Identifier
Values
0 => [2] => [1,1,0,0] => [1,2] => 0
1 => [1,1] => [1,0,1,0] => [2,1] => 0
00 => [3] => [1,1,1,0,0,0] => [1,2,3] => 0
01 => [2,1] => [1,1,0,0,1,0] => [1,3,2] => 0
10 => [1,2] => [1,0,1,1,0,0] => [2,1,3] => 0
11 => [1,1,1] => [1,0,1,0,1,0] => [2,3,1] => 0
000 => [4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 1
010 => [2,2] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 0
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 0
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 1
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Description
The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the second entry is St001557The number of inversions of the second entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.