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Identifier

St000921: Binary words ⟶ ℤ

Values

0 => 0

1 => 0

00 => 0

01 => 0

10 => 0

11 => 0

000 => 0

001 => 0

010 => 0

011 => 0

100 => 1

101 => 0

110 => 1

111 => 0

0000 => 0

0001 => 0

0010 => 0

0011 => 0

0100 => 1

0101 => 0

0110 => 1

0111 => 0

1000 => 2

1001 => 1

1010 => 2

1011 => 0

1100 => 0

1101 => 1

1110 => 2

1111 => 0

00000 => 0

00001 => 0

00010 => 0

00011 => 0

00100 => 1

00101 => 0

00110 => 1

00111 => 0

01000 => 2

01001 => 1

01010 => 2

01011 => 0

01100 => 0

01101 => 1

01110 => 2

01111 => 0

10000 => 3

10001 => 2

10010 => 3

10011 => 1

10100 => 1

10101 => 2

10110 => 3

10111 => 0

11000 => 2

11001 => 0

11010 => 1

11011 => 1

11100 => 2

11101 => 2

11110 => 3

11111 => 0

000000 => 0

000001 => 0

000010 => 0

000011 => 0

000100 => 1

000101 => 0

000110 => 1

000111 => 0

001000 => 2

001001 => 1

001010 => 2

001011 => 0

001100 => 0

001101 => 1

001110 => 2

001111 => 0

010000 => 3

010001 => 2

010010 => 3

010011 => 1

010100 => 1

010101 => 2

010110 => 3

010111 => 0

011000 => 2

011001 => 0

011010 => 1

011011 => 1

011100 => 2

011101 => 2

011110 => 3

011111 => 0

100000 => 4

100001 => 3

100010 => 4

100011 => 2

100100 => 2

100101 => 3

100110 => 4

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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.

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

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