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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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* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *
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-----------------------------------------------------------------------------
Statistic identifier: St000290
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
Description: The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(x):
return x.major_index()
-----------------------------------------------------------------------------
Statistic values:
0 => 0
1 => 0
00 => 0
01 => 0
10 => 1
11 => 0
000 => 0
001 => 0
010 => 2
011 => 0
100 => 1
101 => 1
110 => 2
111 => 0
0000 => 0
0001 => 0
0010 => 3
0011 => 0
0100 => 2
0101 => 2
0110 => 3
0111 => 0
1000 => 1
1001 => 1
1010 => 4
1011 => 1
1100 => 2
1101 => 2
1110 => 3
1111 => 0
00000 => 0
00001 => 0
00010 => 4
00011 => 0
00100 => 3
00101 => 3
00110 => 4
00111 => 0
01000 => 2
01001 => 2
01010 => 6
01011 => 2
01100 => 3
01101 => 3
01110 => 4
01111 => 0
10000 => 1
10001 => 1
10010 => 5
10011 => 1
10100 => 4
10101 => 4
10110 => 5
10111 => 1
11000 => 2
11001 => 2
11010 => 6
11011 => 2
11100 => 3
11101 => 3
11110 => 4
11111 => 0
000000 => 0
000001 => 0
000010 => 5
000011 => 0
000100 => 4
000101 => 4
000110 => 5
000111 => 0
001000 => 3
001001 => 3
001010 => 8
001011 => 3
001100 => 4
001101 => 4
001110 => 5
001111 => 0
010000 => 2
010001 => 2
010010 => 7
010011 => 2
010100 => 6
010101 => 6
010110 => 7
010111 => 2
011000 => 3
011001 => 3
011010 => 8
011011 => 3
011100 => 4
011101 => 4
011110 => 5
011111 => 0
100000 => 1
100001 => 1
100010 => 6
100011 => 1
100100 => 5
100101 => 5
100110 => 6
100111 => 1
101000 => 4
101001 => 4
101010 => 9
101011 => 4
101100 => 5
101101 => 5
101110 => 6
101111 => 1
110000 => 2
110001 => 2
110010 => 7
110011 => 2
110100 => 6
110101 => 6
110110 => 7
110111 => 2
111000 => 3
111001 => 3
111010 => 8
111011 => 3
111100 => 4
111101 => 4
111110 => 5
111111 => 0
0000000 => 0
0000001 => 0
0000010 => 6
0000011 => 0
0000100 => 5
0000101 => 5
0000110 => 6
0000111 => 0
0001000 => 4
0001001 => 4
0001010 => 10
0001011 => 4
0001100 => 5
0001101 => 5
0001110 => 6
0001111 => 0
0010000 => 3
0010001 => 3
0010010 => 9
0010011 => 3
0010100 => 8
0010101 => 8
0010110 => 9
0010111 => 3
0011000 => 4
0011001 => 4
0011010 => 10
0011011 => 4
0011100 => 5
0011101 => 5
0011110 => 6
0011111 => 0
0100000 => 2
0100001 => 2
0100010 => 8
0100011 => 2
0100100 => 7
0100101 => 7
0100110 => 8
0100111 => 2
0101000 => 6
0101001 => 6
0101010 => 12
0101011 => 6
0101100 => 7
0101101 => 7
0101110 => 8
0101111 => 2
0110000 => 3
0110001 => 3
0110010 => 9
0110011 => 3
0110100 => 8
0110101 => 8
0110110 => 9
0110111 => 3
0111000 => 4
0111001 => 4
0111010 => 10
0111011 => 4
0111100 => 5
0111101 => 5
0111110 => 6
0111111 => 0
1000000 => 1
1000001 => 1
1000010 => 7
1000011 => 1
1000100 => 6
1000101 => 6
1000110 => 7
1000111 => 1
1001000 => 5
1001001 => 5
1001010 => 11
1001011 => 5
1001100 => 6
1001101 => 6
1001110 => 7
1001111 => 1
1010000 => 4
1010001 => 4
1010010 => 10
1010011 => 4
1010100 => 9
1010101 => 9
1010110 => 10
1010111 => 4
1011000 => 5
1011001 => 5
1011010 => 11
1011011 => 5
1011100 => 6
1011101 => 6
1011110 => 7
1011111 => 1
1100000 => 2
1100001 => 2
1100010 => 8
1100011 => 2
1100100 => 7
1100101 => 7
1100110 => 8
1100111 => 2
1101000 => 6
1101001 => 6
1101010 => 12
1101011 => 6
1101100 => 7
1101101 => 7
1101110 => 8
1101111 => 2
1110000 => 3
1110001 => 3
1110010 => 9
1110011 => 3
1110100 => 8
1110101 => 8
1110110 => 9
1110111 => 3
1111000 => 4
1111001 => 4
1111010 => 10
1111011 => 4
1111100 => 5
1111101 => 5
1111110 => 6
1111111 => 0
00000000 => 0
00000001 => 0
00000010 => 7
00000011 => 0
00000100 => 6
00000101 => 6
00000110 => 7
00000111 => 0
00001000 => 5
00001001 => 5
00001010 => 12
00001011 => 5
00001100 => 6
00001101 => 6
00001110 => 7
00001111 => 0
00010000 => 4
00010001 => 4
00010010 => 11
00010011 => 4
00010100 => 10
00010101 => 10
00010110 => 11
00010111 => 4
00011000 => 5
00011001 => 5
00011010 => 12
00011011 => 5
00011100 => 6
00011101 => 6
00011110 => 7
00011111 => 0
00100000 => 3
00100001 => 3
00100010 => 10
00100011 => 3
00100100 => 9
00100101 => 9
00100110 => 10
00100111 => 3
00101000 => 8
00101001 => 8
00101010 => 15
00101011 => 8
00101100 => 9
00101101 => 9
00101110 => 10
00101111 => 3
00110000 => 4
00110001 => 4
00110010 => 11
00110011 => 4
00110100 => 10
00110101 => 10
00110110 => 11
00110111 => 4
00111000 => 5
00111001 => 5
00111010 => 12
00111011 => 5
00111100 => 6
00111101 => 6
00111110 => 7
00111111 => 0
01000000 => 2
01000001 => 2
01000010 => 9
01000011 => 2
01000100 => 8
01000101 => 8
01000110 => 9
01000111 => 2
01001000 => 7
01001001 => 7
01001010 => 14
01001011 => 7
01001100 => 8
01001101 => 8
01001110 => 9
01001111 => 2
01010000 => 6
01010001 => 6
01010010 => 13
01010011 => 6
01010100 => 12
01010101 => 12
01010110 => 13
01010111 => 6
01011000 => 7
01011001 => 7
01011010 => 14
01011011 => 7
01011100 => 8
01011101 => 8
01011110 => 9
01011111 => 2
01100000 => 3
01100001 => 3
01100010 => 10
01100011 => 3
01100100 => 9
01100101 => 9
01100110 => 10
01100111 => 3
01101000 => 8
01101001 => 8
01101010 => 15
01101011 => 8
01101100 => 9
01101101 => 9
01101110 => 10
01101111 => 3
01110000 => 4
01110001 => 4
01110010 => 11
01110011 => 4
01110100 => 10
01110101 => 10
01110110 => 11
01110111 => 4
01111000 => 5
01111001 => 5
01111010 => 12
01111011 => 5
01111100 => 6
01111101 => 6
01111110 => 7
01111111 => 0
10000000 => 1
10000001 => 1
10000010 => 8
10000011 => 1
10000100 => 7
10000101 => 7
10000110 => 8
10000111 => 1
10001000 => 6
10001001 => 6
10001010 => 13
10001011 => 6
10001100 => 7
10001101 => 7
10001110 => 8
10001111 => 1
10010000 => 5
10010001 => 5
10010010 => 12
10010011 => 5
10010100 => 11
10010101 => 11
10010110 => 12
10010111 => 5
10011000 => 6
10011001 => 6
10011010 => 13
10011011 => 6
10011100 => 7
10011101 => 7
10011110 => 8
10011111 => 1
10100000 => 4
10100001 => 4
10100010 => 11
10100011 => 4
10100100 => 10
10100101 => 10
10100110 => 11
10100111 => 4
10101000 => 9
10101001 => 9
10101010 => 16
10101011 => 9
10101100 => 10
10101101 => 10
10101110 => 11
10101111 => 4
10110000 => 5
10110001 => 5
10110010 => 12
10110011 => 5
10110100 => 11
10110101 => 11
10110110 => 12
10110111 => 5
10111000 => 6
10111001 => 6
10111010 => 13
10111011 => 6
10111100 => 7
10111101 => 7
10111110 => 8
10111111 => 1
11000000 => 2
11000001 => 2
11000010 => 9
11000011 => 2
11000100 => 8
11000101 => 8
11000110 => 9
11000111 => 2
11001000 => 7
11001001 => 7
11001010 => 14
11001011 => 7
11001100 => 8
11001101 => 8
11001110 => 9
11001111 => 2
11010000 => 6
11010001 => 6
11010010 => 13
11010011 => 6
11010100 => 12
11010101 => 12
11010110 => 13
11010111 => 6
11011000 => 7
11011001 => 7
11011010 => 14
11011011 => 7
11011100 => 8
11011101 => 8
11011110 => 9
11011111 => 2
11100000 => 3
11100001 => 3
11100010 => 10
11100011 => 3
11100100 => 9
11100101 => 9
11100110 => 10
11100111 => 3
11101000 => 8
11101001 => 8
11101010 => 15
11101011 => 8
11101100 => 9
11101101 => 9
11101110 => 10
11101111 => 3
11110000 => 4
11110001 => 4
11110010 => 11
11110011 => 4
11110100 => 10
11110101 => 10
11110110 => 11
11110111 => 4
11111000 => 5
11111001 => 5
11111010 => 12
11111011 => 5
11111100 => 6
11111101 => 6
11111110 => 7
11111111 => 0
000000000 => 0
000000001 => 0
000000010 => 8
000000011 => 0
000000100 => 7
000000101 => 7
000000110 => 8
000000111 => 0
000001000 => 6
000001001 => 6
000001010 => 14
000001011 => 6
000001100 => 7
000001101 => 7
000001110 => 8
000001111 => 0
000010000 => 5
000010001 => 5
000010010 => 13
000010011 => 5
000010100 => 12
000010101 => 12
000010110 => 13
000010111 => 5
000011000 => 6
000011001 => 6
000011010 => 14
000011011 => 6
000011100 => 7
000011101 => 7
000011110 => 8
000011111 => 0
000100000 => 4
000100001 => 4
000100010 => 12
000100011 => 4
000100100 => 11
000100101 => 11
000100110 => 12
000100111 => 4
000101000 => 10
000101001 => 10
000101010 => 18
000101011 => 10
000101100 => 11
000101101 => 11
000101110 => 12
000101111 => 4
000110000 => 5
000110001 => 5
000110010 => 13
000110011 => 5
000110100 => 12
000110101 => 12
000110110 => 13
000110111 => 5
000111000 => 6
000111001 => 6
000111010 => 14
000111011 => 6
000111100 => 7
000111101 => 7
000111110 => 8
000111111 => 0
001000000 => 3
001000001 => 3
001000010 => 11
001000011 => 3
001000100 => 10
001000101 => 10
001000110 => 11
001000111 => 3
001001000 => 9
001001001 => 9
001001010 => 17
001001011 => 9
001001100 => 10
001001101 => 10
001001110 => 11
001001111 => 3
001010000 => 8
001010001 => 8
001010010 => 16
001010011 => 8
001010100 => 15
001010101 => 15
001010110 => 16
001010111 => 8
001011000 => 9
001011001 => 9
001011010 => 17
001011011 => 9
001011100 => 10
001011101 => 10
001011110 => 11
001011111 => 3
001100000 => 4
001100001 => 4
001100010 => 12
001100011 => 4
001100100 => 11
001100101 => 11
001100110 => 12
001100111 => 4
001101000 => 10
001101001 => 10
001101010 => 18
001101011 => 10
001101100 => 11
001101101 => 11
001101110 => 12
001101111 => 4
001110000 => 5
001110001 => 5
001110010 => 13
001110011 => 5
001110100 => 12
001110101 => 12
001110110 => 13
001110111 => 5
001111000 => 6
001111001 => 6
001111010 => 14
001111011 => 6
001111100 => 7
001111101 => 7
001111110 => 8
001111111 => 0
010000000 => 2
010000001 => 2
010000010 => 10
010000011 => 2
010000100 => 9
010000101 => 9
010000110 => 10
010000111 => 2
010001000 => 8
010001001 => 8
010001010 => 16
010001011 => 8
010001100 => 9
010001101 => 9
010001110 => 10
010001111 => 2
010010000 => 7
010010001 => 7
010010010 => 15
010010011 => 7
010010100 => 14
010010101 => 14
010010110 => 15
010010111 => 7
010011000 => 8
010011001 => 8
010011010 => 16
010011011 => 8
010011100 => 9
010011101 => 9
010011110 => 10
010011111 => 2
010100000 => 6
010100001 => 6
010100010 => 14
010100011 => 6
010100100 => 13
010100101 => 13
010100110 => 14
010100111 => 6
010101000 => 12
010101001 => 12
010101010 => 20
010101011 => 12
010101100 => 13
010101101 => 13
010101110 => 14
010101111 => 6
010110000 => 7
010110001 => 7
010110010 => 15
010110011 => 7
010110100 => 14
010110101 => 14
010110110 => 15
010110111 => 7
010111000 => 8
010111001 => 8
010111010 => 16
010111011 => 8
010111100 => 9
010111101 => 9
010111110 => 10
010111111 => 2
011000000 => 3
011000001 => 3
011000010 => 11
011000011 => 3
011000100 => 10
011000101 => 10
011000110 => 11
011000111 => 3
011001000 => 9
011001001 => 9
011001010 => 17
011001011 => 9
011001100 => 10
011001101 => 10
011001110 => 11
011001111 => 3
011010000 => 8
011010001 => 8
011010010 => 16
011010011 => 8
011010100 => 15
011010101 => 15
011010110 => 16
011010111 => 8
011011000 => 9
011011001 => 9
011011010 => 17
011011011 => 9
011011100 => 10
011011101 => 10
011011110 => 11
011011111 => 3
011100000 => 4
011100001 => 4
011100010 => 12
011100011 => 4
011100100 => 11
011100101 => 11
011100110 => 12
011100111 => 4
011101000 => 10
011101001 => 10
011101010 => 18
011101011 => 10
011101100 => 11
011101101 => 11
011101110 => 12
011101111 => 4
011110000 => 5
011110001 => 5
011110010 => 13
011110011 => 5
011110100 => 12
011110101 => 12
011110110 => 13
011110111 => 5
011111000 => 6
011111001 => 6
011111010 => 14
011111011 => 6
011111100 => 7
011111101 => 7
011111110 => 8
011111111 => 0
100000000 => 1
100000001 => 1
100000010 => 9
100000011 => 1
100000100 => 8
100000101 => 8
100000110 => 9
100000111 => 1
100001000 => 7
100001001 => 7
100001010 => 15
100001011 => 7
100001100 => 8
100001101 => 8
100001110 => 9
100001111 => 1
100010000 => 6
100010001 => 6
100010010 => 14
100010011 => 6
100010100 => 13
100010101 => 13
100010110 => 14
100010111 => 6
100011000 => 7
100011001 => 7
100011010 => 15
100011011 => 7
100011100 => 8
100011101 => 8
100011110 => 9
100011111 => 1
100100000 => 5
100100001 => 5
100100010 => 13
100100011 => 5
100100100 => 12
100100101 => 12
100100110 => 13
100100111 => 5
100101000 => 11
100101001 => 11
100101010 => 19
100101011 => 11
100101100 => 12
100101101 => 12
100101110 => 13
100101111 => 5
100110000 => 6
100110001 => 6
100110010 => 14
100110011 => 6
100110100 => 13
100110101 => 13
100110110 => 14
100110111 => 6
100111000 => 7
100111001 => 7
100111010 => 15
100111011 => 7
100111100 => 8
100111101 => 8
100111110 => 9
100111111 => 1
101000000 => 4
101000001 => 4
101000010 => 12
101000011 => 4
101000100 => 11
101000101 => 11
101000110 => 12
101000111 => 4
101001000 => 10
101001001 => 10
101001010 => 18
101001011 => 10
101001100 => 11
101001101 => 11
101001110 => 12
101001111 => 4
101010000 => 9
101010001 => 9
101010010 => 17
101010011 => 9
101010100 => 16
101010101 => 16
101010110 => 17
101010111 => 9
101011000 => 10
101011001 => 10
101011010 => 18
101011011 => 10
101011100 => 11
101011101 => 11
101011110 => 12
101011111 => 4
101100000 => 5
101100001 => 5
101100010 => 13
101100011 => 5
101100100 => 12
101100101 => 12
101100110 => 13
101100111 => 5
101101000 => 11
101101001 => 11
101101010 => 19
101101011 => 11
101101100 => 12
101101101 => 12
101101110 => 13
101101111 => 5
101110000 => 6
101110001 => 6
101110010 => 14
101110011 => 6
101110100 => 13
101110101 => 13
101110110 => 14
101110111 => 6
101111000 => 7
101111001 => 7
101111010 => 15
101111011 => 7
101111100 => 8
101111101 => 8
101111110 => 9
101111111 => 1
110000000 => 2
110000001 => 2
110000010 => 10
110000011 => 2
110000100 => 9
110000101 => 9
110000110 => 10
110000111 => 2
110001000 => 8
110001001 => 8
110001010 => 16
110001011 => 8
110001100 => 9
110001101 => 9
110001110 => 10
110001111 => 2
110010000 => 7
110010001 => 7
110010010 => 15
110010011 => 7
110010100 => 14
110010101 => 14
110010110 => 15
110010111 => 7
110011000 => 8
110011001 => 8
110011010 => 16
110011011 => 8
110011100 => 9
110011101 => 9
110011110 => 10
110011111 => 2
110100000 => 6
110100001 => 6
110100010 => 14
110100011 => 6
110100100 => 13
110100101 => 13
110100110 => 14
110100111 => 6
110101000 => 12
110101001 => 12
110101010 => 20
110101011 => 12
110101100 => 13
110101101 => 13
110101110 => 14
110101111 => 6
110110000 => 7
110110001 => 7
110110010 => 15
110110011 => 7
110110100 => 14
110110101 => 14
110110110 => 15
110110111 => 7
110111000 => 8
110111001 => 8
110111010 => 16
110111011 => 8
110111100 => 9
110111101 => 9
110111110 => 10
110111111 => 2
111000000 => 3
111000001 => 3
111000010 => 11
111000011 => 3
111000100 => 10
111000101 => 10
111000110 => 11
111000111 => 3
111001000 => 9
111001001 => 9
111001010 => 17
111001011 => 9
111001100 => 10
111001101 => 10
111001110 => 11
111001111 => 3
111010000 => 8
111010001 => 8
111010010 => 16
111010011 => 8
111010100 => 15
111010101 => 15
111010110 => 16
111010111 => 8
111011000 => 9
111011001 => 9
111011010 => 17
111011011 => 9
111011100 => 10
111011101 => 10
111011110 => 11
111011111 => 3
111100000 => 4
111100001 => 4
111100010 => 12
111100011 => 4
111100100 => 11
111100101 => 11
111100110 => 12
111100111 => 4
111101000 => 10
111101001 => 10
111101010 => 18
111101011 => 10
111101100 => 11
111101101 => 11
111101110 => 12
111101111 => 4
111110000 => 5
111110001 => 5
111110010 => 13
111110011 => 5
111110100 => 12
111110101 => 12
111110110 => 13
111110111 => 5
111111000 => 6
111111001 => 6
111111010 => 14
111111011 => 6
111111100 => 7
111111101 => 7
111111110 => 8
111111111 => 0
1010101010 => 25
1010101100 => 17
1010110010 => 19
1010110100 => 18
1010111000 => 11
1011001010 => 21
1011001100 => 13
1011010010 => 20
1011010100 => 19
1011011000 => 12
1011100010 => 15
1011100100 => 14
1011101000 => 13
1011110000 => 7
1100101010 => 23
1100101100 => 15
1100110010 => 17
1100110100 => 16
1100111000 => 9
1101001010 => 22
1101001100 => 14
1101010010 => 21
1101010100 => 20
1101011000 => 13
1101100010 => 16
1101100100 => 15
1101101000 => 14
1101110000 => 8
1110001010 => 19
1110001100 => 11
1110010010 => 18
1110010100 => 17
1110011000 => 10
1110100010 => 17
1110100100 => 16
1110101000 => 15
1110110000 => 9
1111000010 => 13
1111000100 => 12
1111001000 => 11
1111010000 => 10
1111100000 => 5
1111111110 => 9
1111111101 => 8
1111000001 => 4
1110111110 => 12
1110100001 => 8
1110011110 => 12
1110010001 => 9
1110001001 => 10
1110000101 => 11
1101111110 => 11
1101100001 => 7
1101010001 => 12
1101001110 => 15
1101001001 => 13
1101000101 => 14
1100110001 => 8
1100101001 => 14
1100100110 => 16
1100100101 => 15
1100011001 => 9
1100010101 => 16
1100010010 => 17
1100001000 => 9
1011111110 => 10
1011111101 => 9
1011100001 => 6
1011010001 => 11
1011001001 => 12
1011000101 => 13
1010111110 => 13
1010110001 => 10
1010101001 => 16
1010100110 => 18
1010100101 => 17
1010011010 => 20
1010011001 => 11
1010010110 => 19
1010010101 => 18
1010001110 => 13
1001111110 => 10
1001111000 => 8
1001110100 => 15
1001110001 => 7
1001101100 => 14
1001101010 => 22
1001101001 => 13
1001100110 => 15
1001100101 => 14
1001011110 => 14
1001011010 => 21
1001011001 => 12
1001010110 => 20
1001010101 => 19
1001001110 => 14
1000111110 => 10
1000111100 => 9
1000111010 => 17
1000110110 => 16
1000101110 => 15
1000011110 => 10
1000010110 => 16
1000001110 => 10
1000001010 => 17
1000000110 => 10
1000000100 => 9
1000000010 => 10
1000000000 => 1
0111111111 => 0
0111111101 => 8
0111111001 => 7
0111110001 => 6
0111100001 => 5
0111000001 => 4
0110000001 => 3
0100000001 => 2
0000000001 => 0
0111111011 => 7
0111110101 => 14
0111101001 => 12
0111010001 => 10
0110100001 => 8
0101000001 => 6
0010000001 => 3
0101010101 => 20
0101010011 => 12
0101001101 => 14
0101001011 => 13
0101000111 => 6
0100110101 => 16
0100110011 => 8
0100101101 => 15
0100101011 => 14
0100100111 => 7
0100011101 => 10
0100011011 => 9
0100010111 => 8
0100001111 => 2
0011010101 => 18
0011010011 => 10
0011001101 => 12
0011001011 => 11
0011000111 => 4
0010110101 => 17
0010110011 => 9
0010101101 => 16
0010101011 => 15
0010100111 => 8
0010011101 => 11
0010011011 => 10
0010010111 => 9
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-----------------------------------------------------------------------------
Created: Nov 19, 2015 at 11:57 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Nov 17, 2019 at 18:31 by Martin Rubey