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* www.FindStat.org - The Combinatorial Statistic Finder *
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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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-----------------------------------------------------------------------------
Statistic identifier: St001838
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
Description: The number of nonempty primitive factors of a binary word.
A word $u$ is a factor of a word $w$ if $w = p u s$ for words $p$ and $s$. A word is primitive, if it is not of the form $u^k$ for a word $u$ and an integer $k\geq 2$.
Apparently, the maximal number of nonempty primitive factors a binary word of length $n$ can have is given by [[oeis:A131673]].
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(w):
return sum(1 for f in w.factor_iterator() if f and f.is_primitive())
-----------------------------------------------------------------------------
Statistic values:
0 => 1
1 => 1
00 => 1
01 => 3
10 => 3
11 => 1
000 => 1
001 => 4
010 => 5
011 => 4
100 => 4
101 => 5
110 => 4
111 => 1
0000 => 1
0001 => 5
0010 => 7
0011 => 6
0100 => 7
0101 => 6
0110 => 7
0111 => 5
1000 => 5
1001 => 7
1010 => 6
1011 => 7
1100 => 6
1101 => 7
1110 => 5
1111 => 1
00000 => 1
00001 => 6
00010 => 9
00011 => 8
00100 => 10
00101 => 9
00110 => 10
00111 => 8
01000 => 9
01001 => 10
01010 => 7
01011 => 9
01100 => 10
01101 => 10
01110 => 9
01111 => 6
10000 => 6
10001 => 9
10010 => 10
10011 => 10
10100 => 9
10101 => 7
10110 => 10
10111 => 9
11000 => 8
11001 => 10
11010 => 9
11011 => 10
11100 => 8
11101 => 9
11110 => 6
11111 => 1
000000 => 1
000001 => 7
000010 => 11
000011 => 10
000100 => 13
000101 => 12
000110 => 13
000111 => 11
001000 => 13
001001 => 12
001010 => 11
001011 => 13
001100 => 14
001101 => 14
001110 => 13
001111 => 10
010000 => 11
010001 => 13
010010 => 12
010011 => 14
010100 => 11
010101 => 8
010110 => 13
010111 => 12
011000 => 13
011001 => 14
011010 => 13
011011 => 12
011100 => 13
011101 => 13
011110 => 11
011111 => 7
100000 => 7
100001 => 11
100010 => 13
100011 => 13
100100 => 12
100101 => 13
100110 => 14
100111 => 13
101000 => 12
101001 => 13
101010 => 8
101011 => 11
101100 => 14
101101 => 12
101110 => 13
101111 => 11
110000 => 10
110001 => 13
110010 => 14
110011 => 14
110100 => 13
110101 => 11
110110 => 12
110111 => 13
111000 => 11
111001 => 13
111010 => 12
111011 => 13
111100 => 10
111101 => 11
111110 => 7
111111 => 1
0000000 => 1
0000001 => 8
0000010 => 13
0000011 => 12
0000100 => 16
0000101 => 15
0000110 => 16
0000111 => 14
0001000 => 17
0001001 => 16
0001010 => 15
0001011 => 17
0001100 => 18
0001101 => 18
0001110 => 17
0001111 => 14
0010000 => 16
0010001 => 17
0010010 => 14
0010011 => 17
0010100 => 16
0010101 => 13
0010110 => 18
0010111 => 17
0011000 => 18
0011001 => 18
0011010 => 18
0011011 => 17
0011100 => 18
0011101 => 18
0011110 => 16
0011111 => 12
0100000 => 13
0100001 => 16
0100010 => 17
0100011 => 18
0100100 => 14
0100101 => 16
0100110 => 19
0100111 => 18
0101000 => 15
0101001 => 16
0101010 => 9
0101011 => 13
0101100 => 18
0101101 => 16
0101110 => 17
0101111 => 15
0110000 => 16
0110001 => 18
0110010 => 19
0110011 => 18
0110100 => 18
0110101 => 16
0110110 => 14
0110111 => 16
0111000 => 17
0111001 => 18
0111010 => 17
0111011 => 17
0111100 => 16
0111101 => 16
0111110 => 13
0111111 => 8
1000000 => 8
1000001 => 13
1000010 => 16
1000011 => 16
1000100 => 17
1000101 => 17
1000110 => 18
1000111 => 17
1001000 => 16
1001001 => 14
1001010 => 16
1001011 => 18
1001100 => 18
1001101 => 19
1001110 => 18
1001111 => 16
1010000 => 15
1010001 => 17
1010010 => 16
1010011 => 18
1010100 => 13
1010101 => 9
1010110 => 16
1010111 => 15
1011000 => 18
1011001 => 19
1011010 => 16
1011011 => 14
1011100 => 18
1011101 => 17
1011110 => 16
1011111 => 13
1100000 => 12
1100001 => 16
1100010 => 18
1100011 => 18
1100100 => 17
1100101 => 18
1100110 => 18
1100111 => 18
1101000 => 17
1101001 => 18
1101010 => 13
1101011 => 16
1101100 => 17
1101101 => 14
1101110 => 17
1101111 => 16
1110000 => 14
1110001 => 17
1110010 => 18
1110011 => 18
1110100 => 17
1110101 => 15
1110110 => 16
1110111 => 17
1111000 => 14
1111001 => 16
1111010 => 15
1111011 => 16
1111100 => 12
1111101 => 13
1111110 => 8
1111111 => 1
00000000 => 1
00000001 => 9
00000010 => 15
00000011 => 14
00000100 => 19
00000101 => 18
00000110 => 19
00000111 => 17
00001000 => 21
00001001 => 20
00001010 => 19
00001011 => 21
00001100 => 22
00001101 => 22
00001110 => 21
00001111 => 18
00010000 => 21
00010001 => 20
00010010 => 19
00010011 => 22
00010100 => 21
00010101 => 18
00010110 => 23
00010111 => 22
00011000 => 23
00011001 => 23
00011010 => 23
00011011 => 22
00011100 => 23
00011101 => 23
00011110 => 21
00011111 => 17
00100000 => 19
00100001 => 21
00100010 => 20
00100011 => 23
00100100 => 16
00100101 => 19
00100110 => 23
00100111 => 22
00101000 => 21
00101001 => 21
00101010 => 15
00101011 => 19
00101100 => 24
00101101 => 22
00101110 => 23
00101111 => 21
00110000 => 22
00110001 => 23
00110010 => 24
00110011 => 21
00110100 => 24
00110101 => 22
00110110 => 20
00110111 => 22
00111000 => 23
00111001 => 23
00111010 => 23
00111011 => 23
00111100 => 22
00111101 => 22
00111110 => 19
00111111 => 14
01000000 => 15
01000001 => 19
01000010 => 21
01000011 => 22
01000100 => 20
01000101 => 22
01000110 => 24
01000111 => 23
01001000 => 19
01001001 => 16
01001010 => 20
01001011 => 22
01001100 => 24
01001101 => 25
01001110 => 24
01001111 => 22
01010000 => 19
01010001 => 21
01010010 => 20
01010011 => 22
01010100 => 15
01010101 => 10
01010110 => 19
01010111 => 18
01011000 => 23
01011001 => 24
01011010 => 20
01011011 => 19
01011100 => 23
01011101 => 22
01011110 => 21
01011111 => 18
01100000 => 19
01100001 => 22
01100010 => 24
01100011 => 23
01100100 => 23
01100101 => 24
01100110 => 21
01100111 => 23
01101000 => 23
01101001 => 24
01101010 => 19
01101011 => 21
01101100 => 20
01101101 => 16
01101110 => 21
01101111 => 20
01110000 => 21
01110001 => 23
01110010 => 24
01110011 => 23
01110100 => 23
01110101 => 21
01110110 => 21
01110111 => 20
01111000 => 21
01111001 => 22
01111010 => 21
01111011 => 21
01111100 => 19
01111101 => 19
01111110 => 15
01111111 => 9
10000000 => 9
10000001 => 15
10000010 => 19
10000011 => 19
10000100 => 21
10000101 => 21
10000110 => 22
10000111 => 21
10001000 => 20
10001001 => 21
10001010 => 21
10001011 => 23
10001100 => 23
10001101 => 24
10001110 => 23
10001111 => 21
10010000 => 20
10010001 => 21
10010010 => 16
10010011 => 20
10010100 => 21
10010101 => 19
10010110 => 24
10010111 => 23
10011000 => 23
10011001 => 21
10011010 => 24
10011011 => 23
10011100 => 23
10011101 => 24
10011110 => 22
10011111 => 19
10100000 => 18
10100001 => 21
10100010 => 22
10100011 => 23
10100100 => 19
10100101 => 20
10100110 => 24
10100111 => 23
10101000 => 18
10101001 => 19
10101010 => 10
10101011 => 15
10101100 => 22
10101101 => 20
10101110 => 21
10101111 => 19
10110000 => 22
10110001 => 24
10110010 => 25
10110011 => 24
10110100 => 22
10110101 => 20
10110110 => 16
10110111 => 19
10111000 => 23
10111001 => 24
10111010 => 22
10111011 => 20
10111100 => 22
10111101 => 21
10111110 => 19
10111111 => 15
11000000 => 14
11000001 => 19
11000010 => 22
11000011 => 22
11000100 => 23
11000101 => 23
11000110 => 23
11000111 => 23
11001000 => 22
11001001 => 20
11001010 => 22
11001011 => 24
11001100 => 21
11001101 => 24
11001110 => 23
11001111 => 22
11010000 => 21
11010001 => 23
11010010 => 22
11010011 => 24
11010100 => 19
11010101 => 15
11010110 => 21
11010111 => 21
11011000 => 22
11011001 => 23
11011010 => 19
11011011 => 16
11011100 => 23
11011101 => 20
11011110 => 21
11011111 => 19
11100000 => 17
11100001 => 21
11100010 => 23
11100011 => 23
11100100 => 22
11100101 => 23
11100110 => 23
11100111 => 23
11101000 => 22
11101001 => 23
11101010 => 18
11101011 => 21
11101100 => 22
11101101 => 19
11101110 => 20
11101111 => 21
11110000 => 18
11110001 => 21
11110010 => 22
11110011 => 22
11110100 => 21
11110101 => 19
11110110 => 20
11110111 => 21
11111000 => 17
11111001 => 19
11111010 => 18
11111011 => 19
11111100 => 14
11111101 => 15
11111110 => 9
11111111 => 1
000000000 => 1
000000001 => 10
000000010 => 17
000000011 => 16
000000100 => 22
000000101 => 21
000000110 => 22
000000111 => 20
000001000 => 25
000001001 => 24
000001010 => 23
000001011 => 25
000001100 => 26
000001101 => 26
000001110 => 25
000001111 => 22
000010000 => 26
000010001 => 25
000010010 => 24
000010011 => 27
000010100 => 26
000010101 => 23
000010110 => 28
000010111 => 27
000011000 => 28
000011001 => 28
000011010 => 28
000011011 => 27
000011100 => 28
000011101 => 28
000011110 => 26
000011111 => 22
000100000 => 25
000100001 => 26
000100010 => 23
000100011 => 27
000100100 => 22
000100101 => 25
000100110 => 29
000100111 => 28
000101000 => 27
000101001 => 27
000101010 => 21
000101011 => 25
000101100 => 30
000101101 => 28
000101110 => 29
000101111 => 27
000110000 => 28
000110001 => 28
000110010 => 30
000110011 => 27
000110100 => 30
000110101 => 28
000110110 => 26
000110111 => 28
000111000 => 29
000111001 => 29
000111010 => 29
000111011 => 29
000111100 => 28
000111101 => 28
000111110 => 25
000111111 => 20
001000000 => 22
001000001 => 25
001000010 => 26
001000011 => 28
001000100 => 23
001000101 => 26
001000110 => 30
001000111 => 29
001001000 => 22
001001001 => 18
001001010 => 24
001001011 => 26
001001100 => 29
001001101 => 30
001001110 => 29
001001111 => 27
001010000 => 26
001010001 => 27
001010010 => 25
001010011 => 28
001010100 => 22
001010101 => 17
001010110 => 26
001010111 => 25
001011000 => 30
001011001 => 30
001011010 => 27
001011011 => 26
001011100 => 30
001011101 => 29
001011110 => 28
001011111 => 25
001100000 => 26
001100001 => 28
001100010 => 30
001100011 => 28
001100100 => 29
001100101 => 30
001100110 => 24
001100111 => 27
001101000 => 30
001101001 => 30
001101010 => 26
001101011 => 28
001101100 => 27
001101101 => 23
001101110 => 28
001101111 => 27
001110000 => 28
001110001 => 29
001110010 => 30
001110011 => 28
001110100 => 30
001110101 => 28
001110110 => 28
001110111 => 27
001111000 => 28
001111001 => 28
001111010 => 28
001111011 => 28
001111100 => 26
001111101 => 26
001111110 => 22
001111111 => 16
010000000 => 17
010000001 => 22
010000010 => 25
010000011 => 26
010000100 => 26
010000101 => 27
010000110 => 29
010000111 => 28
010001000 => 23
010001001 => 25
010001010 => 27
010001011 => 29
010001100 => 30
010001101 => 31
010001110 => 30
010001111 => 28
010010000 => 24
010010001 => 25
010010010 => 18
010010011 => 23
010010100 => 25
010010101 => 24
010010110 => 29
010010111 => 28
010011000 => 30
010011001 => 28
010011010 => 30
010011011 => 30
010011100 => 30
010011101 => 31
010011110 => 29
010011111 => 26
010100000 => 23
010100001 => 26
010100010 => 27
010100011 => 28
010100100 => 24
010100101 => 25
010100110 => 29
010100111 => 28
010101000 => 21
010101001 => 22
010101010 => 11
010101011 => 17
010101100 => 26
010101101 => 24
010101110 => 25
010101111 => 23
010110000 => 28
010110001 => 30
010110010 => 30
010110011 => 30
010110100 => 27
010110101 => 25
010110110 => 22
010110111 => 25
010111000 => 29
010111001 => 30
010111010 => 27
010111011 => 26
010111100 => 28
010111101 => 27
010111110 => 25
010111111 => 21
011000000 => 22
011000001 => 26
011000010 => 29
011000011 => 28
011000100 => 30
011000101 => 30
011000110 => 28
011000111 => 29
011001000 => 29
011001001 => 27
011001010 => 29
011001011 => 30
011001100 => 24
011001101 => 28
011001110 => 29
011001111 => 28
011010000 => 28
011010001 => 30
011010010 => 29
011010011 => 30
011010100 => 26
011010101 => 22
011010110 => 26
011010111 => 27
011011000 => 26
011011001 => 27
011011010 => 22
011011011 => 18
011011100 => 28
011011101 => 25
011011110 => 26
011011111 => 24
011100000 => 25
011100001 => 28
011100010 => 30
011100011 => 29
011100100 => 29
011100101 => 30
011100110 => 29
011100111 => 28
011101000 => 29
011101001 => 30
011101010 => 25
011101011 => 27
011101100 => 28
011101101 => 25
011101110 => 23
011101111 => 25
011110000 => 26
011110001 => 28
011110010 => 29
011110011 => 28
011110100 => 28
011110101 => 26
011110110 => 26
011110111 => 26
011111000 => 25
011111001 => 26
011111010 => 25
011111011 => 25
011111100 => 22
011111101 => 22
011111110 => 17
011111111 => 10
100000000 => 10
100000001 => 17
100000010 => 22
100000011 => 22
100000100 => 25
100000101 => 25
100000110 => 26
100000111 => 25
100001000 => 26
100001001 => 26
100001010 => 26
100001011 => 28
100001100 => 28
100001101 => 29
100001110 => 28
100001111 => 26
100010000 => 25
100010001 => 23
100010010 => 25
100010011 => 28
100010100 => 27
100010101 => 25
100010110 => 30
100010111 => 29
100011000 => 28
100011001 => 29
100011010 => 30
100011011 => 29
100011100 => 29
100011101 => 30
100011110 => 28
100011111 => 25
100100000 => 24
100100001 => 26
100100010 => 25
100100011 => 28
100100100 => 18
100100101 => 22
100100110 => 27
100100111 => 26
100101000 => 27
100101001 => 26
100101010 => 22
100101011 => 26
100101100 => 30
100101101 => 29
100101110 => 30
100101111 => 28
100110000 => 28
100110001 => 29
100110010 => 28
100110011 => 24
100110100 => 30
100110101 => 29
100110110 => 27
100110111 => 29
100111000 => 29
100111001 => 28
100111010 => 30
100111011 => 30
100111100 => 28
100111101 => 29
100111110 => 26
100111111 => 22
101000000 => 21
101000001 => 25
101000010 => 27
101000011 => 28
101000100 => 26
101000101 => 27
101000110 => 30
101000111 => 29
101001000 => 25
101001001 => 22
101001010 => 25
101001011 => 27
101001100 => 30
101001101 => 30
101001110 => 30
101001111 => 28
101010000 => 23
101010001 => 25
101010010 => 24
101010011 => 26
101010100 => 17
101010101 => 11
101010110 => 22
101010111 => 21
101011000 => 28
101011001 => 29
101011010 => 25
101011011 => 24
101011100 => 28
101011101 => 27
101011110 => 26
101011111 => 23
101100000 => 26
101100001 => 29
101100010 => 31
101100011 => 30
101100100 => 30
101100101 => 30
101100110 => 28
101100111 => 30
101101000 => 28
101101001 => 29
101101010 => 24
101101011 => 25
101101100 => 23
101101101 => 18
101101110 => 25
101101111 => 24
101110000 => 28
101110001 => 30
101110010 => 31
101110011 => 30
101110100 => 29
101110101 => 27
101110110 => 25
101110111 => 23
101111000 => 28
101111001 => 29
101111010 => 27
101111011 => 26
101111100 => 26
101111101 => 25
101111110 => 22
101111111 => 17
110000000 => 16
110000001 => 22
110000010 => 26
110000011 => 26
110000100 => 28
110000101 => 28
110000110 => 28
110000111 => 28
110001000 => 27
110001001 => 28
110001010 => 28
110001011 => 30
110001100 => 28
110001101 => 30
110001110 => 29
110001111 => 28
110010000 => 27
110010001 => 28
110010010 => 23
110010011 => 27
110010100 => 28
110010101 => 26
110010110 => 30
110010111 => 30
110011000 => 27
110011001 => 24
110011010 => 30
110011011 => 29
110011100 => 28
110011101 => 30
110011110 => 28
110011111 => 26
110100000 => 25
110100001 => 28
110100010 => 29
110100011 => 30
110100100 => 26
110100101 => 27
110100110 => 30
110100111 => 30
110101000 => 25
110101001 => 26
110101010 => 17
110101011 => 22
110101100 => 28
110101101 => 25
110101110 => 27
110101111 => 26
110110000 => 27
110110001 => 29
110110010 => 30
110110011 => 29
110110100 => 26
110110101 => 24
110110110 => 18
110110111 => 22
110111000 => 29
110111001 => 30
110111010 => 26
110111011 => 23
110111100 => 28
110111101 => 26
110111110 => 25
110111111 => 22
111000000 => 20
111000001 => 25
111000010 => 28
111000011 => 28
111000100 => 29
111000101 => 29
111000110 => 29
111000111 => 29
111001000 => 28
111001001 => 26
111001010 => 28
111001011 => 30
111001100 => 27
111001101 => 30
111001110 => 28
111001111 => 28
111010000 => 27
111010001 => 29
111010010 => 28
111010011 => 30
111010100 => 25
111010101 => 21
111010110 => 27
111010111 => 27
111011000 => 28
111011001 => 29
111011010 => 25
111011011 => 22
111011100 => 27
111011101 => 23
111011110 => 26
111011111 => 25
111100000 => 22
111100001 => 26
111100010 => 28
111100011 => 28
111100100 => 27
111100101 => 28
111100110 => 28
111100111 => 28
111101000 => 27
111101001 => 28
111101010 => 23
111101011 => 26
111101100 => 27
111101101 => 24
111101110 => 25
111101111 => 26
111110000 => 22
111110001 => 25
111110010 => 26
111110011 => 26
111110100 => 25
111110101 => 23
111110110 => 24
111110111 => 25
111111000 => 20
111111001 => 22
111111010 => 21
111111011 => 22
111111100 => 16
111111101 => 17
111111110 => 10
111111111 => 1
-----------------------------------------------------------------------------
Created: Sep 16, 2022 at 10:06 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Sep 16, 2022 at 10:06 by Martin Rubey