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* 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. *
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-----------------------------------------------------------------------------
Statistic identifier: St000008
-----------------------------------------------------------------------------
Collection: Integer compositions
-----------------------------------------------------------------------------
Description: The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(x):
return x.major_index()
-----------------------------------------------------------------------------
Statistic values:
[1] => 0
[1,1] => 1
[2] => 0
[1,1,1] => 3
[1,2] => 1
[2,1] => 2
[3] => 0
[1,1,1,1] => 6
[1,1,2] => 3
[1,2,1] => 4
[1,3] => 1
[2,1,1] => 5
[2,2] => 2
[3,1] => 3
[4] => 0
[1,1,1,1,1] => 10
[1,1,1,2] => 6
[1,1,2,1] => 7
[1,1,3] => 3
[1,2,1,1] => 8
[1,2,2] => 4
[1,3,1] => 5
[1,4] => 1
[2,1,1,1] => 9
[2,1,2] => 5
[2,2,1] => 6
[2,3] => 2
[3,1,1] => 7
[3,2] => 3
[4,1] => 4
[5] => 0
[1,1,1,1,1,1] => 15
[1,1,1,1,2] => 10
[1,1,1,2,1] => 11
[1,1,1,3] => 6
[1,1,2,1,1] => 12
[1,1,2,2] => 7
[1,1,3,1] => 8
[1,1,4] => 3
[1,2,1,1,1] => 13
[1,2,1,2] => 8
[1,2,2,1] => 9
[1,2,3] => 4
[1,3,1,1] => 10
[1,3,2] => 5
[1,4,1] => 6
[1,5] => 1
[2,1,1,1,1] => 14
[2,1,1,2] => 9
[2,1,2,1] => 10
[2,1,3] => 5
[2,2,1,1] => 11
[2,2,2] => 6
[2,3,1] => 7
[2,4] => 2
[3,1,1,1] => 12
[3,1,2] => 7
[3,2,1] => 8
[3,3] => 3
[4,1,1] => 9
[4,2] => 4
[5,1] => 5
[6] => 0
[1,1,1,1,1,1,1] => 21
[1,1,1,1,1,2] => 15
[1,1,1,1,2,1] => 16
[1,1,1,1,3] => 10
[1,1,1,2,1,1] => 17
[1,1,1,2,2] => 11
[1,1,1,3,1] => 12
[1,1,1,4] => 6
[1,1,2,1,1,1] => 18
[1,1,2,1,2] => 12
[1,1,2,2,1] => 13
[1,1,2,3] => 7
[1,1,3,1,1] => 14
[1,1,3,2] => 8
[1,1,4,1] => 9
[1,1,5] => 3
[1,2,1,1,1,1] => 19
[1,2,1,1,2] => 13
[1,2,1,2,1] => 14
[1,2,1,3] => 8
[1,2,2,1,1] => 15
[1,2,2,2] => 9
[1,2,3,1] => 10
[1,2,4] => 4
[1,3,1,1,1] => 16
[1,3,1,2] => 10
[1,3,2,1] => 11
[1,3,3] => 5
[1,4,1,1] => 12
[1,4,2] => 6
[1,5,1] => 7
[1,6] => 1
[2,1,1,1,1,1] => 20
[2,1,1,1,2] => 14
[2,1,1,2,1] => 15
[2,1,1,3] => 9
[2,1,2,1,1] => 16
[2,1,2,2] => 10
[2,1,3,1] => 11
[2,1,4] => 5
[2,2,1,1,1] => 17
[2,2,1,2] => 11
[2,2,2,1] => 12
[2,2,3] => 6
[2,3,1,1] => 13
[2,3,2] => 7
[2,4,1] => 8
[2,5] => 2
[3,1,1,1,1] => 18
[3,1,1,2] => 12
[3,1,2,1] => 13
[3,1,3] => 7
[3,2,1,1] => 14
[3,2,2] => 8
[3,3,1] => 9
[3,4] => 3
[4,1,1,1] => 15
[4,1,2] => 9
[4,2,1] => 10
[4,3] => 4
[5,1,1] => 11
[5,2] => 5
[6,1] => 6
[7] => 0
[1,1,1,1,1,1,1,1] => 28
[1,1,1,1,1,1,2] => 21
[1,1,1,1,1,2,1] => 22
[1,1,1,1,1,3] => 15
[1,1,1,1,2,1,1] => 23
[1,1,1,1,2,2] => 16
[1,1,1,1,3,1] => 17
[1,1,1,1,4] => 10
[1,1,1,2,1,1,1] => 24
[1,1,1,2,1,2] => 17
[1,1,1,2,2,1] => 18
[1,1,1,2,3] => 11
[1,1,1,3,1,1] => 19
[1,1,1,3,2] => 12
[1,1,1,4,1] => 13
[1,1,1,5] => 6
[1,1,2,1,1,1,1] => 25
[1,1,2,1,1,2] => 18
[1,1,2,1,2,1] => 19
[1,1,2,1,3] => 12
[1,1,2,2,1,1] => 20
[1,1,2,2,2] => 13
[1,1,2,3,1] => 14
[1,1,2,4] => 7
[1,1,3,1,1,1] => 21
[1,1,3,1,2] => 14
[1,1,3,2,1] => 15
[1,1,3,3] => 8
[1,1,4,1,1] => 16
[1,1,4,2] => 9
[1,1,5,1] => 10
[1,1,6] => 3
[1,2,1,1,1,1,1] => 26
[1,2,1,1,1,2] => 19
[1,2,1,1,2,1] => 20
[1,2,1,1,3] => 13
[1,2,1,2,1,1] => 21
[1,2,1,2,2] => 14
[1,2,1,3,1] => 15
[1,2,1,4] => 8
[1,2,2,1,1,1] => 22
[1,2,2,1,2] => 15
[1,2,2,2,1] => 16
[1,2,2,3] => 9
[1,2,3,1,1] => 17
[1,2,3,2] => 10
[1,2,4,1] => 11
[1,2,5] => 4
[1,3,1,1,1,1] => 23
[1,3,1,1,2] => 16
[1,3,1,2,1] => 17
[1,3,1,3] => 10
[1,3,2,1,1] => 18
[1,3,2,2] => 11
[1,3,3,1] => 12
[1,3,4] => 5
[1,4,1,1,1] => 19
[1,4,1,2] => 12
[1,4,2,1] => 13
[1,4,3] => 6
[1,5,1,1] => 14
[1,5,2] => 7
[1,6,1] => 8
[1,7] => 1
[2,1,1,1,1,1,1] => 27
[2,1,1,1,1,2] => 20
[2,1,1,1,2,1] => 21
[2,1,1,1,3] => 14
[2,1,1,2,1,1] => 22
[2,1,1,2,2] => 15
[2,1,1,3,1] => 16
[2,1,1,4] => 9
[2,1,2,1,1,1] => 23
[2,1,2,1,2] => 16
[2,1,2,2,1] => 17
[2,1,2,3] => 10
[2,1,3,1,1] => 18
[2,1,3,2] => 11
[2,1,4,1] => 12
[2,1,5] => 5
[2,2,1,1,1,1] => 24
[2,2,1,1,2] => 17
[2,2,1,2,1] => 18
[2,2,1,3] => 11
[2,2,2,1,1] => 19
[2,2,2,2] => 12
[2,2,3,1] => 13
[2,2,4] => 6
[2,3,1,1,1] => 20
[2,3,1,2] => 13
[2,3,2,1] => 14
[2,3,3] => 7
[2,4,1,1] => 15
[2,4,2] => 8
[2,5,1] => 9
[2,6] => 2
[3,1,1,1,1,1] => 25
[3,1,1,1,2] => 18
[3,1,1,2,1] => 19
[3,1,1,3] => 12
[3,1,2,1,1] => 20
[3,1,2,2] => 13
[3,1,3,1] => 14
[3,1,4] => 7
[3,2,1,1,1] => 21
[3,2,1,2] => 14
[3,2,2,1] => 15
[3,2,3] => 8
[3,3,1,1] => 16
[3,3,2] => 9
[3,4,1] => 10
[3,5] => 3
[4,1,1,1,1] => 22
[4,1,1,2] => 15
[4,1,2,1] => 16
[4,1,3] => 9
[4,2,1,1] => 17
[4,2,2] => 10
[4,3,1] => 11
[4,4] => 4
[5,1,1,1] => 18
[5,1,2] => 11
[5,2,1] => 12
[5,3] => 5
[6,1,1] => 13
[6,2] => 6
[7,1] => 7
[8] => 0
[1,1,1,1,4,1] => 18
[1,1,1,2,3,1] => 19
[1,1,2,1,3,1] => 20
[1,1,6,1] => 11
[1,2,1,1,3,1] => 21
[1,2,1,2,2,1] => 22
[1,2,2,1,2,1] => 23
[1,2,5,1] => 12
[1,3,1,1,2,1] => 24
[1,3,1,2,1,1] => 25
[1,3,2,1,1,1] => 26
[1,3,4,1] => 13
[1,4,1,1,1,1] => 27
[1,4,1,3] => 12
[1,4,2,2] => 13
[1,4,3,1] => 14
[1,5,1,2] => 14
[1,5,2,1] => 15
[1,6,1,1] => 16
[1,6,2] => 8
[1,7,1] => 9
[1,8] => 1
[2,1,5,1] => 13
[2,2,4,1] => 14
[3,1,4,1] => 15
[6,2,1] => 14
[7,1,1] => 15
[8,1] => 8
[1,1,1,1,1,1,1,1,1,1] => 45
[1,1,1,1,1,1,2,2] => 29
[1,1,1,1,2,1,1,2] => 31
[1,1,1,1,2,2,1,1] => 33
[1,1,1,1,3,3] => 17
[1,1,1,1,5,1] => 19
[1,1,2,1,1,1,1,2] => 33
[1,1,2,1,1,2,1,1] => 35
[1,1,2,1,2,3] => 19
[1,1,2,2,1,1,1,1] => 37
[1,1,2,2,2,2] => 21
[1,1,3,1,1,3] => 21
[1,1,3,2,1,2] => 23
[1,1,3,3,1,1] => 25
[1,1,4,4] => 9
[1,1,7,1] => 12
[1,2,1,1,4,1] => 22
[1,2,6,1] => 13
[1,3,1,1,3,1] => 25
[1,3,5,1] => 14
[1,4,1,1,2,1] => 28
[1,4,4,1] => 15
[1,5,1,1,1,1] => 31
[1,5,3,1] => 16
[1,6,1,2] => 16
[1,6,2,1] => 17
[1,7,1,1] => 18
[1,8,1] => 10
[1,9] => 1
[2,1,1,1,1,1,1,2] => 35
[2,1,1,1,1,2,1,1] => 37
[2,1,1,1,2,3] => 21
[2,1,1,2,1,1,1,1] => 39
[2,1,1,2,2,2] => 23
[2,1,2,1,1,3] => 23
[2,1,2,2,1,2] => 25
[2,1,2,3,1,1] => 27
[2,1,3,4] => 11
[2,1,6,1] => 14
[2,2,1,1,1,1,1,1] => 41
[2,2,1,1,2,2] => 25
[2,2,2,1,1,2] => 27
[2,2,2,2,1,1] => 29
[2,2,3,3] => 13
[3,1,1,1,1,3] => 25
[3,1,1,2,1,2] => 27
[3,1,1,3,1,1] => 29
[3,1,2,4] => 13
[3,2,1,1,1,2] => 29
[3,2,1,2,1,1] => 31
[3,2,2,3] => 15
[3,3,1,1,1,1] => 33
[3,3,2,2] => 17
[4,1,1,4] => 15
[4,2,1,3] => 17
[4,3,1,2] => 19
[4,4,1,1] => 21
[5,5] => 5
[8,1,1] => 17
[9,1] => 9
[1,10] => 1
[1,8,1,1] => 20
[1,7,2,1] => 19
[1,6,3,1] => 18
[1,5,4,1] => 17
[1,4,5,1] => 16
[1,3,6,1] => 15
[1,2,7,1] => 14
[1,1,8,1] => 13
[10,1] => 10
[1,1,1,1,1,1,1,1,1,1,1,1] => 66
[1,1,1,1,1,1,1,1,2,2] => 46
[1,1,1,1,1,1,2,2,1,1] => 50
[1,1,1,1,1,1,2,1,1,2] => 48
[1,1,1,1,1,1,3,3] => 30
[1,1,1,1,2,2,1,1,1,1] => 54
[1,1,1,1,2,2,2,2] => 34
[1,1,1,1,2,1,1,2,1,1] => 52
[1,1,1,1,2,1,1,1,1,2] => 50
[1,1,1,1,2,1,2,3] => 32
[1,1,1,1,3,3,1,1] => 38
[1,1,1,1,3,2,1,2] => 36
[1,1,1,1,3,1,1,3] => 34
[1,1,1,1,4,4] => 18
[1,1,2,2,1,1,1,1,1,1] => 58
[1,1,2,2,1,1,2,2] => 38
[1,1,2,2,2,2,1,1] => 42
[1,1,2,2,2,1,1,2] => 40
[1,1,2,2,3,3] => 22
[1,1,2,1,1,2,1,1,1,1] => 56
[1,1,2,1,1,2,2,2] => 36
[1,1,2,1,1,1,1,2,1,1] => 54
[1,1,2,1,1,1,1,1,1,2] => 52
[1,1,2,1,1,1,2,3] => 34
[1,1,2,1,2,3,1,1] => 40
[1,1,2,1,2,2,1,2] => 38
[1,1,2,1,2,1,1,3] => 36
[1,1,2,1,3,4] => 20
[1,1,3,3,1,1,1,1] => 46
[1,1,3,3,2,2] => 26
[1,1,3,2,1,2,1,1] => 44
[1,1,3,2,1,1,1,2] => 42
[1,1,3,2,2,3] => 24
[1,1,3,1,1,3,1,1] => 42
[1,1,3,1,1,2,1,2] => 40
[1,1,3,1,1,1,1,3] => 38
[1,1,3,1,2,4] => 22
[1,1,4,4,1,1] => 30
[1,1,4,3,1,2] => 28
[1,1,4,2,1,3] => 26
[1,1,4,1,1,4] => 24
[1,1,5,5] => 10
[2,2,1,1,1,1,1,1,1,1] => 62
[2,2,1,1,1,1,2,2] => 42
[2,2,1,1,2,2,1,1] => 46
[2,2,1,1,2,1,1,2] => 44
[2,2,1,1,3,3] => 26
[2,2,2,2,1,1,1,1] => 50
[2,2,2,2,2,2] => 30
[2,2,2,1,1,2,1,1] => 48
[2,2,2,1,1,1,1,2] => 46
[2,2,2,1,2,3] => 28
[2,2,3,3,1,1] => 34
[2,2,3,2,1,2] => 32
[2,2,3,1,1,3] => 30
[2,2,4,4] => 14
[2,1,1,2,1,1,1,1,1,1] => 60
[2,1,1,2,1,1,2,2] => 40
[2,1,1,2,2,2,1,1] => 44
[2,1,1,2,2,1,1,2] => 42
[2,1,1,2,3,3] => 24
[2,1,1,1,1,2,1,1,1,1] => 58
[2,1,1,1,1,2,2,2] => 38
[2,1,1,1,1,1,1,2,1,1] => 56
[2,1,1,1,1,1,1,1,1,2] => 54
[2,1,1,1,1,1,2,3] => 36
[2,1,1,1,2,3,1,1] => 42
[2,1,1,1,2,2,1,2] => 40
[2,1,1,1,2,1,1,3] => 38
[2,1,1,1,3,4] => 22
[2,1,2,3,1,1,1,1] => 48
[2,1,2,3,2,2] => 28
[2,1,2,2,1,2,1,1] => 46
[2,1,2,2,1,1,1,2] => 44
[2,1,2,2,2,3] => 26
[2,1,2,1,1,3,1,1] => 44
[2,1,2,1,1,2,1,2] => 42
[2,1,2,1,1,1,1,3] => 40
[2,1,2,1,2,4] => 24
[2,1,3,4,1,1] => 32
[2,1,3,3,1,2] => 30
[2,1,3,2,1,3] => 28
[2,1,3,1,1,4] => 26
[2,1,4,5] => 12
[3,3,1,1,1,1,1,1] => 54
[3,3,1,1,2,2] => 34
[3,3,2,2,1,1] => 38
[3,3,2,1,1,2] => 36
[3,3,3,3] => 18
[3,2,1,2,1,1,1,1] => 52
[3,2,1,2,2,2] => 32
[3,2,1,1,1,2,1,1] => 50
[3,2,1,1,1,1,1,2] => 48
[3,2,1,1,2,3] => 30
[3,2,2,3,1,1] => 36
[3,2,2,2,1,2] => 34
[3,2,2,1,1,3] => 32
[3,2,3,4] => 16
[3,1,1,3,1,1,1,1] => 50
[3,1,1,3,2,2] => 30
[3,1,1,2,1,2,1,1] => 48
[3,1,1,2,1,1,1,2] => 46
[3,1,1,2,2,3] => 28
[3,1,1,1,1,3,1,1] => 46
[3,1,1,1,1,2,1,2] => 44
[3,1,1,1,1,1,1,3] => 42
[3,1,1,1,2,4] => 26
[3,1,2,4,1,1] => 34
[3,1,2,3,1,2] => 32
[3,1,2,2,1,3] => 30
[3,1,2,1,1,4] => 28
[3,1,3,5] => 14
[4,4,1,1,1,1] => 42
[4,4,2,2] => 22
[4,3,1,2,1,1] => 40
[4,3,1,1,1,2] => 38
[4,3,2,3] => 20
[4,2,1,3,1,1] => 38
[4,2,1,2,1,2] => 36
[4,2,1,1,1,3] => 34
[4,2,2,4] => 18
[4,1,1,4,1,1] => 36
[4,1,1,3,1,2] => 34
[4,1,1,2,1,3] => 32
[4,1,1,1,1,4] => 30
[4,1,2,5] => 16
[5,5,1,1] => 26
[5,4,1,2] => 24
[5,3,1,3] => 22
[5,2,1,4] => 20
[5,1,1,5] => 18
[6,6] => 6
-----------------------------------------------------------------------------
Created: Sep 21, 2011 at 16:41 by Chris Berg
-----------------------------------------------------------------------------
Last Updated: Jan 26, 2018 at 09:17 by Martin Rubey