Your data matches 1 statistic following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000921
(load all 2 compositions to match this statistic)
Mapping
St000921: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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.