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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, *
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-----------------------------------------------------------------------------
Statistic identifier: St001437
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
Description: The flex of a binary word.
This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].
-----------------------------------------------------------------------------
References: [1] Ahlbach, C., Swanson, J. P. Refined cyclic sieving on words for the major index statistic [[MathSciNet:3836732]]
-----------------------------------------------------------------------------
Code:
def lex(w):
orbit = [ tuple(w[i:] + w[:i]) for i in range(len(w)) ]
return sorted(set(orbit)).index(tuple(w))+1
def freq(w):
w = list(w)
n = len(w)
for i in sorted(Integer(n).divisors()):
if w == w[:i]*(n//i):
return n // i
def statistic(w):
return lex(w)*freq(w)
-----------------------------------------------------------------------------
Statistic values:
0 => 1
1 => 1
00 => 2
01 => 1
10 => 2
11 => 2
000 => 3
001 => 1
010 => 2
011 => 1
100 => 3
101 => 2
110 => 3
111 => 3
0000 => 4
0001 => 1
0010 => 2
0011 => 1
0100 => 3
0101 => 2
0110 => 2
0111 => 1
1000 => 4
1001 => 3
1010 => 4
1011 => 2
1100 => 4
1101 => 3
1110 => 4
1111 => 4
00000 => 5
00001 => 1
00010 => 2
00011 => 1
00100 => 3
00101 => 1
00110 => 2
00111 => 1
01000 => 4
01001 => 2
01010 => 3
01011 => 1
01100 => 3
01101 => 2
01110 => 2
01111 => 1
10000 => 5
10001 => 4
10010 => 4
10011 => 3
10100 => 5
10101 => 3
10110 => 4
10111 => 2
11000 => 5
11001 => 4
11010 => 5
11011 => 3
11100 => 5
11101 => 4
11110 => 5
11111 => 5
000000 => 6
000001 => 1
000010 => 2
000011 => 1
000100 => 3
000101 => 1
000110 => 2
000111 => 1
001000 => 4
001001 => 2
001010 => 2
001011 => 1
001100 => 3
001101 => 1
001110 => 2
001111 => 1
010000 => 5
010001 => 3
010010 => 4
010011 => 2
010100 => 4
010101 => 3
010110 => 2
010111 => 1
011000 => 4
011001 => 3
011010 => 3
011011 => 2
011100 => 3
011101 => 2
011110 => 2
011111 => 1
100000 => 6
100001 => 5
100010 => 5
100011 => 4
100100 => 6
100101 => 4
100110 => 4
100111 => 3
101000 => 6
101001 => 5
101010 => 6
101011 => 3
101100 => 5
101101 => 4
101110 => 4
101111 => 2
110000 => 6
110001 => 5
110010 => 6
110011 => 4
110100 => 6
110101 => 5
110110 => 6
110111 => 3
111000 => 6
111001 => 5
111010 => 6
111011 => 4
111100 => 6
111101 => 5
111110 => 6
111111 => 6
0000000 => 7
0000001 => 1
0000010 => 2
0000011 => 1
0000100 => 3
0000101 => 1
0000110 => 2
0000111 => 1
0001000 => 4
0001001 => 1
0001010 => 2
0001011 => 1
0001100 => 3
0001101 => 1
0001110 => 2
0001111 => 1
0010000 => 5
0010001 => 2
0010010 => 3
0010011 => 1
0010100 => 3
0010101 => 1
0010110 => 2
0010111 => 1
0011000 => 4
0011001 => 2
0011010 => 2
0011011 => 1
0011100 => 3
0011101 => 1
0011110 => 2
0011111 => 1
0100000 => 6
0100001 => 4
0100010 => 4
0100011 => 3
0100100 => 5
0100101 => 2
0100110 => 3
0100111 => 2
0101000 => 5
0101001 => 3
0101010 => 4
0101011 => 1
0101100 => 3
0101101 => 2
0101110 => 2
0101111 => 1
0110000 => 5
0110001 => 4
0110010 => 4
0110011 => 2
0110100 => 4
0110101 => 3
0110110 => 3
0110111 => 1
0111000 => 4
0111001 => 3
0111010 => 3
0111011 => 2
0111100 => 3
0111101 => 2
0111110 => 2
0111111 => 1
1000000 => 7
1000001 => 6
1000010 => 6
1000011 => 5
1000100 => 6
1000101 => 5
1000110 => 5
1000111 => 4
1001000 => 7
1001001 => 5
1001010 => 5
1001011 => 4
1001100 => 6
1001101 => 4
1001110 => 4
1001111 => 3
1010000 => 7
1010001 => 6
1010010 => 6
1010011 => 5
1010100 => 7
1010101 => 4
1010110 => 5
1010111 => 3
1011000 => 6
1011001 => 5
1011010 => 6
1011011 => 3
1011100 => 5
1011101 => 4
1011110 => 4
1011111 => 2
1100000 => 7
1100001 => 6
1100010 => 7
1100011 => 5
1100100 => 7
1100101 => 6
1100110 => 6
1100111 => 4
1101000 => 7
1101001 => 6
1101010 => 7
1101011 => 5
1101100 => 7
1101101 => 5
1101110 => 6
1101111 => 3
1110000 => 7
1110001 => 6
1110010 => 7
1110011 => 5
1110100 => 7
1110101 => 6
1110110 => 7
1110111 => 4
1111000 => 7
1111001 => 6
1111010 => 7
1111011 => 5
1111100 => 7
1111101 => 6
1111110 => 7
1111111 => 7
00000000 => 8
00000001 => 1
00000010 => 2
00000011 => 1
00000100 => 3
00000101 => 1
00000110 => 2
00000111 => 1
00001000 => 4
00001001 => 1
00001010 => 2
00001011 => 1
00001100 => 3
00001101 => 1
00001110 => 2
00001111 => 1
00010000 => 5
00010001 => 2
00010010 => 2
00010011 => 1
00010100 => 3
00010101 => 1
00010110 => 2
00010111 => 1
00011000 => 4
00011001 => 1
00011010 => 2
00011011 => 1
00011100 => 3
00011101 => 1
00011110 => 2
00011111 => 1
00100000 => 6
00100001 => 3
00100010 => 4
00100011 => 2
00100100 => 4
00100101 => 1
00100110 => 2
00100111 => 1
00101000 => 4
00101001 => 2
00101010 => 2
00101011 => 1
00101100 => 3
00101101 => 1
00101110 => 2
00101111 => 1
00110000 => 5
00110001 => 3
00110010 => 3
00110011 => 2
00110100 => 3
00110101 => 1
00110110 => 2
00110111 => 1
00111000 => 4
00111001 => 2
00111010 => 2
00111011 => 1
00111100 => 3
00111101 => 1
00111110 => 2
00111111 => 1
01000000 => 7
01000001 => 5
01000010 => 5
01000011 => 4
01000100 => 6
01000101 => 3
01000110 => 4
01000111 => 3
01001000 => 6
01001001 => 3
01001010 => 4
01001011 => 2
01001100 => 4
01001101 => 2
01001110 => 3
01001111 => 2
01010000 => 6
01010001 => 4
01010010 => 5
01010011 => 3
01010100 => 5
01010101 => 4
01010110 => 2
01010111 => 1
01011000 => 4
01011001 => 3
01011010 => 3
01011011 => 1
01011100 => 3
01011101 => 2
01011110 => 2
01011111 => 1
01100000 => 6
01100001 => 5
01100010 => 5
01100011 => 3
01100100 => 5
01100101 => 4
01100110 => 4
01100111 => 2
01101000 => 5
01101001 => 4
01101010 => 4
01101011 => 2
01101100 => 4
01101101 => 3
01101110 => 2
01101111 => 1
01110000 => 5
01110001 => 4
01110010 => 4
01110011 => 3
01110100 => 4
01110101 => 3
01110110 => 3
01110111 => 2
01111000 => 4
01111001 => 3
01111010 => 3
01111011 => 2
01111100 => 3
01111101 => 2
01111110 => 2
01111111 => 1
10000000 => 8
10000001 => 7
10000010 => 7
10000011 => 6
10000100 => 7
10000101 => 6
10000110 => 6
10000111 => 5
10001000 => 8
10001001 => 6
10001010 => 6
10001011 => 5
10001100 => 6
10001101 => 5
10001110 => 5
10001111 => 4
10010000 => 8
10010001 => 7
10010010 => 6
10010011 => 5
10010100 => 7
10010101 => 5
10010110 => 5
10010111 => 4
10011000 => 7
10011001 => 6
10011010 => 5
10011011 => 4
10011100 => 6
10011101 => 4
10011110 => 4
10011111 => 3
10100000 => 8
10100001 => 7
10100010 => 7
10100011 => 6
10100100 => 8
10100101 => 6
10100110 => 6
10100111 => 5
10101000 => 8
10101001 => 7
10101010 => 8
10101011 => 4
10101100 => 6
10101101 => 4
10101110 => 5
10101111 => 3
10110000 => 7
10110001 => 6
10110010 => 7
10110011 => 5
10110100 => 7
10110101 => 5
10110110 => 6
10110111 => 3
10111000 => 6
10111001 => 5
10111010 => 6
10111011 => 4
10111100 => 5
10111101 => 4
10111110 => 4
10111111 => 2
11000000 => 8
11000001 => 7
11000010 => 8
11000011 => 6
11000100 => 8
11000101 => 7
11000110 => 7
11000111 => 5
11001000 => 8
11001001 => 7
11001010 => 8
11001011 => 6
11001100 => 8
11001101 => 6
11001110 => 6
11001111 => 4
11010000 => 8
11010001 => 7
11010010 => 8
11010011 => 6
11010100 => 8
11010101 => 7
11010110 => 7
11010111 => 5
11011000 => 8
11011001 => 7
11011010 => 8
11011011 => 5
11011100 => 7
11011101 => 6
11011110 => 6
11011111 => 3
11100000 => 8
11100001 => 7
11100010 => 8
11100011 => 6
11100100 => 8
11100101 => 7
11100110 => 8
11100111 => 5
11101000 => 8
11101001 => 7
11101010 => 8
11101011 => 6
11101100 => 8
11101101 => 7
11101110 => 8
11101111 => 4
11110000 => 8
11110001 => 7
11110010 => 8
11110011 => 6
11110100 => 8
11110101 => 7
11110110 => 8
11110111 => 5
11111000 => 8
11111001 => 7
11111010 => 8
11111011 => 6
11111100 => 8
11111101 => 7
11111110 => 8
11111111 => 8
000000000 => 9
000000001 => 1
000000010 => 2
000000011 => 1
000000100 => 3
000000101 => 1
000000110 => 2
000000111 => 1
000001000 => 4
000001001 => 1
000001010 => 2
000001011 => 1
000001100 => 3
000001101 => 1
000001110 => 2
000001111 => 1
000010000 => 5
000010001 => 1
000010010 => 2
000010011 => 1
000010100 => 3
000010101 => 1
000010110 => 2
000010111 => 1
000011000 => 4
000011001 => 1
000011010 => 2
000011011 => 1
000011100 => 3
000011101 => 1
000011110 => 2
000011111 => 1
000100000 => 6
000100001 => 2
000100010 => 3
000100011 => 1
000100100 => 3
000100101 => 1
000100110 => 2
000100111 => 1
000101000 => 4
000101001 => 1
000101010 => 2
000101011 => 1
000101100 => 3
000101101 => 1
000101110 => 2
000101111 => 1
000110000 => 5
000110001 => 2
000110010 => 2
000110011 => 1
000110100 => 3
000110101 => 1
000110110 => 2
000110111 => 1
000111000 => 4
000111001 => 1
000111010 => 2
000111011 => 1
000111100 => 3
000111101 => 1
000111110 => 2
000111111 => 1
001000000 => 7
001000001 => 4
001000010 => 4
001000011 => 3
001000100 => 5
001000101 => 2
001000110 => 3
001000111 => 2
001001000 => 5
001001001 => 3
001001010 => 2
001001011 => 1
001001100 => 3
001001101 => 1
001001110 => 2
001001111 => 1
001010000 => 5
001010001 => 3
001010010 => 3
001010011 => 1
001010100 => 3
001010101 => 1
001010110 => 2
001010111 => 1
001011000 => 4
001011001 => 2
001011010 => 2
001011011 => 1
001011100 => 3
001011101 => 1
001011110 => 2
001011111 => 1
001100000 => 6
001100001 => 4
001100010 => 4
001100011 => 2
001100100 => 4
001100101 => 2
001100110 => 3
001100111 => 1
001101000 => 4
001101001 => 2
001101010 => 2
001101011 => 1
001101100 => 3
001101101 => 1
001101110 => 2
001101111 => 1
001110000 => 5
001110001 => 3
001110010 => 3
001110011 => 2
001110100 => 3
001110101 => 1
001110110 => 2
001110111 => 1
001111000 => 4
001111001 => 2
001111010 => 2
001111011 => 1
001111100 => 3
001111101 => 1
001111110 => 2
001111111 => 1
010000000 => 8
010000001 => 6
010000010 => 6
010000011 => 5
010000100 => 6
010000101 => 4
010000110 => 5
010000111 => 4
010001000 => 7
010001001 => 4
010001010 => 4
010001011 => 3
010001100 => 5
010001101 => 3
010001110 => 4
010001111 => 3
010010000 => 7
010010001 => 5
010010010 => 6
010010011 => 3
010010100 => 5
010010101 => 2
010010110 => 3
010010111 => 2
010011000 => 5
010011001 => 3
010011010 => 4
010011011 => 2
010011100 => 4
010011101 => 2
010011110 => 3
010011111 => 2
010100000 => 7
010100001 => 5
010100010 => 6
010100011 => 4
010100100 => 6
010100101 => 3
010100110 => 4
010100111 => 3
010101000 => 6
010101001 => 4
010101010 => 5
010101011 => 1
010101100 => 3
010101101 => 2
010101110 => 2
010101111 => 1
010110000 => 5
010110001 => 4
010110010 => 4
010110011 => 2
010110100 => 4
010110101 => 3
010110110 => 2
010110111 => 1
010111000 => 4
010111001 => 3
010111010 => 3
010111011 => 1
010111100 => 3
010111101 => 2
010111110 => 2
010111111 => 1
011000000 => 7
011000001 => 6
011000010 => 6
011000011 => 4
011000100 => 6
011000101 => 5
011000110 => 4
011000111 => 3
011001000 => 6
011001001 => 5
011001010 => 5
011001011 => 3
011001100 => 5
011001101 => 3
011001110 => 3
011001111 => 2
011010000 => 6
011010001 => 5
011010010 => 5
011010011 => 3
011010100 => 5
011010101 => 4
011010110 => 4
011010111 => 2
011011000 => 5
011011001 => 4
011011010 => 4
011011011 => 3
011011100 => 3
011011101 => 2
011011110 => 2
011011111 => 1
011100000 => 6
011100001 => 5
011100010 => 5
011100011 => 4
011100100 => 5
011100101 => 4
011100110 => 4
011100111 => 2
011101000 => 5
011101001 => 4
011101010 => 4
011101011 => 3
011101100 => 4
011101101 => 3
011101110 => 3
011101111 => 1
011110000 => 5
011110001 => 4
011110010 => 4
011110011 => 3
011110100 => 4
011110101 => 3
011110110 => 3
011110111 => 2
011111000 => 4
011111001 => 3
011111010 => 3
011111011 => 2
011111100 => 3
011111101 => 2
011111110 => 2
011111111 => 1
100000000 => 9
100000001 => 8
100000010 => 8
100000011 => 7
100000100 => 8
100000101 => 7
100000110 => 7
100000111 => 6
100001000 => 8
100001001 => 7
100001010 => 7
100001011 => 6
100001100 => 7
100001101 => 6
100001110 => 6
100001111 => 5
100010000 => 9
100010001 => 7
100010010 => 7
100010011 => 6
100010100 => 7
100010101 => 6
100010110 => 6
100010111 => 5
100011000 => 8
100011001 => 6
100011010 => 6
100011011 => 5
100011100 => 6
100011101 => 5
100011110 => 5
100011111 => 4
100100000 => 9
100100001 => 8
100100010 => 8
100100011 => 7
100100100 => 9
100100101 => 6
100100110 => 6
100100111 => 5
100101000 => 8
100101001 => 6
100101010 => 6
100101011 => 5
100101100 => 7
100101101 => 5
100101110 => 5
100101111 => 4
100110000 => 8
100110001 => 7
100110010 => 7
100110011 => 5
100110100 => 7
100110101 => 5
100110110 => 5
100110111 => 4
100111000 => 7
100111001 => 6
100111010 => 5
100111011 => 4
100111100 => 6
100111101 => 4
100111110 => 4
100111111 => 3
101000000 => 9
101000001 => 8
101000010 => 8
101000011 => 7
101000100 => 9
101000101 => 7
101000110 => 7
101000111 => 6
101001000 => 9
101001001 => 8
101001010 => 7
101001011 => 6
101001100 => 8
101001101 => 6
101001110 => 6
101001111 => 5
101010000 => 9
101010001 => 8
101010010 => 8
101010011 => 7
101010100 => 9
101010101 => 5
101010110 => 6
101010111 => 4
101011000 => 7
101011001 => 6
101011010 => 7
101011011 => 4
101011100 => 6
101011101 => 4
101011110 => 5
101011111 => 3
101100000 => 8
101100001 => 7
101100010 => 8
101100011 => 6
101100100 => 8
101100101 => 6
101100110 => 7
101100111 => 5
101101000 => 8
101101001 => 7
101101010 => 8
101101011 => 5
101101100 => 7
101101101 => 6
101101110 => 5
101101111 => 3
101110000 => 7
101110001 => 6
101110010 => 7
101110011 => 5
101110100 => 7
101110101 => 6
101110110 => 6
101110111 => 3
101111000 => 6
101111001 => 5
101111010 => 6
101111011 => 4
101111100 => 5
101111101 => 4
101111110 => 4
101111111 => 2
110000000 => 9
110000001 => 8
110000010 => 9
110000011 => 7
110000100 => 9
110000101 => 8
110000110 => 8
110000111 => 6
110001000 => 9
110001001 => 8
110001010 => 9
110001011 => 7
110001100 => 8
110001101 => 7
110001110 => 7
110001111 => 5
110010000 => 9
110010001 => 8
110010010 => 9
110010011 => 7
110010100 => 9
110010101 => 8
110010110 => 8
110010111 => 6
110011000 => 9
110011001 => 7
110011010 => 8
110011011 => 6
110011100 => 8
110011101 => 6
110011110 => 6
110011111 => 4
110100000 => 9
110100001 => 8
110100010 => 9
110100011 => 7
110100100 => 9
110100101 => 8
110100110 => 8
110100111 => 6
110101000 => 9
110101001 => 8
110101010 => 9
110101011 => 7
110101100 => 9
110101101 => 7
110101110 => 7
110101111 => 5
110110000 => 9
110110001 => 8
110110010 => 9
110110011 => 7
110110100 => 9
110110101 => 8
110110110 => 9
110110111 => 5
110111000 => 8
110111001 => 7
110111010 => 8
110111011 => 5
110111100 => 7
110111101 => 6
110111110 => 6
110111111 => 3
111000000 => 9
111000001 => 8
111000010 => 9
111000011 => 7
111000100 => 9
111000101 => 8
111000110 => 9
111000111 => 6
111001000 => 9
111001001 => 8
111001010 => 9
111001011 => 7
111001100 => 9
111001101 => 8
111001110 => 8
111001111 => 5
111010000 => 9
111010001 => 8
111010010 => 9
111010011 => 7
111010100 => 9
111010101 => 8
111010110 => 9
111010111 => 6
111011000 => 9
111011001 => 8
111011010 => 9
111011011 => 7
111011100 => 9
111011101 => 7
111011110 => 8
111011111 => 4
111100000 => 9
111100001 => 8
111100010 => 9
111100011 => 7
111100100 => 9
111100101 => 8
111100110 => 9
111100111 => 6
111101000 => 9
111101001 => 8
111101010 => 9
111101011 => 7
111101100 => 9
111101101 => 8
111101110 => 9
111101111 => 5
111110000 => 9
111110001 => 8
111110010 => 9
111110011 => 7
111110100 => 9
111110101 => 8
111110110 => 9
111110111 => 6
111111000 => 9
111111001 => 8
111111010 => 9
111111011 => 7
111111100 => 9
111111101 => 8
111111110 => 9
111111111 => 9
1010101010 => 10
1010101100 => 7
1010110010 => 8
1010110100 => 8
1010111000 => 7
1011001010 => 9
1011001100 => 8
1011010010 => 9
1011010100 => 9
1011011000 => 8
1011100010 => 8
1011100100 => 8
1011101000 => 8
1011110000 => 7
1100101010 => 10
1100101100 => 9
1100110010 => 10
1100110100 => 9
1100111000 => 9
1101001010 => 10
1101001100 => 10
1101010010 => 10
1101010100 => 10
1101011000 => 10
1101100010 => 10
1101100100 => 10
1101101000 => 10
1101110000 => 9
1110001010 => 10
1110001100 => 10
1110010010 => 10
1110010100 => 10
1110011000 => 10
1110100010 => 10
1110100100 => 10
1110101000 => 10
1110110000 => 10
1111000010 => 10
1111000100 => 10
1111001000 => 10
1111010000 => 10
1111100000 => 10
1111111110 => 10
1111111101 => 9
1111000001 => 9
1110111110 => 8
1110100001 => 9
1110011110 => 8
1110010001 => 9
1110001001 => 9
1110000101 => 9
1101111110 => 6
1101100001 => 9
1101010001 => 9
1101001110 => 8
1101001001 => 9
1101000101 => 9
1100110001 => 9
1100101001 => 9
1100100110 => 9
1100100101 => 9
1100011001 => 8
1100010101 => 9
1100010010 => 10
1100001000 => 10
1011111110 => 4
1011111101 => 4
1011100001 => 7
1011010001 => 8
1011001001 => 8
1011000101 => 7
1010111110 => 5
1010110001 => 7
1010101001 => 9
1010100110 => 8
1010100101 => 8
1010011010 => 7
1010011001 => 8
1010010110 => 7
1010010101 => 7
1010001110 => 7
1001111110 => 4
1001111000 => 7
1001110100 => 7
1001110001 => 7
1001101100 => 7
1001101010 => 6
1001101001 => 7
1001100110 => 6
1001100101 => 7
1001011110 => 5
1001011010 => 6
1001011001 => 6
1001010110 => 6
1001010101 => 6
1001001110 => 6
1000111110 => 5
1000111100 => 6
1000111010 => 6
1000110110 => 6
1000101110 => 6
1000011110 => 6
1000010110 => 7
1000001110 => 7
1000001010 => 8
1000000110 => 8
1000000100 => 9
1000000010 => 9
1000000000 => 10
10000000000 => 11
10000000010 => 10
10000000110 => 9
10000001110 => 8
10000011110 => 7
10000111110 => 6
10001111110 => 5
10011111110 => 4
10111111110 => 4
11111111110 => 11
10000000100 => 10
10000001010 => 9
10000010110 => 8
10000101110 => 7
10001011110 => 6
10010111110 => 5
10101111110 => 5
11011111110 => 6
10000010000 => 10
10000100010 => 9
10001000010 => 10
10000100100 => 9
10001001010 => 8
10001000100 => 9
10000101000 => 9
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-----------------------------------------------------------------------------
Created: Jul 02, 2019 at 11:52 by Christian Stump
-----------------------------------------------------------------------------
Last Updated: Jul 04, 2019 at 22:29 by Martin Rubey