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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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-----------------------------------------------------------------------------
Statistic identifier: St001414
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
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 $2k+1$.
-----------------------------------------------------------------------------
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])
-----------------------------------------------------------------------------
Statistic 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
0010111 => 0
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
1001001 => 3
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
000011001 => 1
000011010 => 1
000011011 => 1
000011100 => 1
000011101 => 1
000011110 => 1
000011111 => 1
000100000 => 3
000100001 => 3
000100010 => 3
000100011 => 3
000100100 => 1
000100101 => 1
000100110 => 1
000100111 => 1
000101000 => 4
000101001 => 1
000101010 => 1
000101011 => 1
000101100 => 1
000101101 => 1
000101110 => 1
000101111 => 1
000110000 => 1
000110001 => 1
000110010 => 1
000110011 => 1
000110100 => 1
000110101 => 1
000110110 => 1
000110111 => 1
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
001000111 => 2
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
001011001 => 0
001011010 => 0
001011011 => 0
001011100 => 0
001011101 => 0
001011110 => 0
001011111 => 0
001100000 => 0
001100001 => 0
001100010 => 0
001100011 => 0
001100100 => 0
001100101 => 0
001100110 => 0
001100111 => 0
001101000 => 0
001101001 => 0
001101010 => 0
001101011 => 0
001101100 => 4
001101101 => 0
001101110 => 0
001101111 => 0
001110000 => 3
001110001 => 3
001110010 => 3
001110011 => 3
001110100 => 0
001110101 => 0
001110110 => 0
001110111 => 0
001111000 => 0
001111001 => 0
001111010 => 0
001111011 => 0
001111100 => 4
001111101 => 0
001111110 => 0
001111111 => 0
010000000 => 1
010000001 => 1
010000010 => 4
010000011 => 1
010000100 => 1
010000101 => 1
010000110 => 1
010000111 => 1
010001000 => 3
010001001 => 3
010001010 => 3
010001011 => 3
010001100 => 1
010001101 => 1
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
010011011 => 1
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
011000111 => 0
011001000 => 0
011001001 => 0
011001010 => 0
011001011 => 0
011001100 => 0
011001101 => 0
011001110 => 0
011001111 => 0
011010000 => 0
011010001 => 0
011010010 => 0
011010011 => 0
011010100 => 0
011010101 => 0
011010110 => 4
011010111 => 0
011011000 => 3
011011001 => 3
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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Created: Jun 11, 2019 at 07:28 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Mar 05, 2020 at 14:09 by Martin Rubey