Identifier
- St000921: Binary words ⟶ ℤ
Values
=>
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111001010001=>9
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10110111100000=>14
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10111011100000=>15
10111100110000=>14
10111101001000=>14
10111101010000=>15
10111101100000=>16
10111110000100=>3
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10111110010000=>5
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11001111100000=>14
11011110000010=>3
11101101000010=>14
11101110000010=>4
11110011000010=>14
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11111000010010=>14
11111000100010=>15
11111001000010=>16
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1011111011000000=>19
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1011111110000000=>8
1111011100000010=>5
1111101010000010=>18
1111101100000010=>6
1111110001000010=>18
1111110010000010=>19
1111110100000010=>7
1111111000000010=>8
111111110000000010=>9
101111111100000000=>9
111111101000000010=>8
101111111010000000=>8
111111100100000010=>22
111111011000000010=>7
101111111001000000=>7
101111110110000000=>22
11111111100000000010=>10
10111111111000000000=>10
11111111010000000010=>9
10111111110100000000=>9
1111111111000000000010=>11
1011111111110000000000=>11
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Description
The number of internal inversions of a binary word.
Let $\bar w$ be the non-decreasing rearrangement of $w$, that is, $\bar w$ is sorted.
An internal inversion is a pair $i < j$ such that $w_i > w_j$ and $\bar w_i = \bar w_j$. For example, the word $110$ has two inversions, but only the second is internal.
Let $\bar w$ be the non-decreasing rearrangement of $w$, that is, $\bar w$ is sorted.
An internal inversion is a pair $i < j$ such that $w_i > w_j$ and $\bar w_i = \bar w_j$. For example, the word $110$ has two inversions, but only the second is internal.
References
[1] Foata, D., Han, G.-N. Inverses of words MathSciNet:1611988
[2] Hohlweg, C., Reutenauer, C. Inverses of words and the parabolic structure of the symmetric group MathSciNet:1861050
[2] Hohlweg, C., Reutenauer, C. Inverses of words and the parabolic structure of the symmetric group MathSciNet:1861050
Code
def statistic(w): w_bar = sorted(w) return len([(i,j) for (i, j) in w.inversions() if w_bar[i] == w_bar[j]])
Created
Aug 01, 2017 at 08:16 by Martin Rubey
Updated
Apr 19, 2019 at 22:18 by Martin Rubey
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