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Identifier

St001931: Integer compositions ⟶ ℤ

Values

[1] => 0

[1,1] => 1

[2] => 0

[1,1,1] => 3

[1,2] => 0

[2,1] => 1

[3] => 0

[1,1,1,1] => 6

[1,1,2] => 1

[1,2,1] => 2

[1,3] => 0

[2,1,1] => 3

[2,2] => 1

[3,1] => 1

[4] => 0

[1,1,1,1,1] => 10

[1,1,1,2] => 3

[1,1,2,1] => 4

[1,1,3] => 1

[1,2,1,1] => 5

[1,2,2] => 2

[1,3,1] => 2

[1,4] => 0

[2,1,1,1] => 6

[2,1,2] => 1

[2,2,1] => 3

[2,3] => 0

[3,1,1] => 3

[3,2] => 1

[4,1] => 1

[5] => 0

[1,1,1,1,1,1] => 15

[1,1,1,1,2] => 6

[1,1,1,2,1] => 7

[1,1,1,3] => 3

[1,1,2,1,1] => 8

[1,1,2,2] => 4

[1,1,3,1] => 4

[1,1,4] => 1

[1,2,1,1,1] => 9

[1,2,1,2] => 2

[1,2,2,1] => 5

[1,2,3] => 0

[1,3,1,1] => 5

[1,3,2] => 2

[1,4,1] => 2

[1,5] => 0

[2,1,1,1,1] => 10

[2,1,1,2] => 3

[2,1,2,1] => 4

[2,1,3] => 1

[2,2,1,1] => 6

[2,2,2] => 3

[2,3,1] => 2

[2,4] => 0

[3,1,1,1] => 6

[3,1,2] => 1

[3,2,1] => 3

[3,3] => 1

[4,1,1] => 3

[4,2] => 1

[5,1] => 1

[6] => 0

[1,1,1,1,1,1,1] => 21

[1,1,1,1,1,2] => 10

[1,1,1,1,2,1] => 11

[1,1,1,1,3] => 6

[1,1,1,2,1,1] => 12

[1,1,1,2,2] => 7

[1,1,1,3,1] => 7

[1,1,1,4] => 3

[1,1,2,1,1,1] => 13

[1,1,2,1,2] => 4

[1,1,2,2,1] => 8

[1,1,2,3] => 1

[1,1,3,1,1] => 8

[1,1,3,2] => 4

[1,1,4,1] => 4

[1,1,5] => 1

[1,2,1,1,1,1] => 14

[1,2,1,1,2] => 5

[1,2,1,2,1] => 6

[1,2,1,3] => 2

[1,2,2,1,1] => 9

[1,2,2,2] => 5

[1,2,3,1] => 3

[1,2,4] => 0

[1,3,1,1,1] => 9

[1,3,1,2] => 2

[1,3,2,1] => 5

[1,3,3] => 2

[1,4,1,1] => 5

[1,4,2] => 2

[1,5,1] => 2

[1,6] => 0

[2,1,1,1,1,1] => 15

[2,1,1,1,2] => 6

[2,1,1,2,1] => 7

[2,1,1,3] => 3

[2,1,2,1,1] => 8

[2,1,2,2] => 4

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Description

The weak major index of an integer composition regarded as a word.
This is the sum of the positions of the weak descents, regarding the composition as a word. That is, for a composition $c = (c_1,\dots,c_n)$,
$$ \sum_{\substack{1\leq i < n\\ c_i\geq c_{i+1}}} i. $$

Code

def statistic(w):
    return sum(i for i in range(1, len(w)) if w[i-1] >= w[i])

Created

Nov 05, 2023 at 16:02 by Martin Rubey

Updated

Nov 05, 2023 at 16:02 by Martin Rubey