*****************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* 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. *
*****************************************************************************
-----------------------------------------------------------------------------
Statistic identifier: St001930
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
Description: The weak major index of a binary word.
This is the sum of the positions of the weak descents, that is, for a word $w = (w_1,\dots,w_n)$,
$$
\sum_{\substack{1\leq i < n\\ w_i\geq w_{i+1}}} i.
$$
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(w):
return sum(i for i in range(1, len(w)) if w[i-1] >= w[i])
-----------------------------------------------------------------------------
Statistic values:
0 => 0
1 => 0
00 => 1
01 => 0
10 => 1
11 => 1
000 => 3
001 => 1
010 => 2
011 => 2
100 => 3
101 => 1
110 => 3
111 => 3
0000 => 6
0001 => 3
0010 => 4
0011 => 4
0100 => 5
0101 => 2
0110 => 5
0111 => 5
1000 => 6
1001 => 3
1010 => 4
1011 => 4
1100 => 6
1101 => 3
1110 => 6
1111 => 6
00000 => 10
00001 => 6
00010 => 7
00011 => 7
00100 => 8
00101 => 4
00110 => 8
00111 => 8
01000 => 9
01001 => 5
01010 => 6
01011 => 6
01100 => 9
01101 => 5
01110 => 9
01111 => 9
10000 => 10
10001 => 6
10010 => 7
10011 => 7
10100 => 8
10101 => 4
10110 => 8
10111 => 8
11000 => 10
11001 => 6
11010 => 7
11011 => 7
11100 => 10
11101 => 6
11110 => 10
11111 => 10
000000 => 15
000001 => 10
000010 => 11
000011 => 11
000100 => 12
000101 => 7
000110 => 12
000111 => 12
001000 => 13
001001 => 8
001010 => 9
001011 => 9
001100 => 13
001101 => 8
001110 => 13
001111 => 13
010000 => 14
010001 => 9
010010 => 10
010011 => 10
010100 => 11
010101 => 6
010110 => 11
010111 => 11
011000 => 14
011001 => 9
011010 => 10
011011 => 10
011100 => 14
011101 => 9
011110 => 14
011111 => 14
100000 => 15
100001 => 10
100010 => 11
100011 => 11
100100 => 12
100101 => 7
100110 => 12
100111 => 12
101000 => 13
101001 => 8
101010 => 9
101011 => 9
101100 => 13
101101 => 8
101110 => 13
101111 => 13
110000 => 15
110001 => 10
110010 => 11
110011 => 11
110100 => 12
110101 => 7
110110 => 12
110111 => 12
111000 => 15
111001 => 10
111010 => 11
111011 => 11
111100 => 15
111101 => 10
111110 => 15
111111 => 15
0000000 => 21
0000001 => 15
0000010 => 16
0000011 => 16
0000100 => 17
0000101 => 11
0000110 => 17
0000111 => 17
0001000 => 18
0001001 => 12
0001010 => 13
0001011 => 13
0001100 => 18
0001101 => 12
0001110 => 18
0001111 => 18
0010000 => 19
0010001 => 13
0010010 => 14
0010011 => 14
0010100 => 15
0010101 => 9
0010110 => 15
0010111 => 15
0011000 => 19
0011001 => 13
0011010 => 14
0011011 => 14
0011100 => 19
0011101 => 13
0011110 => 19
0011111 => 19
0100000 => 20
0100001 => 14
0100010 => 15
0100011 => 15
0100100 => 16
0100101 => 10
0100110 => 16
0100111 => 16
0101000 => 17
0101001 => 11
0101010 => 12
0101011 => 12
0101100 => 17
0101101 => 11
0101110 => 17
0101111 => 17
0110000 => 20
0110001 => 14
0110010 => 15
0110011 => 15
0110100 => 16
0110101 => 10
0110110 => 16
0110111 => 16
0111000 => 20
0111001 => 14
0111010 => 15
0111011 => 15
0111100 => 20
0111101 => 14
0111110 => 20
0111111 => 20
1000000 => 21
1000001 => 15
1000010 => 16
1000011 => 16
1000100 => 17
1000101 => 11
1000110 => 17
1000111 => 17
1001000 => 18
1001001 => 12
1001010 => 13
1001011 => 13
1001100 => 18
1001101 => 12
1001110 => 18
1001111 => 18
1010000 => 19
1010001 => 13
1010010 => 14
1010011 => 14
1010100 => 15
1010101 => 9
1010110 => 15
1010111 => 15
1011000 => 19
1011001 => 13
1011010 => 14
1011011 => 14
1011100 => 19
1011101 => 13
1011110 => 19
1011111 => 19
1100000 => 21
1100001 => 15
1100010 => 16
1100011 => 16
1100100 => 17
1100101 => 11
1100110 => 17
1100111 => 17
1101000 => 18
1101001 => 12
1101010 => 13
1101011 => 13
1101100 => 18
1101101 => 12
1101110 => 18
1101111 => 18
1110000 => 21
1110001 => 15
1110010 => 16
1110011 => 16
1110100 => 17
1110101 => 11
1110110 => 17
1110111 => 17
1111000 => 21
1111001 => 15
1111010 => 16
1111011 => 16
1111100 => 21
1111101 => 15
1111110 => 21
1111111 => 21
00000000 => 28
00000001 => 21
00000010 => 22
00000011 => 22
00000100 => 23
00000101 => 16
00000110 => 23
00000111 => 23
00001000 => 24
00001001 => 17
00001010 => 18
00001011 => 18
00001100 => 24
00001101 => 17
00001110 => 24
00001111 => 24
00010000 => 25
00010001 => 18
00010010 => 19
00010011 => 19
00010100 => 20
00010101 => 13
00010110 => 20
00010111 => 20
00011000 => 25
00011001 => 18
00011010 => 19
00011011 => 19
00011100 => 25
00011101 => 18
00011110 => 25
00011111 => 25
00100000 => 26
00100001 => 19
00100010 => 20
00100011 => 20
00100100 => 21
00100101 => 14
00100110 => 21
00100111 => 21
00101000 => 22
00101001 => 15
00101010 => 16
00101011 => 16
00101100 => 22
00101101 => 15
00101110 => 22
00101111 => 22
00110000 => 26
00110001 => 19
00110010 => 20
00110011 => 20
00110100 => 21
00110101 => 14
00110110 => 21
00110111 => 21
00111000 => 26
00111001 => 19
00111010 => 20
00111011 => 20
00111100 => 26
00111101 => 19
00111110 => 26
00111111 => 26
01000000 => 27
01000001 => 20
01000010 => 21
01000011 => 21
01000100 => 22
01000101 => 15
01000110 => 22
01000111 => 22
01001000 => 23
01001001 => 16
01001010 => 17
01001011 => 17
01001100 => 23
01001101 => 16
01001110 => 23
01001111 => 23
01010000 => 24
01010001 => 17
01010010 => 18
01010011 => 18
01010100 => 19
01010101 => 12
01010110 => 19
01010111 => 19
01011000 => 24
01011001 => 17
01011010 => 18
01011011 => 18
01011100 => 24
01011101 => 17
01011110 => 24
01011111 => 24
01100000 => 27
01100001 => 20
01100010 => 21
01100011 => 21
01100100 => 22
01100101 => 15
01100110 => 22
01100111 => 22
01101000 => 23
01101001 => 16
01101010 => 17
01101011 => 17
01101100 => 23
01101101 => 16
01101110 => 23
01101111 => 23
01110000 => 27
01110001 => 20
01110010 => 21
01110011 => 21
01110100 => 22
01110101 => 15
01110110 => 22
01110111 => 22
01111000 => 27
01111001 => 20
01111010 => 21
01111011 => 21
01111100 => 27
01111101 => 20
01111110 => 27
01111111 => 27
10000000 => 28
10000001 => 21
10000010 => 22
10000011 => 22
10000100 => 23
10000101 => 16
10000110 => 23
10000111 => 23
10001000 => 24
10001001 => 17
10001010 => 18
10001011 => 18
10001100 => 24
10001101 => 17
10001110 => 24
10001111 => 24
10010000 => 25
10010001 => 18
10010010 => 19
10010011 => 19
10010100 => 20
10010101 => 13
10010110 => 20
10010111 => 20
10011000 => 25
10011001 => 18
10011010 => 19
10011011 => 19
10011100 => 25
10011101 => 18
10011110 => 25
10011111 => 25
10100000 => 26
10100001 => 19
10100010 => 20
10100011 => 20
10100100 => 21
10100101 => 14
10100110 => 21
10100111 => 21
10101000 => 22
10101001 => 15
10101010 => 16
10101011 => 16
10101100 => 22
10101101 => 15
10101110 => 22
10101111 => 22
10110000 => 26
10110001 => 19
10110010 => 20
10110011 => 20
10110100 => 21
10110101 => 14
10110110 => 21
10110111 => 21
10111000 => 26
10111001 => 19
10111010 => 20
10111011 => 20
10111100 => 26
10111101 => 19
10111110 => 26
10111111 => 26
11000000 => 28
11000001 => 21
11000010 => 22
11000011 => 22
11000100 => 23
11000101 => 16
11000110 => 23
11000111 => 23
11001000 => 24
11001001 => 17
11001010 => 18
11001011 => 18
11001100 => 24
11001101 => 17
11001110 => 24
11001111 => 24
11010000 => 25
11010001 => 18
11010010 => 19
11010011 => 19
11010100 => 20
11010101 => 13
11010110 => 20
11010111 => 20
11011000 => 25
11011001 => 18
11011010 => 19
11011011 => 19
11011100 => 25
11011101 => 18
11011110 => 25
11011111 => 25
11100000 => 28
11100001 => 21
11100010 => 22
11100011 => 22
11100100 => 23
11100101 => 16
11100110 => 23
11100111 => 23
11101000 => 24
11101001 => 17
11101010 => 18
11101011 => 18
11101100 => 24
11101101 => 17
11101110 => 24
11101111 => 24
11110000 => 28
11110001 => 21
11110010 => 22
11110011 => 22
11110100 => 23
11110101 => 16
11110110 => 23
11110111 => 23
11111000 => 28
11111001 => 21
11111010 => 22
11111011 => 22
11111100 => 28
11111101 => 21
11111110 => 28
11111111 => 28
000000000 => 36
000000001 => 28
000000010 => 29
000000011 => 29
000000100 => 30
000000101 => 22
000000110 => 30
000000111 => 30
000001000 => 31
000001001 => 23
000001010 => 24
000001011 => 24
000001100 => 31
000001101 => 23
000001110 => 31
000001111 => 31
000010000 => 32
000010001 => 24
000010010 => 25
000010011 => 25
000010100 => 26
000010101 => 18
000010110 => 26
000010111 => 26
000011000 => 32
000011001 => 24
000011010 => 25
000011011 => 25
000011100 => 32
000011101 => 24
000011110 => 32
000011111 => 32
000100000 => 33
000100001 => 25
000100010 => 26
000100011 => 26
000100100 => 27
000100101 => 19
000100110 => 27
000100111 => 27
000101000 => 28
000101001 => 20
000101010 => 21
000101011 => 21
000101100 => 28
000101101 => 20
000101110 => 28
000101111 => 28
000110000 => 33
000110001 => 25
000110010 => 26
000110011 => 26
000110100 => 27
000110101 => 19
000110110 => 27
000110111 => 27
000111000 => 33
000111001 => 25
000111010 => 26
000111011 => 26
000111100 => 33
000111101 => 25
000111110 => 33
000111111 => 33
001000000 => 34
001000001 => 26
001000010 => 27
001000011 => 27
001000100 => 28
001000101 => 20
001000110 => 28
001000111 => 28
001001000 => 29
001001001 => 21
001001010 => 22
001001011 => 22
001001100 => 29
001001101 => 21
001001110 => 29
001001111 => 29
001010000 => 30
001010001 => 22
001010010 => 23
001010011 => 23
001010100 => 24
001010101 => 16
001010110 => 24
001010111 => 24
001011000 => 30
001011001 => 22
001011010 => 23
001011011 => 23
001011100 => 30
001011101 => 22
001011110 => 30
001011111 => 30
001100000 => 34
001100001 => 26
001100010 => 27
001100011 => 27
001100100 => 28
001100101 => 20
001100110 => 28
001100111 => 28
001101000 => 29
001101001 => 21
001101010 => 22
001101011 => 22
001101100 => 29
001101101 => 21
001101110 => 29
001101111 => 29
001110000 => 34
001110001 => 26
001110010 => 27
001110011 => 27
001110100 => 28
001110101 => 20
001110110 => 28
001110111 => 28
001111000 => 34
001111001 => 26
001111010 => 27
001111011 => 27
001111100 => 34
001111101 => 26
001111110 => 34
001111111 => 34
010000000 => 35
010000001 => 27
010000010 => 28
010000011 => 28
010000100 => 29
010000101 => 21
010000110 => 29
010000111 => 29
010001000 => 30
010001001 => 22
010001010 => 23
010001011 => 23
010001100 => 30
010001101 => 22
010001110 => 30
010001111 => 30
010010000 => 31
010010001 => 23
010010010 => 24
010010011 => 24
010010100 => 25
010010101 => 17
010010110 => 25
010010111 => 25
010011000 => 31
010011001 => 23
010011010 => 24
010011011 => 24
010011100 => 31
010011101 => 23
010011110 => 31
010011111 => 31
010100000 => 32
010100001 => 24
010100010 => 25
010100011 => 25
010100100 => 26
010100101 => 18
010100110 => 26
010100111 => 26
010101000 => 27
010101001 => 19
010101010 => 20
010101011 => 20
010101100 => 27
010101101 => 19
010101110 => 27
010101111 => 27
010110000 => 32
010110001 => 24
010110010 => 25
010110011 => 25
010110100 => 26
010110101 => 18
010110110 => 26
010110111 => 26
010111000 => 32
010111001 => 24
010111010 => 25
010111011 => 25
010111100 => 32
010111101 => 24
010111110 => 32
010111111 => 32
011000000 => 35
011000001 => 27
011000010 => 28
011000011 => 28
011000100 => 29
011000101 => 21
011000110 => 29
011000111 => 29
011001000 => 30
011001001 => 22
011001010 => 23
011001011 => 23
011001100 => 30
011001101 => 22
011001110 => 30
011001111 => 30
011010000 => 31
011010001 => 23
011010010 => 24
011010011 => 24
011010100 => 25
011010101 => 17
011010110 => 25
011010111 => 25
011011000 => 31
011011001 => 23
011011010 => 24
011011011 => 24
011011100 => 31
011011101 => 23
011011110 => 31
011011111 => 31
011100000 => 35
011100001 => 27
011100010 => 28
011100011 => 28
011100100 => 29
011100101 => 21
011100110 => 29
011100111 => 29
011101000 => 30
011101001 => 22
011101010 => 23
011101011 => 23
011101100 => 30
011101101 => 22
011101110 => 30
011101111 => 30
011110000 => 35
011110001 => 27
011110010 => 28
011110011 => 28
011110100 => 29
011110101 => 21
011110110 => 29
011110111 => 29
011111000 => 35
011111001 => 27
011111010 => 28
011111011 => 28
011111100 => 35
011111101 => 27
011111110 => 35
011111111 => 35
100000000 => 36
100000001 => 28
100000010 => 29
100000011 => 29
100000100 => 30
100000101 => 22
100000110 => 30
100000111 => 30
100001000 => 31
100001001 => 23
100001010 => 24
100001011 => 24
100001100 => 31
100001101 => 23
100001110 => 31
100001111 => 31
100010000 => 32
100010001 => 24
100010010 => 25
100010011 => 25
100010100 => 26
100010101 => 18
100010110 => 26
100010111 => 26
100011000 => 32
100011001 => 24
100011010 => 25
100011011 => 25
100011100 => 32
100011101 => 24
100011110 => 32
100011111 => 32
100100000 => 33
100100001 => 25
100100010 => 26
100100011 => 26
100100100 => 27
100100101 => 19
100100110 => 27
100100111 => 27
100101000 => 28
100101001 => 20
100101010 => 21
100101011 => 21
100101100 => 28
100101101 => 20
100101110 => 28
100101111 => 28
100110000 => 33
100110001 => 25
100110010 => 26
100110011 => 26
100110100 => 27
100110101 => 19
100110110 => 27
100110111 => 27
100111000 => 33
100111001 => 25
100111010 => 26
100111011 => 26
100111100 => 33
100111101 => 25
100111110 => 33
100111111 => 33
101000000 => 34
101000001 => 26
101000010 => 27
101000011 => 27
101000100 => 28
101000101 => 20
101000110 => 28
101000111 => 28
101001000 => 29
101001001 => 21
101001010 => 22
101001011 => 22
101001100 => 29
101001101 => 21
101001110 => 29
101001111 => 29
101010000 => 30
101010001 => 22
101010010 => 23
101010011 => 23
101010100 => 24
101010101 => 16
101010110 => 24
101010111 => 24
101011000 => 30
101011001 => 22
101011010 => 23
101011011 => 23
101011100 => 30
101011101 => 22
101011110 => 30
101011111 => 30
101100000 => 34
101100001 => 26
101100010 => 27
101100011 => 27
101100100 => 28
101100101 => 20
101100110 => 28
101100111 => 28
101101000 => 29
101101001 => 21
101101010 => 22
101101011 => 22
101101100 => 29
101101101 => 21
101101110 => 29
101101111 => 29
101110000 => 34
101110001 => 26
101110010 => 27
101110011 => 27
101110100 => 28
101110101 => 20
101110110 => 28
101110111 => 28
101111000 => 34
101111001 => 26
101111010 => 27
101111011 => 27
101111100 => 34
101111101 => 26
101111110 => 34
101111111 => 34
110000000 => 36
110000001 => 28
110000010 => 29
110000011 => 29
110000100 => 30
110000101 => 22
110000110 => 30
110000111 => 30
110001000 => 31
110001001 => 23
110001010 => 24
110001011 => 24
110001100 => 31
110001101 => 23
110001110 => 31
110001111 => 31
110010000 => 32
110010001 => 24
110010010 => 25
110010011 => 25
110010100 => 26
110010101 => 18
110010110 => 26
110010111 => 26
110011000 => 32
110011001 => 24
110011010 => 25
110011011 => 25
110011100 => 32
110011101 => 24
110011110 => 32
110011111 => 32
110100000 => 33
110100001 => 25
110100010 => 26
110100011 => 26
110100100 => 27
110100101 => 19
110100110 => 27
110100111 => 27
110101000 => 28
110101001 => 20
110101010 => 21
110101011 => 21
110101100 => 28
110101101 => 20
110101110 => 28
110101111 => 28
110110000 => 33
110110001 => 25
110110010 => 26
110110011 => 26
110110100 => 27
110110101 => 19
110110110 => 27
110110111 => 27
110111000 => 33
110111001 => 25
110111010 => 26
110111011 => 26
110111100 => 33
110111101 => 25
110111110 => 33
110111111 => 33
111000000 => 36
111000001 => 28
111000010 => 29
111000011 => 29
111000100 => 30
111000101 => 22
111000110 => 30
111000111 => 30
111001000 => 31
111001001 => 23
111001010 => 24
111001011 => 24
111001100 => 31
111001101 => 23
111001110 => 31
111001111 => 31
111010000 => 32
111010001 => 24
111010010 => 25
111010011 => 25
111010100 => 26
111010101 => 18
111010110 => 26
111010111 => 26
111011000 => 32
111011001 => 24
111011010 => 25
111011011 => 25
111011100 => 32
111011101 => 24
111011110 => 32
111011111 => 32
111100000 => 36
111100001 => 28
111100010 => 29
111100011 => 29
111100100 => 30
111100101 => 22
111100110 => 30
111100111 => 30
111101000 => 31
111101001 => 23
111101010 => 24
111101011 => 24
111101100 => 31
111101101 => 23
111101110 => 31
111101111 => 31
111110000 => 36
111110001 => 28
111110010 => 29
111110011 => 29
111110100 => 30
111110101 => 22
111110110 => 30
111110111 => 30
111111000 => 36
111111001 => 28
111111010 => 29
111111011 => 29
111111100 => 36
111111101 => 28
111111110 => 36
111111111 => 36
-----------------------------------------------------------------------------
Created: Nov 05, 2023 at 15:59 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Nov 05, 2023 at 15:59 by Martin Rubey