
00000111=>2
00001000=>1
00001001=>1
00001010=>1
00001011=>1
00001100=>1
00001101=>1
00001110=>1
00001111=>1
00010000=>3
00010001=>3
00010010=>1
00010011=>1
00010100=>1
00010101=>1
00010110=>1
00010111=>1
00011000=>1
00011001=>1
00011010=>1
00011011=>1
00011100=>1
00011101=>1
00011110=>1
00011111=>1
00100000=>2
00100001=>2
00100010=>2
00100011=>2
00100100=>2
00100101=>2
00100110=>2
00100111=>2
00101000=>3
00101001=>3
00101010=>0
00101011=>0
00101100=>0
00101101=>0
00101110=>0
00101111=>0
00110000=>0
00110001=>0
00110010=>0
00110011=>0
00110100=>0
00110101=>0
00110110=>0
00110111=>0
00111000=>3
00111001=>3
00111010=>0
00111011=>0
00111100=>0
00111101=>0
00111110=>0
00111111=>0
01000000=>1
01000001=>1
01000010=>1
01000011=>1
01000100=>3
01000101=>3
01000110=>1
01000111=>1
01001000=>1
01001001=>1
01001010=>1
01001011=>1
01001100=>1
01001101=>1
01001110=>1
01001111=>1
01010000=>2
01010001=>2
01010010=>2
01010011=>2
01010100=>3
01010101=>3
01010110=>2
01010111=>2
01011000=>1
01011001=>1
01011010=>1
01011011=>1
01011100=>1
01011101=>1
01011110=>1
01011111=>1
01100000=>0
01100001=>0
01100010=>0
01100011=>0
01100100=>0
01100101=>0
01100110=>0
01100111=>0
01101000=>0
01101001=>0
01101010=>0
01101011=>0
01101100=>3
01101101=>3
01101110=>0
01101111=>0
01110000=>2
01110001=>2
01110010=>2
01110011=>2
01110100=>2
01110101=>2
01110110=>2
01110111=>2
01111000=>0
01111001=>0
01111010=>0
01111011=>0
01111100=>3
01111101=>3
01111110=>0
01111111=>0
10000000=>0
10000001=>0
10000010=>3
10000011=>3
10000100=>0
10000101=>0
10000110=>0
10000111=>0
10001000=>2
10001001=>2
10001010=>2
10001011=>2
10001100=>2
10001101=>2
10001110=>2
10001111=>2
10010000=>0
10010001=>0
10010010=>3
10010011=>3
10010100=>0
10010101=>0
10010110=>0
10010111=>0
10011000=>0
10011001=>0
10011010=>0
10011011=>0
10011100=>0
10011101=>0
10011110=>0
10011111=>0
10100000=>1
10100001=>1
10100010=>1
10100011=>1
10100100=>1
10100101=>1
10100110=>1
10100111=>1
10101000=>2
10101001=>2
10101010=>3
10101011=>3
10101100=>2
10101101=>2
10101110=>2
10101111=>2
10110000=>1
10110001=>1
10110010=>1
10110011=>1
10110100=>1
10110101=>1
10110110=>1
10110111=>1
10111000=>1
10111001=>1
10111010=>3
10111011=>3
10111100=>1
10111101=>1
10111110=>1
10111111=>1
11000000=>0
11000001=>0
11000010=>0
11000011=>0
11000100=>0
11000101=>0
11000110=>3
11000111=>3
11001000=>0
11001001=>0
11001010=>0
11001011=>0
11001100=>0
11001101=>0
11001110=>0
11001111=>0
11010000=>0
11010001=>0
11010010=>0
11010011=>0
11010100=>0
11010101=>0
11010110=>3
11010111=>3
11011000=>2
11011001=>2
11011010=>2
11011011=>2
11011100=>2
11011101=>2
11011110=>2
11011111=>2
11100000=>1
11100001=>1
11100010=>1
11100011=>1
11100100=>1
11100101=>1
11100110=>1
11100111=>1
11101000=>1
11101001=>1
11101010=>1
11101011=>1
11101100=>1
11101101=>1
11101110=>3
11101111=>3
11110000=>1
11110001=>1
11110010=>1
11110011=>1
11110100=>1
11110101=>1
11110110=>1
11110111=>1
11111000=>2
11111001=>2
11111010=>2
11111011=>2
11111100=>2
11111101=>2
11111110=>3
11111111=>3
000000000=>4
000000001=>3
000000010=>3
000000011=>3
000000100=>2
000000101=>2
000000110=>2
000000111=>2
000001000=>2
000001001=>2
000001010=>2
000001011=>2
000001100=>2
000001101=>2
000001110=>2
000001111=>2
000010000=>4
000010001=>1
000010010=>1
000010011=>1
000010100=>1
000010101=>1
000010110=>1
000010111=>1
000011000=>1
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000011011=>1
000011100=>1
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000011111=>1
000100000=>3
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000100010=>3
000100011=>3
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000100111=>1
000101000=>4
000101001=>1
000101010=>1
000101011=>1
000101100=>1
000101101=>1
000101110=>1
000101111=>1
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000110010=>1
000110011=>1
000110100=>1
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000111000=>4
000111001=>1
000111010=>1
000111011=>1
000111100=>1
000111101=>1
000111110=>1
000111111=>1
001000000=>2
001000001=>2
001000010=>2
001000011=>2
001000100=>4
001000101=>2
001000110=>2
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001001000=>2
001001001=>2
001001010=>2
001001011=>2
001001100=>2
001001101=>2
001001110=>2
001001111=>2
001010000=>3
001010001=>3
001010010=>3
001010011=>3
001010100=>4
001010101=>0
001010110=>0
001010111=>0
001011000=>0
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001011010=>0
001011011=>0
001011100=>0
001011101=>0
001011110=>0
001011111=>0
001100000=>0
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001101101=>0
001101110=>0
001101111=>0
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001110110=>0
001110111=>0
001111000=>0
001111001=>0
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001111100=>4
001111101=>0
001111110=>0
001111111=>0
010000000=>1
010000001=>1
010000010=>4
010000011=>1
010000100=>1
010000101=>1
010000110=>1
010000111=>1
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010001001=>3
010001010=>3
010001011=>3
010001100=>1
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010001110=>1
010001111=>1
010010000=>1
010010001=>1
010010010=>4
010010011=>1
010010100=>1
010010101=>1
010010110=>1
010010111=>1
010011000=>1
010011001=>1
010011010=>1
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010011100=>1
010011101=>1
010011110=>1
010011111=>1
010100000=>2
010100001=>2
010100010=>2
010100011=>2
010100100=>2
010100101=>2
010100110=>2
010100111=>2
010101000=>3
010101001=>3
010101010=>4
010101011=>3
010101100=>2
010101101=>2
010101110=>2
010101111=>2
010110000=>1
010110001=>1
010110010=>1
010110011=>1
010110100=>1
010110101=>1
010110110=>1
010110111=>1
010111000=>1
010111001=>1
010111010=>4
010111011=>1
010111100=>1
010111101=>1
010111110=>1
010111111=>1
011000000=>0
011000001=>0
011000010=>0
011000011=>0
011000100=>0
011000101=>0
011000110=>4
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011001010=>0
011001011=>0
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011001111=>0
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011011010=>3
011011011=>3
011011100=>0
011011101=>0
011011110=>0
011011111=>0
011100000=>2
011100001=>2
011100010=>2
011100011=>2
011100100=>2
011100101=>2
011100110=>2
011100111=>2
011101000=>2
011101001=>2
011101010=>2
011101011=>2
011101100=>2
011101101=>2
011101110=>4
011101111=>2
011110000=>0
011110001=>0
011110010=>0
011110011=>0
011110100=>0
011110101=>0
011110110=>0
011110111=>0
011111000=>3
011111001=>3
011111010=>3
011111011=>3
011111100=>0
011111101=>0
011111110=>4
011111111=>0
100000000=>0
100000001=>4
100000010=>0
100000011=>0
100000100=>3
100000101=>3
100000110=>3
100000111=>3
100001000=>0
100001001=>0
100001010=>0
100001011=>0
100001100=>0
100001101=>0
100001110=>0
100001111=>0
100010000=>2
100010001=>4
100010010=>2
100010011=>2
100010100=>2
100010101=>2
100010110=>2
100010111=>2
100011000=>2
100011001=>2
100011010=>2
100011011=>2
100011100=>2
100011101=>2
100011110=>2
100011111=>2
100100000=>0
100100001=>0
100100010=>0
100100011=>0
100100100=>3
100100101=>3
100100110=>3
100100111=>3
100101000=>0
100101001=>4
100101010=>0
100101011=>0
100101100=>0
100101101=>0
100101110=>0
100101111=>0
100110000=>0
100110001=>0
100110010=>0
100110011=>0
100110100=>0
100110101=>0
100110110=>0
100110111=>0
100111000=>0
100111001=>4
100111010=>0
100111011=>0
100111100=>0
100111101=>0
100111110=>0
100111111=>0
101000000=>1
101000001=>1
101000010=>1
101000011=>1
101000100=>1
101000101=>4
101000110=>1
101000111=>1
101001000=>1
101001001=>1
101001010=>1
101001011=>1
101001100=>1
101001101=>1
101001110=>1
101001111=>1
101010000=>2
101010001=>2
101010010=>2
101010011=>2
101010100=>3
101010101=>4
101010110=>3
101010111=>3
101011000=>2
101011001=>2
101011010=>2
101011011=>2
101011100=>2
101011101=>2
101011110=>2
101011111=>2
101100000=>1
101100001=>1
101100010=>1
101100011=>1
101100100=>1
101100101=>1
101100110=>1
101100111=>1
101101000=>1
101101001=>1
101101010=>1
101101011=>1
101101100=>1
101101101=>4
101101110=>1
101101111=>1
101110000=>1
101110001=>1
101110010=>1
101110011=>1
101110100=>3
101110101=>3
101110110=>3
101110111=>3
101111000=>1
101111001=>1
101111010=>1
101111011=>1
101111100=>1
101111101=>4
101111110=>1
101111111=>1
110000000=>0
110000001=>0
110000010=>0
110000011=>4
110000100=>0
110000101=>0
110000110=>0
110000111=>0
110001000=>0
110001001=>0
110001010=>0
110001011=>0
110001100=>3
110001101=>3
110001110=>3
110001111=>3
110010000=>0
110010001=>0
110010010=>0
110010011=>4
110010100=>0
110010101=>0
110010110=>0
110010111=>0
110011000=>0
110011001=>0
110011010=>0
110011011=>0
110011100=>0
110011101=>0
110011110=>0
110011111=>0
110100000=>0
110100001=>0
110100010=>0
110100011=>0
110100100=>0
110100101=>0
110100110=>0
110100111=>0
110101000=>0
110101001=>0
110101010=>0
110101011=>4
110101100=>3
110101101=>3
110101110=>3
110101111=>3
110110000=>2
110110001=>2
110110010=>2
110110011=>2
110110100=>2
110110101=>2
110110110=>2
110110111=>2
110111000=>2
110111001=>2
110111010=>2
110111011=>4
110111100=>2
110111101=>2
110111110=>2
110111111=>2
111000000=>1
111000001=>1
111000010=>1
111000011=>1
111000100=>1
111000101=>1
111000110=>1
111000111=>4
111001000=>1
111001001=>1
111001010=>1
111001011=>1
111001100=>1
111001101=>1
111001110=>1
111001111=>1
111010000=>1
111010001=>1
111010010=>1
111010011=>1
111010100=>1
111010101=>1
111010110=>1
111010111=>4
111011000=>1
111011001=>1
111011010=>1
111011011=>1
111011100=>3
111011101=>3
111011110=>3
111011111=>3
111100000=>1
111100001=>1
111100010=>1
111100011=>1
111100100=>1
111100101=>1
111100110=>1
111100111=>1
111101000=>1
111101001=>1
111101010=>1
111101011=>1
111101100=>1
111101101=>1
111101110=>1
111101111=>4
111110000=>2
111110001=>2
111110010=>2
111110011=>2
111110100=>2
111110101=>2
111110110=>2
111110111=>2
111111000=>2
111111001=>2
111111010=>2
111111011=>2
111111100=>3
111111101=>3
111111110=>3
111111111=>4
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Description
Half the length of the longest odd length palindromic prefix of a binary word.
More precisely, this statistic is the largest number $k$ such that the word has a palindromic prefix of length $2k+1$.
More precisely, this statistic is the largest number $k$ such that the word has a palindromic prefix of length $2k+1$.
References
[1] Gabric, D., Shallit, J. Borders, Palindrome Prefixes, and Square Prefixes arXiv:1906.03689
Code
def statistic(w): return max((k-1)//2 for k in range(1,len(w)+1,2) if w[:k] == w[:k][::-1])
Created
Jun 11, 2019 at 07:28 by Martin Rubey
Updated
Mar 05, 2020 at 14:09 by Martin Rubey
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