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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
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-----------------------------------------------------------------------------
Statistic identifier: St000792
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
Description: The Grundy value for the game of ruler on a binary word.
Two players alternately may switch any consecutive sequence of numbers that ends with a 1. The player facing the word which has only 0's looses.
-----------------------------------------------------------------------------
References: [1] Lugo, M. Sprague-Grundy sequence for the ruler game [[MathOverflow:39165]]
-----------------------------------------------------------------------------
Code:
def statistic(w):
two_adic = Zp(2)
res = 0
for i, e in enumerate(w):
if e == 1:
res = res^^(2^ZZ((two_adic(i+1).ordp())))
return res
-----------------------------------------------------------------------------
Statistic values:
0 => 0
1 => 1
00 => 0
01 => 2
10 => 1
11 => 3
000 => 0
001 => 1
010 => 2
011 => 3
100 => 1
101 => 0
110 => 3
111 => 2
0000 => 0
0001 => 4
0010 => 1
0011 => 5
0100 => 2
0101 => 6
0110 => 3
0111 => 7
1000 => 1
1001 => 5
1010 => 0
1011 => 4
1100 => 3
1101 => 7
1110 => 2
1111 => 6
00000 => 0
00001 => 1
00010 => 4
00011 => 5
00100 => 1
00101 => 0
00110 => 5
00111 => 4
01000 => 2
01001 => 3
01010 => 6
01011 => 7
01100 => 3
01101 => 2
01110 => 7
01111 => 6
10000 => 1
10001 => 0
10010 => 5
10011 => 4
10100 => 0
10101 => 1
10110 => 4
10111 => 5
11000 => 3
11001 => 2
11010 => 7
11011 => 6
11100 => 2
11101 => 3
11110 => 6
11111 => 7
000000 => 0
000001 => 2
000010 => 1
000011 => 3
000100 => 4
000101 => 6
000110 => 5
000111 => 7
001000 => 1
001001 => 3
001010 => 0
001011 => 2
001100 => 5
001101 => 7
001110 => 4
001111 => 6
010000 => 2
010001 => 0
010010 => 3
010011 => 1
010100 => 6
010101 => 4
010110 => 7
010111 => 5
011000 => 3
011001 => 1
011010 => 2
011011 => 0
011100 => 7
011101 => 5
011110 => 6
011111 => 4
100000 => 1
100001 => 3
100010 => 0
100011 => 2
100100 => 5
100101 => 7
100110 => 4
100111 => 6
101000 => 0
101001 => 2
101010 => 1
101011 => 3
101100 => 4
101101 => 6
101110 => 5
101111 => 7
110000 => 3
110001 => 1
110010 => 2
110011 => 0
110100 => 7
110101 => 5
110110 => 6
110111 => 4
111000 => 2
111001 => 0
111010 => 3
111011 => 1
111100 => 6
111101 => 4
111110 => 7
111111 => 5
0000000 => 0
0000001 => 1
0000010 => 2
0000011 => 3
0000100 => 1
0000101 => 0
0000110 => 3
0000111 => 2
0001000 => 4
0001001 => 5
0001010 => 6
0001011 => 7
0001100 => 5
0001101 => 4
0001110 => 7
0001111 => 6
0010000 => 1
0010001 => 0
0010010 => 3
0010011 => 2
0010100 => 0
0010101 => 1
0010110 => 2
0010111 => 3
0011000 => 5
0011001 => 4
0011010 => 7
0011011 => 6
0011100 => 4
0011101 => 5
0011110 => 6
0011111 => 7
0100000 => 2
0100001 => 3
0100010 => 0
0100011 => 1
0100100 => 3
0100101 => 2
0100110 => 1
0100111 => 0
0101000 => 6
0101001 => 7
0101010 => 4
0101011 => 5
0101100 => 7
0101101 => 6
0101110 => 5
0101111 => 4
0110000 => 3
0110001 => 2
0110010 => 1
0110011 => 0
0110100 => 2
0110101 => 3
0110110 => 0
0110111 => 1
0111000 => 7
0111001 => 6
0111010 => 5
0111011 => 4
0111100 => 6
0111101 => 7
0111110 => 4
0111111 => 5
1000000 => 1
1000001 => 0
1000010 => 3
1000011 => 2
1000100 => 0
1000101 => 1
1000110 => 2
1000111 => 3
1001000 => 5
1001001 => 4
1001010 => 7
1001011 => 6
1001100 => 4
1001101 => 5
1001110 => 6
1001111 => 7
1010000 => 0
1010001 => 1
1010010 => 2
1010011 => 3
1010100 => 1
1010101 => 0
1010110 => 3
1010111 => 2
1011000 => 4
1011001 => 5
1011010 => 6
1011011 => 7
1011100 => 5
1011101 => 4
1011110 => 7
1011111 => 6
1100000 => 3
1100001 => 2
1100010 => 1
1100011 => 0
1100100 => 2
1100101 => 3
1100110 => 0
1100111 => 1
1101000 => 7
1101001 => 6
1101010 => 5
1101011 => 4
1101100 => 6
1101101 => 7
1101110 => 4
1101111 => 5
1110000 => 2
1110001 => 3
1110010 => 0
1110011 => 1
1110100 => 3
1110101 => 2
1110110 => 1
1110111 => 0
1111000 => 6
1111001 => 7
1111010 => 4
1111011 => 5
1111100 => 7
1111101 => 6
1111110 => 5
1111111 => 4
00000000 => 0
00000001 => 8
00000010 => 1
00000011 => 9
00000100 => 2
00000101 => 10
00000110 => 3
00000111 => 11
00001000 => 1
00001001 => 9
00001010 => 0
00001011 => 8
00001100 => 3
00001101 => 11
00001110 => 2
00001111 => 10
00010000 => 4
00010001 => 12
00010010 => 5
00010011 => 13
00010100 => 6
00010101 => 14
00010110 => 7
00010111 => 15
00011000 => 5
00011001 => 13
00011010 => 4
00011011 => 12
00011100 => 7
00011101 => 15
00011110 => 6
00011111 => 14
00100000 => 1
00100001 => 9
00100010 => 0
00100011 => 8
00100100 => 3
00100101 => 11
00100110 => 2
00100111 => 10
00101000 => 0
00101001 => 8
00101010 => 1
00101011 => 9
00101100 => 2
00101101 => 10
00101110 => 3
00101111 => 11
00110000 => 5
00110001 => 13
00110010 => 4
00110011 => 12
00110100 => 7
00110101 => 15
00110110 => 6
00110111 => 14
00111000 => 4
00111001 => 12
00111010 => 5
00111011 => 13
00111100 => 6
00111101 => 14
00111110 => 7
00111111 => 15
01000000 => 2
01000001 => 10
01000010 => 3
01000011 => 11
01000100 => 0
01000101 => 8
01000110 => 1
01000111 => 9
01001000 => 3
01001001 => 11
01001010 => 2
01001011 => 10
01001100 => 1
01001101 => 9
01001110 => 0
01001111 => 8
01010000 => 6
01010001 => 14
01010010 => 7
01010011 => 15
01010100 => 4
01010101 => 12
01010110 => 5
01010111 => 13
01011000 => 7
01011001 => 15
01011010 => 6
01011011 => 14
01011100 => 5
01011101 => 13
01011110 => 4
01011111 => 12
01100000 => 3
01100001 => 11
01100010 => 2
01100011 => 10
01100100 => 1
01100101 => 9
01100110 => 0
01100111 => 8
01101000 => 2
01101001 => 10
01101010 => 3
01101011 => 11
01101100 => 0
01101101 => 8
01101110 => 1
01101111 => 9
01110000 => 7
01110001 => 15
01110010 => 6
01110011 => 14
01110100 => 5
01110101 => 13
01110110 => 4
01110111 => 12
01111000 => 6
01111001 => 14
01111010 => 7
01111011 => 15
01111100 => 4
01111101 => 12
01111110 => 5
01111111 => 13
10000000 => 1
10000001 => 9
10000010 => 0
10000011 => 8
10000100 => 3
10000101 => 11
10000110 => 2
10000111 => 10
10001000 => 0
10001001 => 8
10001010 => 1
10001011 => 9
10001100 => 2
10001101 => 10
10001110 => 3
10001111 => 11
10010000 => 5
10010001 => 13
10010010 => 4
10010011 => 12
10010100 => 7
10010101 => 15
10010110 => 6
10010111 => 14
10011000 => 4
10011001 => 12
10011010 => 5
10011011 => 13
10011100 => 6
10011101 => 14
10011110 => 7
10011111 => 15
10100000 => 0
10100001 => 8
10100010 => 1
10100011 => 9
10100100 => 2
10100101 => 10
10100110 => 3
10100111 => 11
10101000 => 1
10101001 => 9
10101010 => 0
10101011 => 8
10101100 => 3
10101101 => 11
10101110 => 2
10101111 => 10
10110000 => 4
10110001 => 12
10110010 => 5
10110011 => 13
10110100 => 6
10110101 => 14
10110110 => 7
10110111 => 15
10111000 => 5
10111001 => 13
10111010 => 4
10111011 => 12
10111100 => 7
10111101 => 15
10111110 => 6
10111111 => 14
11000000 => 3
11000001 => 11
11000010 => 2
11000011 => 10
11000100 => 1
11000101 => 9
11000110 => 0
11000111 => 8
11001000 => 2
11001001 => 10
11001010 => 3
11001011 => 11
11001100 => 0
11001101 => 8
11001110 => 1
11001111 => 9
11010000 => 7
11010001 => 15
11010010 => 6
11010011 => 14
11010100 => 5
11010101 => 13
11010110 => 4
11010111 => 12
11011000 => 6
11011001 => 14
11011010 => 7
11011011 => 15
11011100 => 4
11011101 => 12
11011110 => 5
11011111 => 13
11100000 => 2
11100001 => 10
11100010 => 3
11100011 => 11
11100100 => 0
11100101 => 8
11100110 => 1
11100111 => 9
11101000 => 3
11101001 => 11
11101010 => 2
11101011 => 10
11101100 => 1
11101101 => 9
11101110 => 0
11101111 => 8
11110000 => 6
11110001 => 14
11110010 => 7
11110011 => 15
11110100 => 4
11110101 => 12
11110110 => 5
11110111 => 13
11111000 => 7
11111001 => 15
11111010 => 6
11111011 => 14
11111100 => 5
11111101 => 13
11111110 => 4
11111111 => 12
000000000 => 0
000000001 => 1
000000010 => 8
000000011 => 9
000000100 => 1
000000101 => 0
000000110 => 9
000000111 => 8
000001000 => 2
000001001 => 3
000001010 => 10
000001011 => 11
000001100 => 3
000001101 => 2
000001110 => 11
000001111 => 10
000010000 => 1
000010001 => 0
000010010 => 9
000010011 => 8
000010100 => 0
000010101 => 1
000010110 => 8
000010111 => 9
000011000 => 3
000011001 => 2
000011010 => 11
000011011 => 10
000011100 => 2
000011101 => 3
000011110 => 10
000011111 => 11
000100000 => 4
000100001 => 5
000100010 => 12
000100011 => 13
000100100 => 5
000100101 => 4
000100110 => 13
000100111 => 12
000101000 => 6
000101001 => 7
000101010 => 14
000101011 => 15
000101100 => 7
000101101 => 6
000101110 => 15
000101111 => 14
000110000 => 5
000110001 => 4
000110010 => 13
000110011 => 12
000110100 => 4
000110101 => 5
000110110 => 12
000110111 => 13
000111000 => 7
000111001 => 6
000111010 => 15
000111011 => 14
000111100 => 6
000111101 => 7
000111110 => 14
000111111 => 15
001000000 => 1
001000001 => 0
001000010 => 9
001000011 => 8
001000100 => 0
001000101 => 1
001000110 => 8
001000111 => 9
001001000 => 3
001001001 => 2
001001010 => 11
001001011 => 10
001001100 => 2
001001101 => 3
001001110 => 10
001001111 => 11
001010000 => 0
001010001 => 1
001010010 => 8
001010011 => 9
001010100 => 1
001010101 => 0
001010110 => 9
001010111 => 8
001011000 => 2
001011001 => 3
001011010 => 10
001011011 => 11
001011100 => 3
001011101 => 2
001011110 => 11
001011111 => 10
001100000 => 5
001100001 => 4
001100010 => 13
001100011 => 12
001100100 => 4
001100101 => 5
001100110 => 12
001100111 => 13
001101000 => 7
001101001 => 6
001101010 => 15
001101011 => 14
001101100 => 6
001101101 => 7
001101110 => 14
001101111 => 15
001110000 => 4
001110001 => 5
001110010 => 12
001110011 => 13
001110100 => 5
001110101 => 4
001110110 => 13
001110111 => 12
001111000 => 6
001111001 => 7
001111010 => 14
001111011 => 15
001111100 => 7
001111101 => 6
001111110 => 15
001111111 => 14
010000000 => 2
010000001 => 3
010000010 => 10
010000011 => 11
010000100 => 3
010000101 => 2
010000110 => 11
010000111 => 10
010001000 => 0
010001001 => 1
010001010 => 8
010001011 => 9
010001100 => 1
010001101 => 0
010001110 => 9
010001111 => 8
010010000 => 3
010010001 => 2
010010010 => 11
010010011 => 10
010010100 => 2
010010101 => 3
010010110 => 10
010010111 => 11
010011000 => 1
010011001 => 0
010011010 => 9
010011011 => 8
010011100 => 0
010011101 => 1
010011110 => 8
010011111 => 9
010100000 => 6
010100001 => 7
010100010 => 14
010100011 => 15
010100100 => 7
010100101 => 6
010100110 => 15
010100111 => 14
010101000 => 4
010101001 => 5
010101010 => 12
010101011 => 13
010101100 => 5
010101101 => 4
010101110 => 13
010101111 => 12
010110000 => 7
010110001 => 6
010110010 => 15
010110011 => 14
010110100 => 6
010110101 => 7
010110110 => 14
010110111 => 15
010111000 => 5
010111001 => 4
010111010 => 13
010111011 => 12
010111100 => 4
010111101 => 5
010111110 => 12
010111111 => 13
011000000 => 3
011000001 => 2
011000010 => 11
011000011 => 10
011000100 => 2
011000101 => 3
011000110 => 10
011000111 => 11
011001000 => 1
011001001 => 0
011001010 => 9
011001011 => 8
011001100 => 0
011001101 => 1
011001110 => 8
011001111 => 9
011010000 => 2
011010001 => 3
011010010 => 10
011010011 => 11
011010100 => 3
011010101 => 2
011010110 => 11
011010111 => 10
011011000 => 0
011011001 => 1
011011010 => 8
011011011 => 9
011011100 => 1
011011101 => 0
011011110 => 9
011011111 => 8
011100000 => 7
011100001 => 6
011100010 => 15
011100011 => 14
011100100 => 6
011100101 => 7
011100110 => 14
011100111 => 15
011101000 => 5
011101001 => 4
011101010 => 13
011101011 => 12
011101100 => 4
011101101 => 5
011101110 => 12
011101111 => 13
011110000 => 6
011110001 => 7
011110010 => 14
011110011 => 15
011110100 => 7
011110101 => 6
011110110 => 15
011110111 => 14
011111000 => 4
011111001 => 5
011111010 => 12
011111011 => 13
011111100 => 5
011111101 => 4
011111110 => 13
011111111 => 12
100000000 => 1
100000001 => 0
100000010 => 9
100000011 => 8
100000100 => 0
100000101 => 1
100000110 => 8
100000111 => 9
100001000 => 3
100001001 => 2
100001010 => 11
100001011 => 10
100001100 => 2
100001101 => 3
100001110 => 10
100001111 => 11
100010000 => 0
100010001 => 1
100010010 => 8
100010011 => 9
100010100 => 1
100010101 => 0
100010110 => 9
100010111 => 8
100011000 => 2
100011001 => 3
100011010 => 10
100011011 => 11
100011100 => 3
100011101 => 2
100011110 => 11
100011111 => 10
100100000 => 5
100100001 => 4
100100010 => 13
100100011 => 12
100100100 => 4
100100101 => 5
100100110 => 12
100100111 => 13
100101000 => 7
100101001 => 6
100101010 => 15
100101011 => 14
100101100 => 6
100101101 => 7
100101110 => 14
100101111 => 15
100110000 => 4
100110001 => 5
100110010 => 12
100110011 => 13
100110100 => 5
100110101 => 4
100110110 => 13
100110111 => 12
100111000 => 6
100111001 => 7
100111010 => 14
100111011 => 15
100111100 => 7
100111101 => 6
100111110 => 15
100111111 => 14
101000000 => 0
101000001 => 1
101000010 => 8
101000011 => 9
101000100 => 1
101000101 => 0
101000110 => 9
101000111 => 8
101001000 => 2
101001001 => 3
101001010 => 10
101001011 => 11
101001100 => 3
101001101 => 2
101001110 => 11
101001111 => 10
101010000 => 1
101010001 => 0
101010010 => 9
101010011 => 8
101010100 => 0
101010101 => 1
101010110 => 8
101010111 => 9
101011000 => 3
101011001 => 2
101011010 => 11
101011011 => 10
101011100 => 2
101011101 => 3
101011110 => 10
101011111 => 11
101100000 => 4
101100001 => 5
101100010 => 12
101100011 => 13
101100100 => 5
101100101 => 4
101100110 => 13
101100111 => 12
101101000 => 6
101101001 => 7
101101010 => 14
101101011 => 15
101101100 => 7
101101101 => 6
101101110 => 15
101101111 => 14
101110000 => 5
101110001 => 4
101110010 => 13
101110011 => 12
101110100 => 4
101110101 => 5
101110110 => 12
101110111 => 13
101111000 => 7
101111001 => 6
101111010 => 15
101111011 => 14
101111100 => 6
101111101 => 7
101111110 => 14
101111111 => 15
110000000 => 3
110000001 => 2
110000010 => 11
110000011 => 10
110000100 => 2
110000101 => 3
110000110 => 10
110000111 => 11
110001000 => 1
110001001 => 0
110001010 => 9
110001011 => 8
110001100 => 0
110001101 => 1
110001110 => 8
110001111 => 9
110010000 => 2
110010001 => 3
110010010 => 10
110010011 => 11
110010100 => 3
110010101 => 2
110010110 => 11
110010111 => 10
110011000 => 0
110011001 => 1
110011010 => 8
110011011 => 9
110011100 => 1
110011101 => 0
110011110 => 9
110011111 => 8
110100000 => 7
110100001 => 6
110100010 => 15
110100011 => 14
110100100 => 6
110100101 => 7
110100110 => 14
110100111 => 15
110101000 => 5
110101001 => 4
110101010 => 13
110101011 => 12
110101100 => 4
110101101 => 5
110101110 => 12
110101111 => 13
110110000 => 6
110110001 => 7
110110010 => 14
110110011 => 15
110110100 => 7
110110101 => 6
110110110 => 15
110110111 => 14
110111000 => 4
110111001 => 5
110111010 => 12
110111011 => 13
110111100 => 5
110111101 => 4
110111110 => 13
110111111 => 12
111000000 => 2
111000001 => 3
111000010 => 10
111000011 => 11
111000100 => 3
111000101 => 2
111000110 => 11
111000111 => 10
111001000 => 0
111001001 => 1
111001010 => 8
111001011 => 9
111001100 => 1
111001101 => 0
111001110 => 9
111001111 => 8
111010000 => 3
111010001 => 2
111010010 => 11
111010011 => 10
111010100 => 2
111010101 => 3
111010110 => 10
111010111 => 11
111011000 => 1
111011001 => 0
111011010 => 9
111011011 => 8
111011100 => 0
111011101 => 1
111011110 => 8
111011111 => 9
111100000 => 6
111100001 => 7
111100010 => 14
111100011 => 15
111100100 => 7
111100101 => 6
111100110 => 15
111100111 => 14
111101000 => 4
111101001 => 5
111101010 => 12
111101011 => 13
111101100 => 5
111101101 => 4
111101110 => 13
111101111 => 12
111110000 => 7
111110001 => 6
111110010 => 15
111110011 => 14
111110100 => 6
111110101 => 7
111110110 => 14
111110111 => 15
111111000 => 5
111111001 => 4
111111010 => 13
111111011 => 12
111111100 => 4
111111101 => 5
111111110 => 12
111111111 => 13
1010101010 => 1
1010101100 => 8
1010110010 => 2
1010110100 => 11
1010111000 => 2
1011001010 => 4
1011001100 => 13
1011010010 => 7
1011010100 => 14
1011011000 => 7
1011100010 => 4
1011100100 => 13
1011101000 => 4
1011110000 => 7
1100101010 => 2
1100101100 => 11
1100110010 => 1
1100110100 => 8
1100111000 => 1
1101001010 => 7
1101001100 => 14
1101010010 => 4
1101010100 => 13
1101011000 => 4
1101100010 => 7
1101100100 => 14
1101101000 => 7
1101110000 => 4
1110001010 => 2
1110001100 => 11
1110010010 => 1
1110010100 => 8
1110011000 => 1
1110100010 => 2
1110100100 => 11
1110101000 => 2
1110110000 => 1
1111000010 => 7
1111000100 => 14
1111001000 => 7
1111010000 => 4
1111100000 => 7
1111111110 => 13
1111111101 => 14
1111000001 => 4
1110111110 => 9
1110100001 => 1
1110011110 => 8
1110010001 => 2
1110001001 => 1
1110000101 => 8
1101111110 => 12
1101100001 => 4
1101010001 => 7
1101001110 => 15
1101001001 => 4
1101000101 => 13
1100110001 => 2
1100101001 => 1
1100100110 => 11
1100100101 => 8
1100011001 => 2
1100010101 => 11
1100010010 => 0
1100001000 => 2
1011111110 => 15
1011111101 => 12
1011100001 => 7
1011010001 => 4
1011001001 => 7
1011000101 => 14
1010111110 => 11
1010110001 => 1
1010101001 => 2
1010100110 => 8
1010100101 => 11
1010011010 => 2
1010011001 => 1
1010010110 => 11
1010010101 => 8
1010001110 => 8
1001111110 => 14
1001111000 => 7
1001110100 => 14
1001110001 => 4
1001101100 => 13
1001101010 => 4
1001101001 => 7
1001100110 => 13
1001100101 => 14
1001011110 => 15
1001011010 => 7
1001011001 => 4
1001010110 => 14
1001010101 => 13
1001001110 => 13
1000111110 => 10
1000111100 => 11
1000111010 => 2
1000110110 => 11
1000101110 => 8
1000011110 => 11
1000010110 => 10
1000001110 => 9
1000001010 => 1
1000000110 => 8
1000000100 => 9
1000000010 => 0
1000000000 => 1
0111111111 => 14
0111111101 => 15
0111111001 => 7
0111110001 => 6
0111100001 => 4
0111000001 => 5
0110000001 => 1
0100000001 => 0
0000000001 => 2
0111111011 => 6
0111110101 => 14
0111101001 => 5
0111010001 => 7
0110100001 => 0
0101000001 => 4
0010000001 => 3
0101010101 => 14
0101010011 => 7
0101001101 => 13
0101001011 => 4
0101000111 => 13
0100110101 => 11
0100110011 => 2
0100101101 => 8
0100101011 => 1
0100100111 => 8
0100011101 => 11
0100011011 => 2
0100010111 => 11
0100001111 => 8
0011010101 => 13
0011010011 => 4
0011001101 => 14
0011001011 => 7
0011000111 => 14
0010110101 => 8
0010110011 => 1
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-----------------------------------------------------------------------------
Created: May 01, 2017 at 07:36 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: May 02, 2018 at 23:04 by Martin Rubey