Identifier
Values
0 => [2] => [1,1,0,0] => [2,3,1] => 0
1 => [1,1] => [1,0,1,0] => [3,1,2] => 0
00 => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 0
01 => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 0
=> [1] => [1,0] => [2,1] => 0
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Description
The number of stretching pairs of a permutation.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
bounce path
Description
The bounce path determined by an integer composition.
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.